Spectral-admittance and certi cation theory of multiresonator quantum memories Maxim Vyacheslavovich Churilov Independent Researcher, Orenburg, Russian Federation June 2026 Abstract Multiresonator quantum memories are commonly analyzed through speci c combs, cav- ity topologies, or echo protocols. This work formulates them instead as passive spectral- admittance synthesizers with controlled storage dilations. In the weak-excitation regime the prompt re ection coe cient is the Cayley transform of a positive-real self-energy generated by resonator and matter degrees of freedom. For a lossless controlled-dilation device, the write defect of a one-photon wavepacket is exactly the prompt re ected spectral power. Hence the worst-case write e ciency on a continuous signal band is 1 − ess sup | r | 2 , and the vector- port generalization is obtained by replacing | r | with the largest singular value of the prompt scattering matrix. The formulation gives both impossibility results and constructive certi cates. A nite time-independent passive one-port realization cannot be exactly re ectionless on a nonzero continuous band: if a rational re ection coe cient vanishes on such an interval, analyticity forces it to vanish identically, whereas every nite resonator realization has r ( s ) → 1 at high frequency. The associated BodeFano area law gives a stronger bandwidthe ciency ceiling. Conversely, passive minimax matching is a well-posed positive-real approximation problem under explicit linewidth, detuning and oscillator-strength bounds; nite atomic measures are dense, nite-grid continuum certi cates have sparse atomic witnesses, and the xed-pole oscillator-strength subproblem is convex. The paper turns a spectral matching curve into an operational memory certi cate. Finite measurement grids certify a whole band only with a model or derivative margin. Magnitude- only re ection data are insu cient: the complex response must be compatible with a stable Schur function, its inverse Cayley admittance must be positive real, and all-pass phase ambi- guity must be controlled. Absorption is memory-grade only when the non-re ected amplitude enters controlled storage ports. For nite alphabets, the write Gram matrix, reciprocal round- trip matrix, noise covariance, entanglement delity and diamond-distance bound give a linear quantum-channel certi cate. The stationary bounds also identify the resources that bypass them. A known tempo- ral mode can be captured by a gauge-complete dynamic impedance-matching law, and a K -dimensional known alphabet can be captured exactly with K controlled storage modes and a calibrated vector coupling waveform, subject to the explicit bounded-control leakage certi cate developed here. Numerical checks in normalized units κ = 2 give an eleven- internal-mode passive design with max | ω |≤ 1 | r (i ω ) | = 0 . 069464 , certi ed worst-case write e ciency 0 . 995175 , reciprocal round-trip probability at least 0 . 990373 , veri ed imaginary- axis and right-half-plane passivity/Schur margins, data-to-band margins, complex-response uncertainty bounds, robustness tests, capped-control dynamic simulations, thermal-noise con- version and nite-alphabet channel-distance certi cates. A convex weight-optimization cer- ti cate con rms these synthesized designs are optimal for their pole supports, and an op- timal minimax search drives the worst-case re ection systematically toward the BodeFano oor: the eleven-internal-mode design is improved to max | ω |≤ 1 | r | = 0 . 064112 (write e ciency 0 . 995890 ), each optimized design is certi ed globally minimax-optimal by three concordant tests (equioscillation, the convex weight-optimum, and an independent global search), the optimum is shown to be minimum-phase and to saturate the area law, and the gap to e − π is identi ed with the vanishing out-of-band log-area, which the certi ed series closes 1 root-exponentially (a GoncharRakhmanovStahl rate), as conjectured. We give a buildable miniresonator recipe and a fabrication-robust synthesis tolerant of ± 1% parameter errors. 1 Introduction and problem statement Quantum memory for light is a reversible interface between travelling photonic modes and station- ary matter degrees of freedom. It is a central primitive for quantum repeaters, optical networking, photonic processing and microwave quantum processors. The mature families of optical mem- ories include electromagnetically induced transparency, Raman memories, controlled reversible inhomogeneous broadening, gradient echo memory, atomic frequency combs and cavity-enhanced photon echo schemes [1, 2, 3]. The multiresonator quantum memory (MR-QM) programme initiated by Moiseev and collaborators attacks a speci c bottleneck: a single high-Q resonator enhances the lightmatter interaction but ordinarily narrows the working spectral range, whereas many controlled resonators can synthesize a broadband spectral response while keeping high local eld enhancement [7, 8, 11, 13, 16]. The literature contains several important partial answers. The original spatial-frequency comb of microresonators gives a Bragg-type impedance condition, ∆ = π Γ / 2 , for e cient broad- band storage in a waveguide-coupled resonator array [7]. A microwave proof-of-principle demon- strated controllable resonator frequencies and impedance-matched storage, with measured room- temperature e ciencies at the level expected for the design losses [8, 9]. Spectral-topological designs showed that a small number of resonant absorbers can reach extremely high calculated spectral e ciency near parameter values where the resonance spectrum restructures [11]. Super- conducting multiresonator memories have demonstrated on-demand storage and retrieval of weak microwave pulses and time-bin qubits, and recent RF-SQUID-coupled implementations empha- size dynamically switchable storage and pulse-shape preservation [12, 14]. Integrated designs and resonator arrays with atomic ensembles now give a realistic route to chip-scale broadband mem- ories; this direction is reinforced by preparation-free resonator-array proposals, atomic-ensemble MR-QM theory, high-e ciency impedance-matched rare-earth microcavity memories, and recent network-level benchmarking of multimode high- delity memories [13, 15, 16, 18, 19]. A general spectral theory remains missing. Existing results are usually obtained for a par- ticular comb, a particular small topology, a particular optimization functional, or a particular approximation to the atomic ensemble. The following questions remain structurally unresolved. 1. What is the complete frequency-domain object that determines write e ciency and delity? 2. Which spectral functions are physically realizable by passive resonators and atomic ensem- bles? 3. Can a nite passive MR-QM be exactly re ectionless on a continuous signal band? 4. If exact nite-band matching is impossible, what replaces it as the correct universal design principle? 5. How are Bragg, impedance, spectral-dispersion and spectral-topological matching condi- tions uni ed? 6. How does one pass fr om an optimized spectral function to experimentally tunable resonator parameters? 7. How can nite-frequency measurements certify a continuum band rather than only the sampled points? 8. Which measured quantities certify a complete writestoreread quantum channel rather than absorption alone? 2 This paper gives a direct answer. In the linear weak-excitation regime, a one-port MR-QM is a passive linear quantum system. Its prompt re ection coe cient is a scalar Schur function. Equivalently, it is the Cayley transform of a positive-real internal admittance or self-energy. Memory design is therefore a positive-real spectral approximation problem. This point is simple but decisive: it makes clear what can be exact, what can only be approximate, and what data must be optimized. The main results are summarized as follows. The write process is fully characterized by r ( s ) = s − κ − γ c 2 + Σ( s ) s + κ + γ c 2 + Σ( s ) , (1) wh ere s is the Laplace frequency, κ is the external waveguide coupling of the common resonator, γ c its parasitic loss, and Σ is the passive self-energy of the auxiliary resonator matter network. The self-energy is positive real: Re Σ( s ) ≥ 0 , Re s > 0 . (2) For diagonal auxiliary resonators loaded by inhomogeneously broadened ensembles, Σ( s ) = M X m =1 | g m | 2 s + γ b,m / 2 + i∆ m + Φ m ( s ) , (3) with Φ m again positive real. In the absence of parasitic loss, the write e ciency for a one-photon spectrum f ( ω ) is exactly η w [ f ] = 1 − Z | r (i ω + 0) | 2 | f ( ω ) | 2 d ω. (4) The worst-case e ciency over a band B is η min w ( B ) = 1 − ess sup ω ∈B | r (i ω + 0) | 2 . (5) A nite passive time-independent MR-QM cannot have r (i ω + 0) = 0 on a nonzero con- tinuous interval. Exact continuous-band perfect memory requires an in nite/continuum spectral measure, active or time-dependent matching, or restriction to a nite-dimensional signal subspace. Finite devices are nevertheless universal in the physically relevant engineering sense: for any target band and error tolerance, the problem is to synthesize a passive rational positive-real Σ N that minimizes ∥ r N ∥ L ∞ ( B ) . The minimax value is exactly the worst-case memory error. The matching hierarchy Σ( s 0 ) = κ − γ c 2 − s 0 , Σ ′ ( s 0 ) = − 1 , Σ ( q ) ( s 0 ) = 0 ( q = 2 , . . . , p ) (6) is the local at-re ection form of this synthesis problem. It uni es ordinary impedance matching, spectral dispersion compensation and spectral-topological tuning. The resulting conclusion is deliberately sharper than a design recipe. The universal object is not a particular resonator layout, but the positive-real spectral admittance seen by the waveguide. Any optical, microwave, phononic, magnonic or spin-ensemble implementation that realizes the same admittance has the same write e ciency, up to parasitic loss and retrieval control errors. 3 2 Assumptions and scope All results below are formulated in the linear weak-excitation regime. The matter excitations are treated as harmonic bosonic modes, which is the standard single-photon or weak coherent-state approximation to a collectively enhanced ensemble. Saturation, nonlinear photonphoton interac- tions, control-pulse imperfections during the rephasing stage, and long-time spin decoherence are not hidden inside the spectral-admittance model; they enter as additional loss or noise channels and must be certi ed separately. Quantitatively, the HolsteinPrimako linearization that underlies the matter response is controlled by a single small parameter. For a collective mode formed fr om N at two-level emitters carrying n exc stored excitations, the exact lowering operator is σ + = b † p 1 − b † b/N at , so the bosonic commutator and the linear susceptibility hold up to a relative correction of order n exc /N at . In the single-photon or weak-coherent regime n exc = 1 , and a realistically addressed ensemble has N at ≳ 10 6 for microwave spin ensembles and ∼ 10 12 for optical rare-earth ions; the systematic saturation correction to the re ection coe cient, and hence to the certi ed e ciency, is therefore bounded by ≲ 10 − 6 , three or more orders of magnitude below the worst-case margins reported below. The weak-excitation model is thus not a qualitative idealization but a quantitatively negligible approximation for single-photon storage, and the certi cates degrade only linearly in the stored excitation number outside it. The central object of the theory is the coherent write stage. A high write probability is a quantum-memory probability only when the non-re ected amplitude is mapped into controlled storage degrees of freedom. This is why the paper distinguishes controlled storage ports fr om uncontrolled dissipative ports and why it gives a noise certi cate for occupied environmental modes. The retrieval theorems assume an ideal reciprocal read operation on the stored subspace; deviations fr om reciprocity are experimentally measurable channel errors, not modi cations of the write-admittance theorem. The scalar one-port theory is the analytically sharp core. Multiport and vector-mode inter- faces are included through the prompt scattering matrix R (i ω ) . The corresponding certi cate is the largest singular value of R , not a re ection coe cient measured in one selected input po- larization or spatial mode. This distinction is essential for comparing broadband memories with dierent physical encodings. 3 Physical model 3.1 One-port common-resonator architecture We consider the standard one-port architecture shown schematically in Fig. 1. A travelling eld A in ( t ) drives a common resonator mode a . The common mode couples to auxiliary resonator modes b m , and each auxiliary resonator may contain an ensemble of atoms, spins, colour centres, rare-earth ions, quantum dots, or another long-lived matter system. The external waveguide coupling is κ . The intrinsic loss rates of the common and auxiliary resonators are γ c and γ b,m . Detunings are measured fr om the carrier frequency. In a rotating frame and under the rotating-wave approximation, the linearized Heisenberg Langevin equations are ˙ a = − κ + γ c 2 a − i M X m =1 g m b m + √ κ A in + √ γ c c in , (7) ˙ b m = − γ b,m 2 + i∆ m b m − i g ∗ m a − i X α f mα σ mα + √ γ b,m d m, in , (8) ˙ σ mα = − γ a,mα 2 + i δ mα σ mα − i f ∗ mα b m + √ γ a,mα e mα, in . (9) 4 travelling eld common mode a, κ positive-real self-energy Σ( s ) controlled storage register A in g trapping/rephasing A out uncontrolled loss/noise must be reported Figure 1: One-port multiresonator memory as a passive spectral-admittance synthesizer. The travelling eld sees a common mode loaded by a controlled spectral measure. A memory certi - cate must identify which absorbed amplitude enters a reversible register and which part enters uncontrolled loss. Here σ mα is the bosonic weak-excitation approximation to the atomic lowering operator of the α th atom or collective spectral class in the m th auxiliary resonator. This approximation is exactly the usual HolsteinPrimako linearization for a weak signal and an initially unexcited ensemble. The input-output relation is A out ( t ) = A in ( t ) − √ κ a ( t ) . (10) Dierent sign conventions replace r by an overall phase; all e ciencies and matching conditions are unchanged. Let the Laplace transform be x ( s ) = R ∞ 0 e − st x ( t )d t with Re s > 0 . Eliminating matter excitations gives Φ m ( s ) = X α | f mα | 2 s + γ a,mα / 2 + i δ mα , (11) and b m ( s ) = − i g ∗ m a ( s ) s + γ b,m / 2 + i∆ m + Φ m ( s ) + noise terms . (12) The common mode therefore obeys s + κ + γ c 2 + Σ( s ) a ( s ) = √ κ A in ( s ) + noise terms , (13) with self-energy Σ( s ) = M X m =1 | g m | 2 s + γ b,m / 2 + i∆ m + Φ m ( s ) . (14) Neglecting noise inputs when computing the coherent scattering amplitude, Eqs. (13) and (10) yield Eq. (1). 3.2 Continuous atomic broadening For an inhomogeneously broadened ensemble with spectral density ρ m ( δ ) and coupling f m ( δ ) , Φ m ( s ) = Z R | f m ( δ ) | 2 ρ m ( δ ) s + γ a / 2 + i δ d δ. (15) For a Lorentzian pro le ρ m ( δ ) = ∆ a,m /π ∆ 2 a,m + ( δ − ¯ δ m ) 2 , (16) and slowly varying coupling, residue integration gives Φ m ( s ) = N m | f m | 2 s + γ a / 2 + ∆ a,m + i ¯ δ m . (17) 5 This is the analytic mechanism behind the eective linewidth broadening used in recent multires- onator ensemble calculations: matter loading changes both the absorptive part and the dispersion of the auxiliary resonator self-energy [16]. 3.3 Positive-real property A scalar function Y ( s ) is positive real if it is analytic in Re s > 0 , real-symmetric, and Re Y ( s ) ≥ 0 for Re s > 0 . Positive-real functions are the admittances of passive linear systems. Lemma 1 (Matter and resonator self-energies are positive real) . Assume γ b,m , γ a,mα ≥ 0 and all oscillator strengths are nonnegative. Then Φ m ( s ) and Σ( s ) are positive real for Re s > 0 whenever the auxiliary resonator network is passive and stable. Proof. For Eq. (11), Re | f mα | 2 s + γ a,mα / 2 + i δ mα = | f mα | 2 Re s + γ a,mα / 2 | s + γ a,mα / 2 + i δ mα | 2 ≥ 0 . (18) The continuous case follows by monotone convergence. A passive interconnection of positive-real one-port admittances is positive real; equivalently, the state-space realization in Eqs. (7)(9) has a dissipative generator. Thus Eq. (14) is positive real for a stable passive network. In the diagonal form this also follows fr om the Schur complement of an accretive matrix. This lemma is the rst key structural point. It says that a multiresonator memory is not an arbitrary spectral lter. It belongs to the positive-real class. Conversely, passive network synthesis says that rational positive-real admittances can be realized by networks of harmonic modes and lossless couplers; in quantum optics this is the passive linear quantum realization theorem [25, 26, 27]. 3.4 Schurpositive-real equivalence The re ection function and the self-energy contain the same information, but in two dierent analytic classes. The former is a Schur function; the latter is positive real. This equivalence is the inverse-design map. Theorem 1 (Schurpositive-real memory correspondence) . Assume a lossless one-port memory with γ c = 0 and κ > 0 . If Σ is stable and positive real, then r ( s ) = s − κ/ 2 + Σ( s ) s + κ/ 2 + Σ( s ) (19) is analytic and contractive in the right half-plane: | r ( s ) | ≤ 1 , Re s > 0 . (20) Conversely, let r be analytic in Re s > 0 , let 1 − r ( s ) have no zero there, and de ne Σ r ( s ) = κ 2 [1 + r ( s )] − s [1 − r ( s )] 1 − r ( s ) . (21) Then r is realizable by a passive MR-QM of the present one-port form if and only if Σ r is positive real and belongs to the chosen nite or continuum resonatormatter realization class. Proof. Let Re s > 0 and write A ( s ) = s + Σ( s ) . Since Re Σ( s ) ≥ 0 , Re A ( s ) > 0 . The Möbius transformation z 7 → z − κ/ 2 z + κ/ 2 (22) 6 maps the right half-plane into the unit disk; hence | r ( s ) | ≤ 1 . Solving the preceding equation for Σ gives Eq. (21). The converse is therefore immediate: if Eq. (21) is positive real and has a passive realization in the allowed class, substituting it back gives the prescribed r and a passive one-port memory. If Σ r fails positivity, the target Schur function is a passive lter but not a memory of the speci ed resonatormatter admittance type. This theorem is useful because optimization can be stated either as positive-real approximation of the ideal load or as Schur approximation of the zero-re ection target. The positivity of Eq. (21) is the hidden constraint that distinguishes physically writable memories fr om arbitrary re ection lters. 3.5 Controlled-storage dilation A positive-real pole may represent uncontrolled dissipation, but in a memory it must represent a controllable storage channel. The distinction is not semantic: it is the dierence between absorption loss and quantum memory. The following elementary dilation is the operational form used throughout the numerical checks. Theorem 2 (Controlled-storage dilation of a positive-real atom) . Let Σ N ( s ) = N X n =1 w n s + Γ n + i ν n , w n > 0 , Γ n > 0 , (23) and consider the one-port common resonator coupled to auxiliary modes b n with g n = √ w n . Replace the decay Γ n of each b n by a one-sided controlled storage port C n , ˙ b n = − (Γ n + i ν n ) b n − i g n a − p 2Γ n C n, in , (24) C n, out = C n, in + p 2Γ n b n . (25) With all C n, in in the vacuum, the coherent re ection coe cient in the signal port is exactly the Cayley transform with self-energy Σ N . Moreover, on the real frequency axis the full scattering matrix fr om the signal port and all storage ports is unitary, and therefore | r (i ω ) | 2 + N X n =1 | t n (i ω ) | 2 = 1 , (26) wh ere t n is the transfer amplitude into C n, out . Consequently, for every normalized one-photon wavepacket f , η store [ f ] = X n Z | t n (i ω ) | 2 | f ( ω ) | 2 d ω = 1 − Z | r (i ω ) | 2 | f ( ω ) | 2 d ω. (27) Proof. Eliminating the auxiliary modes gives b n ( s ) = − i g n a ( s ) − √ 2Γ n C n, in ( s ) s + Γ n + i ν n , (28) hence the coe cient multiplying a ( s ) in the reduced common-mode equation is Σ N ( s ) . This proves the re ection formula when the storage inputs are vacuum. The enlarged model is a passive Markovian inputoutput system with Hermitian Hamiltonian and coupling operators √ κ a and √ 2Γ n b n ; its transfer matrix is unitary on the imaginary axis. Setting the storage inputs to vacuum and taking the rst column of that unitary matrix gives Eq. (26). Integrating Eq. (26) against | f ( ω ) | 2 gives Eq. (27). 7 Thus the linewidths Γ n in a memory-grade positive-real model must be read as rates into addressable dark spin-wave or rephasing reservoirs, not as uncontrolled heat baths. The theorem is a write-scattering dilation: it proves that the non-re ected probability enters speci ed controlled channels. It does not by itself prove long-lived, on-demand retrieval fr om those channels. A complete memory claim must additionally specify the trapping, rephasing, decoupling, or time- reversal operation that maps the outgoing storage-port wavepacket to a stationary mode and back to the signal port. If a particular experiment cannot reverse, freeze, or rephase those ports, Eq. (27) remains an absorption calculation but not a quantum-memory e ciency. This gives a simple experimental falsi ability criterion for any proposed MR-QM: identify the physical storage ports corresponding to the positive-real atoms of Σ( s ) and report their retrieval map. Theorem 3 (Port-to-register criterion) . Let S = span { f 1 , . . . , f K } be a nite signal alphabet and let T : S → H port be the coherent write map fr om the input alphabet into the controlled storage-port wavepackets of Theorem 2. De ne the write Gram matrix G = T † T . A noiseless register trap that preserves all quantum coherences on the successfully written subspace exists if and only if G has no zero eigenvalue on the subspace to be preserved. In that case there is an isometry C : Ran T → H reg such that the trapped register map is CT , with the same Gram matrix G . If G = I K , CT is an isometry on the whole alphabet. If 0 < G < I , the singular values of CT give the unavoidable mode-dependent write probabilities unless an additional heralding or error- ltering operation is introduced. Proof. This is the polar decomposition of the write map. Write T = V ( G ) 1 / 2 , wh ere V is a partial isometry fr om the support of G onto Ran T . A lossless trap is precisely an isometric embedding C of that range into stationary register modes. Then ( CT ) † ( CT ) = T † T = G . If G = I K , the map preserves inner products and is a unitary encoding of the alphabet into the register. If G has a null vector, the corresponding coherent superposition is never written into the controlled ports and no later register operation can reconstruct it without an additional copy of the lost eld. This criterion is the missing logical bridge between a measured absorption spectrum and a memory claim. The spectral theorem certi es T † T ; the experimental protocol must still im- plement the isometry C and the reciprocal read map. The distinction is especially important for positive-real ts in which part of Σ( s ) is an actual long-lived register and part is ordinary irreversible damping. Theorem 4 (Loss-separation identity) . Suppose the same one-port memory also has uncontrolled Markovian loss channels L q with transfer amplitudes ℓ q ( s ) , in addition to controlled storage-port amplitudes t n ( s ) . On the imaginary axis the enlarged scattering matrix gives | r (i ω ) | 2 + X n | t n (i ω ) | 2 + X q | ℓ q (i ω ) | 2 = 1 . (29) Therefore a normalized input spectrum f is stored with probability η store [ f ] = 1 − Z | r (i ω ) | 2 | f ( ω ) | 2 d ω − Z X q | ℓ q (i ω ) | 2 | f ( ω ) | 2 d ω. (30) Proof. Equation (29) is the rst-column norm identity for the unitary scattering matrix of the signal, storage and uncontrolled-loss ports. Multiplying by | f ( ω ) | 2 and integrating gives Eq. (30). This identity is the operational distinction between high absorption and high quantum- memory e ciency. A spectrum can be impedance matched and still fail as a quantum memory if most of the non-re ected amplitude exits through uncontrolled loss ports. 8 4 Storage e ciency as spectral re ection defect 4.1 Single-photon signal space Let | 1 f ⟩ = Z R f ( ω ) A † in ( ω ) | vac ⟩ d ω, Z | f ( ω ) | 2 d ω = 1 (31) be an incoming one-photon wavepacket. A signal band is a measurable set B ⊂ R , and the band-limited single-photon subspace is H B = f ∈ L 2 ( R ) : supp f ⊂ B . (32) We de ne write e ciency at the end of the absorption interval as the probability that the excitation is in the controlled memory degrees of freedom rather than in the prompt output or uncontrolled loss channels. In a lossless passive device these are the only two alternatives. Theorem 5 (Exact write-e ciency identity) . For a stable, lossless, one-port passive MR-QM in the weak-excitation regime, η w [ f ] = 1 − Z R | r (i ω + 0) | 2 | f ( ω ) | 2 d ω. (33) Consequently the worst-case write e ciency on a band B is η min w ( B ) = 1 − ess sup ω ∈B | r (i ω + 0) | 2 . (34) With parasitic loss channels ℓ j ( ω ) , the identity becomes η w [ f ] = 1 − Z | r (i ω + 0) | 2 + X j | ℓ j (i ω + 0) | 2 | f ( ω ) | 2 d ω. (35) Proof. The full eld plus internal oscillator evolution generated by Eqs. (7)(9) is unitary when all loss channels are included as input-output elds. For each frequency, the linear input-output map is a unitary scattering matrix from the incoming waveguide channel to the outgoing waveguide, controlled memory modes and environmental channels. In the one-port lossless case this gives | r (i ω + 0) | 2 + ∥ w ( ω ) ∥ 2 = 1 , (36) wh ere w ( ω ) is the frequency-domain transfer vector into controlled memory modes at the chosen write time. Parseval's theorem applied to the one-photon spectral amplitude gives Eq. (33). Including environmental channels gives | r | 2 + X j | ℓ j | 2 + ∥ w ∥ 2 = 1 , (37) and hence Eq. (35). The worst-case formula (34) follows because multiplication by | r | 2 on L 2 ( B ) has operator norm ess sup B | r | 2 . This theorem is the operational reduction of the write problem. A memory paper may report an echo e ciency, a cavity linewidth, an optical depth, or a comb nesse, but the write part of the universal interface is completely determined by r ( ω ) , uncontrolled loss, and an explicit identi cation of the controlled storage channels. Retrieval requires the additional inverse map described in Sec. 13, not merely the disappearance of prompt re ection. 9 4.2 Perfect memory criterion Corollary 1 (Perfect stationary write criterion) . A lossless one-port passive MR-QM writes every photon in H B with unit e ciency if and only if r (i ω + 0) = 0 for almost every ω ∈ B . (38) For a nite-dimensional stationary signal subspace S = span { f 1 , . . . , f K } ⊂ L 2 ( R ) , perfect write is equivalent to | r (i ω + 0) | f ( ω ) = 0 for every f ∈ S and almost every ω. (39) Equivalently, after diagonalizing the positive multiplication form on S , all nonzero eigenvectors must be supported inside the zero set of r . For a nonzero rational r , the boundary zero set on the real-frequency axis is nite unless r van- ishes identically. Hence a nite time-independent passive rational device cannot perfectly write any nonzero square-integrable wavepacket whose spectrum occupies a set of positive measure. Exact stationary interpolation is possible only for idealized monochromatic frequency-bin distri- butions or for non-rational/continuum limits. Exact capture of genuine temporal wavepackets is instead a dynamic-control statement, treated in Theorem 12 and Theorem 13. The nite-alphabet Gram certi cate below is therefore an exact e ciency certi cate, not a hidden proof of stationary perfect storage for ordinary L 2 pulse modes. 5 Vector-port and multimode prompt-scattering certi cate Many practical interfaces have more than one prompt optical or microwave channel: polarization, counterpropagating modes, spatial modes, frequency bins, or deliberately engineered multiport couplers. The scalar re ection theorem has an immediate operator-valued extension. Let the incoming signal on a band B be a vector-valued spectrum f ( ω ) ∈ C p , ∥ f ∥ 2 = Z B ∥ f ( ω ) ∥ 2 C p d ω = 1 . Let R (i ω ) ∈ C p × p be the coherent prompt scattering matrix fr om the signal input channels back to the signal output channels after all controlled storage ports have been eliminated. Theorem 6 (Vector-port storage certi cate) . For a lossless controlled-dilation multiresonator memory, η w [ f ] = 1 − Z B f ( ω ) † R (i ω ) † R (i ω ) f ( ω ) d ω. (40) Hence the worst-case write e ciency over all normalized vector wavepackets supported in B is η min w ( B ) = 1 − ess sup ω ∈B σ max ( R (i ω )) 2 . (41) For a nite vector-valued alphabet f 1 , . . . , f K , the exact write Gram matrix is G ij = δ ij − Z B f i ( ω ) † R (i ω ) † R (i ω ) f j ( ω ) d ω. (42) Proof. At each real frequency the enlarged scattering matrix of the prompt channels, controlled storage ports, and any explicitly retained reversible auxiliary ports is unitary. Taking the block column corresponding to the p signal inputs gives R † R + T † T = I p , 10 wh ere T is the transfer matrix into controlled storage ports. Multiplication by f ( ω ) , integration over the band, and normalization give Eq. (40). The worst-case value of the quadratic form Z f † R † Rf d ω over unit vectors in L 2 ( B ; C p ) is the essential supremum of the largest eigenvalue of R † R , which is ess sup σ max ( R ) 2 . Restricting the same quadratic form to a nite orthonormal alphabet gives Eq. (42). The one-port formula used below is the special case p = 1 . The vector theorem is operationally important: a device can have small re ection in one polarization or direction but still fail as a universal interface if σ max ( R ) is large in another prompt channel. Conversely, a deliberately multiport design should be certi ed by the singular-value envelope of R , not by one selected scalar re ection trace. 6 Data-to-band certi cation fr om nite measurements Equations (34) and (41) are continuum statements. They are not certi ed by plotting a nite set of frequency samples unless a rational model, a passivity-constrained t, or a derivative margin is also reported. This point is operationally important: a narrow under-sampled re ection spike can dominate the worst-case memory error even when all measured points look impedance matched. Theorem 7 (Finite-sampling storage certi cate) . Let B = [ ω − , ω + ] , let Ω L = { ω ℓ } L ℓ =1 ⊂ B , and de ne the ll distance h = sup ω ∈B min ℓ | ω − ω ℓ | . (43) If the scalar re ection coe cient is C 1 on B and L r = sup ω ∈B d d ω r (i ω ) < ∞ , (44) then ess sup ω ∈B | r (i ω ) | ≤ ρ grid + L r h, ρ grid = max ℓ | r (i ω ℓ ) | . (45) Consequently every band-limited one-photon mode satis es η w [ f ] ≥ 1 − ( ρ grid + L r h ) 2 . (46) For a vector interface with prompt matrix R (i ω ) , the same statement holds with ρ grid = max ℓ σ max R (i ω ℓ ) , L R = sup ω ∈B d d ω R (i ω ) op , and L r replaced by L R . Proof. For every ω ∈ B , choose a nearest sample ω ℓ . The fundamental theorem of calculus gives | r (i ω ) − r (i ω ℓ ) | ≤ L r | ω − ω ℓ | ≤ L r h, and Eq. (45) follows from the triangle inequality. For matrices, Weyl's singular-value inequality gives σ max R (i ω ) ≤ σ max R (i ω ℓ ) + ∥ R (i ω ) − R (i ω ℓ ) ∥ op ≤ ρ grid + L R h. The e ciency statements are Theorems 5 and 6. The derivative margin can be obtained analytically from a tted passive rational model, by interval arithmetic on the tted transfer function, or conservatively from a dense calibrated scan. Without such a margin a nite plot is a diagnostic, not a certi cate. In the numerical section the eleven-mode design is checked both by dense continuum evaluation and by the derivative-aware nite-grid bound (46). 11 7 Complex-response and causal-realizability certi cation A memory certi cate cannot be based on a magnitude plot alone. The re ection data must be compatible with a causal passive device and must determine the phase convention used by the writeread channel. Theorem 8 (Schuradmittance realizability test) . Let r ( s ) be a scalar rational function analytic in the open right half-plane and let r ( s ) ̸ = 1 there. In the normalized lossless one-port convention κ = 2 , the only self-energy compatible with r is Σ r ( s ) = 1 + r ( s ) 1 − r ( s ) − s. (47) A necessary condition for passive MR-QM realizability is that Σ r is positive real, Re Σ r ( s ) ≥ 0 (Re s > 0) , (48) and has no unstable pole. Conversely, if a rational Σ is strictly proper, stable and positive real, then r Σ ( s ) = s − 1 + Σ( s ) s + 1 + Σ( s ) (49) is a stable Schur re ection coe cient, | r Σ (i ω ) | ≤ 1 on the frequency axis. If the partial-fraction residues of Σ are positive semide nite in a passive oscillator realization, Eq. (49) is realized by a nite controlled-dilation MR-QM. Proof. Solving Eq. (49) for Σ gives Eq. (47). A passive internal network has a positive-real driving-point admittance; hence Eq. (48) and stability are necessary. Conversely, for Re s > 0 write z = s + Σ( s ) . Since Re z = Re s + Re Σ( s ) > 0 , z − 1 z + 1 2 = | z | 2 − 2 Re z + 1 | z | 2 + 2 Re z + 1 ≤ 1 , which proves Schur contractivity on the boundary by continuity. The nal realization statement is the passive-realization lemma applied to the positive-real self-energy; the oscillator-residue condition is precisely the MR-QM subclass of such passive realizations. Proposition 1 (Magnitude-only ambiguity) . Let r ( s ) be any stable Schur re ection coe cient and let a > 0 . The Blaschke factor B a ( s ) = s − a s + a (50) is stable and all-pass on the frequency axis: | B a (i ω ) | = 1 . Therefore e r ( s ) = B a ( s ) r ( s ) has exactly the same re ection magnitude and the same band write-e ciency bound as r , but its phase is shifted by arg B a (i ω ) = π − 2 arctan( ω/a ) (51) up to branch convention, with group-delay contribution τ a ( ω ) = − d d ω arg B a (i ω ) = 2 a a 2 + ω 2 . (52) Thus magnitude data alone cannot certify temporal-mode preservation, reciprocal readout phase, or causal realizability within a declared memory architecture. Proof. The pole of B a is at − a , so the factor is stable, and for s = i ω the numerator and de- nominator have the same modulus. The phase and group-delay formulas follow by dierentiating the boundary value. Multiplication by B a leaves | r | unchanged but changes the causal impulse response and the phase of the written mode. A quantum memory certi cate must therefore report complex scattering data or an equivalent passive state-space model, not only | r | . 12 Proposition 2 (Minimum-phase phase recovery and all-pass obstruction) . Let r ( s ) be a ratio- nal stable Schur re ection coe cient with no pole or zero on the imaginary axis. Its boundary magnitude determines only its outer, or minimum-phase, factor. More explicitly, r ( s ) = e i θ B ( s ) r out ( s ) , (53) wh ere B is a nite product of right-half-plane Blaschke factors and | B (i ω ) | = 1 . If B = 1 , the boundary phase of r is xed, up to the constant θ , by the Hilbert transform of log | r (i ω ) | . If B ̸ = 1 , | r | is unchanged but the group delay and the temporal readout mode change. Proof. This is the rational right-half-plane innerouter factorization. The zeros of r in the open right half-plane generate stable all-pass Blaschke factors of the form ( s − a ) / ( s + ¯ a ) , which have unit modulus on s = i ω . After removing these factors the remaining zero-free stable factor has an analytic logarithm in the right half-plane. The real boundary value of this logarithm is log | r | , and the harmonic conjugate gives the phase by the Hilbert transform, up to an additive constant. Hence magnitude xes phase only after the all-pass content has been independently excluded. Theorem 9 (Complex-response uncertainty certi cate) . Let R : S → L 2 ( B ) and b R : S → L 2 ( B ) be the true and tted prompt scattering maps on a declared nite alphabet S . De ne the write Gram matrices by the full prompt-output norm, G = I − R † R, b G = I − b R † b R. If ∥ R − b R ∥ S→ L 2 ( B ) ≤ ϵ, ∥ b R ∥ S→ L 2 ( B ) ≤ b ρ, (54) then ∥ G − b G ∥ ≤ ϵ (2 b ρ + ϵ ) , (55) and therefore λ min ( G ) ≥ λ min ( b G ) − ϵ (2 b ρ + ϵ ) . (56) For a scalar one-port model acting by multiplication with r (i ω ) and an orthonormal alphabet f 1 , . . . , f K , b G ij = δ ij − Z B | b r (i ω ) | 2 f ∗ i ( ω ) f j ( ω ) d ω. (57) The compressed matrix R f ∗ i b rf j d ω is a same-alphabet coherent-leakage diagnostic; it must not replace Eq. (57) in a write-e ciency certi cate, because prompt photons scattered into modes outside the chosen alphabet are still lost fr om the memory. The su cient pointwise condition ess sup ω ∈B | r (i ω ) − b r (i ω ) | ≤ ϵ (58) implies the operator hypothesis on every subspace of L 2 ( B ) . If the complex error is measured on a nite grid, the same derivative-margin argument as in Theorem 7 gives ϵ ≤ ϵ grid + L δr h. Thus calibrated amplitude and phase errors propagate directly into the nite-alphabet write and reciprocal-readout certi cates without hiding out-of-alphabet prompt leakage. Proof. Write ∆ = R − b R . Then G − b G = − b R † ∆ − ∆ † b R − ∆ † ∆ . 13 Taking operator norms gives Eq. (55). Weyl's eigenvalue inequality gives Eq. (56). In the scalar multiplication case the quadratic form of R † R on coe cients c is Z B | r (i ω ) | 2 X j c j f j ( ω ) 2 d ω, which gives Eq. (57). The operator norm of multiplication by r − b r on L 2 ( B ) is its essential supremum, and the nite-grid statement is the scalar sampling theorem applied to the complex error function. The data-to-band theorem controls under-sampling of a known complex model; the present test controls whether the model itself is physically compatible with a passive memory. In practice one should report the complex S -parameter t, its poles, the positivity margin of Σ r , and the uncertainty of the phase calibration used for the reciprocal read operation. 8 No-go theorem for exact nite passive broadband matching Finite MR-QM models have rational transfer functions. This is true whether the nite modes are empty resonators, single atoms in resonators, or nite collective atomic spectral bins. The next theorem states a limitation that is often hidden in numerical optimization. Theorem 10 (No nite exact continuous-band perfect memory) . Let r ( s ) be the prompt re ection coe cient of a nite, stable, time-independent, passive one-port MR-QM. Suppose that r ( s ) is analytic in a neighbourhood of the imaginary-axis interval i B = { i ω : ω ∈ ( ω 1 , ω 2 ) } and that ω 1 < ω 2 . If r (i ω ) = 0 for all ω ∈ ( ω 1 , ω 2 ) , (59) then r ( s ) ≡ 0 as a meromorphic function. But any nite resonator realization satis es r ( s ) → 1 as | s | → ∞ . Hence no such nite passive device can be exactly re ectionless on a nonzero continuous band. Proof. A nite stable realization has poles only in the open left half-plane, after including positive damping, so r is analytic across the speci ed imaginary-axis interval. The set of zeros has an accumulation point inside the analytic domain. By the identity theorem, r vanishes identically on the connected analytic domain and therefore as a meromorphic function. On the other hand, Eq. (1) has lim | s |→∞ r ( s ) = 1 (60) because Σ( s ) = O (1 /s ) for a nite passive oscillator network. This contradiction proves the result. Corollary 2. Claims of exactly unit memory e ciency over a nite continuous bandwidth in a - nite passive time-independent multiresonator system must involve at least one of the following: an approximation, a nite-dimensional signal alphabet, a continuum/in nite spectral measure, active or time-dependent control during absorption, non-Markovian external coupling, or a convention that omits prompt re ection/loss. This theorem is not pessimistic. It is the correct starting point for design. It says that nite MR-QM is a broadband matching problem of the BodeFano/Darlington type, not an exact algebraic cancellation problem [28, 29]. The gure of merit is the best achievable re ection norm on the target signal set. 14 9 BodeFano area law The no-go theorem forbids exact zero re ection on a continuous band. A stronger statement is available for passive one-port memories with xed external coupling: the total logarithmic matching area is nite. This is the memory form of the BodeFano limitation [20, 21]. Theorem 11 (BodeFano area law for passive MR-QM) . Consider a rational, lossless, passive one-port MR-QM with external coupling κ , re ection coe cient r ( s ) = 1 − κ s + O ( s − 2 ) , s → ∞ , (61) and no singular inner factor. Boundary zeros on the imaginary axis are allowed only when the logarithmic singularity is integrable, as in the limiting examples used below. Let z j be the zeros of r in the open right half-plane, counted with multiplicity. Then Z ∞ −∞ log 1 | r (i ω ) | d ω = π κ − 2 X j Re z j ≤ πκ. (62) Consequently, if | r (i ω ) | ≤ ρ for all | ω | ≤ B , then ρ ≥ exp h − πκ 2 B i , η min w ≤ 1 − exp h − πκ B i . (63) Proof. The passive one-port re ection coe cient is a rational Schur function in the right half- plane. First assume that it has no boundary zeros; the stated formula with integrable boundary zeros follows by moving the contour a distance ϵ > 0 into the right half-plane and then taking ϵ ↓ 0 . Factor its right-half-plane zeros by the Blaschke product B r ( s ) = Y j s − z j s + ¯ z j , (64) which has unit modulus on the imaginary axis. The quotient g ( s ) = r ( s ) /B r ( s ) is zero-free and Schur in the right half-plane. Applying the Poisson representation to log | g | at a real point x > 0 gives log | r ( x ) | − X j log x − z j x + ¯ z j = x π Z ∞ −∞ log | r (i ω ) | x 2 + ω 2 d ω. (65) Using r ( x ) = 1 − κ/x + O ( x − 2 ) and log x − z j x + ¯ z j = − 2 Re z j x + O ( x − 2 ) and then multiplying by πx and taking x → ∞ yields Eq. (62). Since | r (i ω ) | ≤ 1 for a lossless passive one-port, the left hand side is nonnegative and bounded above by πκ . If | r | ≤ ρ on an interval of length 2 B , then 2 B log(1 /ρ ) ≤ Z ∞ −∞ log | r (i ω ) | − 1 d ω ≤ πκ, which gives Eq. (63). For the normalized numerical examples below, κ = 2 and B = 1 , so Eq. (63) gives ρ ≥ e − π = 0 . 043214 and η min w ≤ 1 − e − 2 π = 0 . 998133 . Thus the remaining gap between the nite-mode designs and unity is not merely a numerical imperfection; it is bounded from below by a one-port matching area law unless time-dependence, gain, additional external ports, or a dierent resource class is introduced. 15 10 Dynamic bypass of the stationary bound The BodeFano area law applies to linear time-independent passive one-port matching. It does not forbid exact absorption of a known temporal mode when a control waveform changes the coupling in time. This distinction is central for interpreting universal interface claims. Theorem 12 (Gauge-complete exact dynamic capture of one known temporal mode) . Let f be a normalized, piecewise C 1 input envelope on ( −∞ , T ) , and de ne E ( t ) = Z t −∞ | f ( s ) | 2 d s. (66) On every interval wh ere E ( t ) > 0 and f ( t ) ̸ = 0 , write f ( t ) = | f ( t ) | e i ϕ ( t ) . Consider a one-sided memory mode with tunable external coupling and detuning, ˙ a ( t ) = − κ ( t ) 2 + i∆( t ) a ( t ) − p κ ( t ) f in ( t ) , (67) f out ( t ) = f in ( t ) + p κ ( t ) a ( t ) . (68) Choose κ ( t ) = | f ( t ) | 2 E ( t ) , ∆( t ) = − ˙ ϕ ( t ) , (69) with arbitrary bounded interpolation through isolated zeros of f . Then the input f in = f is captured with f out ( t ) = 0 for all t < T , and a ( t ) = − p E ( t ) e i ϕ ( t ) , | a ( t ) | 2 = E ( t ) . (70) Thus lim t → T | a ( t ) | 2 = 1 if the pulse is fully contained before T . For merely L 2 envelopes the statement holds by approximating f by smooth compactly supported envelopes and taking the L 2 - lim it of the inputoutput map. Proof. The zero-output condition is √ κa = − f . With Eq. (69) it gives the displayed trajectory and | a | 2 = E . Dierentiating a = − √ E e i ϕ gives ˙ a = ˙ E 2 E + i ˙ ϕ ! a = κ 2 + i ˙ ϕ a, because ˙ E = | f | 2 = κE . The right hand side of Eq. (67) evaluated on the zero-output trajectory is − κ 2 + i∆ a − √ κf = − κ 2 + i∆ a + κa = κ 2 − i∆ a. The choice ∆ = − ˙ ϕ makes the two expressions identical. The energy identity then follows either fr om the trajectory or fr om the lossless inputoutput balance d d t | a ( t ) | 2 = | f in ( t ) | 2 − | f out ( t ) | 2 . Equation (69) is the time-domain counterpart of impedance matching, but the theorem now exposes the full control requirement: the amplitude law xes the storage probability and the detuning or coupling phase xes the optical phase. It is exact for a known pulse but depends on the pulse shape, the causal accumulated energy, and the calibrated phase convention. Related dynamic absorption and linewidth-modulation ideas appear in single-photon absorption, short- pulse impedance matching, and recent spin-ensemble absorptionemission optimization [23, 24, 16 22]. For pulses with an in nite leading tail, E ( t ) is exponentially small in the far past and the ideal κ ( t ) can become unbounded; practical schedules therefore require either a regularized leading-tail construction or an explicit bounded-control leakage certi cate. Thus dynamic matching does not provide a time-independent broadband interface for arbitrary unknown waveforms. It is the rst element of the hierarchy: dynamic control gives exact matching on a speci ed one-dimensional signal space, while the positive-real minimax theory solves the stationary nite-band problem. Proposition 3 (Capped dynamic-control certi cate) . Let f be a normalized input mode and let bounded measurable controls κ c ( t ) ≥ 0 and ∆ c ( t ) be used in Eq. (67) instead of the ideal controls (69) . If the memory mode is initially empty and there is no internal loss, then the actually achieved write e ciency is exactly η c ( T ) = | a ( T ) | 2 = 1 − Z T −∞ | f out ( t ) | 2 d t − Z ∞ T | f in ( t ) | 2 d t. (71) In particular, a hardware-limited capped approximation to the singular ideal law is certi ed by di- rect integration of the same inputoutput equations; it is not certi ed by the formal ideal waveform alone. Proof. For Eq. (67) with arbitrary real κ c , ∆ c , d d t | a ( t ) | 2 = | f in ( t ) | 2 − | f out ( t ) | 2 . Integrating from the remote past, wh ere a = 0 , to T gives Eq. (71). The detuning changes phase but not the energy balance. Thus every bounded-control schedule has an experimentally checkable leakage certi cate. 10.1 Exact dynamic capture of a nite temporal alphabet The one-mode construction has a matrix generalization. It is the constructive counterpart of the rank-capacity theorem. Theorem 13 (Exact dynamic capture of a nite alphabet) . Let f 1 , . . . , f K be orthonormal tem- poral modes and de ne the column vector f ( t ) = ( f 1 ( t ) , . . . , f K ( t )) T . (72) Assume that the accumulated Gram matrix S ( t ) = Z t −∞ f ( s ) f ( s ) † d s (73) is nonsingular on the interval of interest; otherwise one uses the regularized seed construction of Appendix D. Consider a passive time-dependent one-input memory with K storage modes, ˙ a ( t ) = − 1 2 L ( t ) L ( t ) † + i H ( t ) a ( t ) − L ( t ) f in ( t ) , (74) f out ( t ) = f in ( t ) + L ( t ) † a ( t ) , (75) wh ere H ( t ) = H ( t ) † is an optional internal Hamiltonian gauge. There exists a complex coupling waveform L ( t ) and, if desired, an equivalent real-coupling representation with a Hamiltonian gauge, such that every input f in ( t ) = f ( t ) † c , c ∈ C K , (76) is captured with f out ( t ) = 0 and nal storage amplitude a ( ∞ ) = U c , (77) 17 − 4 − 3 − 2 − 1 0 1 2 3 4 0 0 . 5 1 time eigenvalues of S ( t ) λ min S ( t ) λ max S ( t ) Figure 2: Accumulated Gram eigenvalues for a two-mode HermiteGaussian alphabet. Exact dynamic alphabet capture becomes well conditioned only after the accumulated Gram matrix is nonsingular; the regularized seed construction controls the leading singular interval. wh ere U is a unitary matrix determined by the storage gauge. One explicit zero-output construc- tion is ˙ A ( t ) = 1 2 A ( t ) −† f ( t ) f ( t ) † , L ( t ) = − A ( t ) −† f ( t ) , H ( t ) = 0 , (78) with A ( t ) † A ( t ) = S ( t ) and a ( t ) = A ( t ) c . Equivalently, a prescribed dierentiable storage-frame rotation A ( t ) 7 → A ( t ) U ( t ) is implemented by the corresponding Hermitian gauge Hamiltonian. Proof. For a = A c and f in = f † c , the zero-output condition is L † A = − f † , (79) which is satis ed by L = − A −† f . Substitution into Eq. (74) with H = 0 gives ˙ A = − 1 2 LL † A − Lf † = 1 2 A −† f f † , which is Eq. (78). The stored Gram matrix obeys d d t ( A † A ) = ˙ A † A + A † ˙ A = f f † . Thus A † A = S whenever this holds at the initial regularized time. Since the modes are or- thonormal, S ( ∞ ) = I K , so A ( ∞ ) is unitary. A time-dependent storage-frame rotation can be implemented after capture or, in an orthonormal register basis, by adding the corresponding Her- mitian control Hamiltonian. The zero-output identity and the Gram evolution are unchanged. This shows that phase and basis gauges are physical control requirements, not hidden assump- tions. The theorem is deliberately resource-explicit. Exact nite-alphabet universality requires a storage dimension at least equal to the alphabet dimension and a calibrated vector coupling waveform. Without those resources, one returns to the stationary positive-real approximation problem and to the BodeFano area law. Corollary 3 (No contradiction with BodeFano) . The exact dynamic capture law (69) does not violate Theorem 11, because the system is not time independent and has no stationary re ection coe cient r (i ω ) to which the BodeFano integral applies. 18 − 3 − 2 − 1 0 1 2 3 0 0 . 5 1 time normalized quantity E ( t ) κ ( t ) / 8 Figure 3: Exact dynamic capture of a chirped Gaussian temporal mode. The accumulated stored energy is E ( t ) and the impedance-matching law is κ ( t ) = | f ( t ) | 2 /E ( t ) ; the plotted coupling is rescaled by a factor eight. nite time-independent passive interface: approx- imate continuous-band matching; certi ed by positive-real minimax norm and BodeFano area time-dependent scalar control: exact capture of one known temporal mode; control waveform is signal-space dependent time-dependent vector control with K storage modes: exact capture of a known K -dimensional alphabet active/non-Foster or measurement-feedback resources: not covered by passive BodeFano; require explicit quantum-noise certi cation Figure 4: Operational hierarchy. The resources that evade stationary BodeFano limitations are precisely the resources that must be reported: time dependence, alphabet dimension, active elements, and added noise. 11 Spectral-admittance synthesis 11.1 Admittance target Equation (1) can be written as r ( s ) = N ( s ) D ( s ) , N ( s ) = s − κ − γ c 2 + Σ( s ) , D ( s ) = s + κ + γ c 2 + Σ( s ) . (80) Broadband matching is the problem of making N (i ω ) small while keeping D (i ω ) stable and away fr om zero. The ideal boundary condition is Σ(i ω + 0) = κ − γ c 2 − i ω, ω ∈ B . (81) A nite positive-real self-energy cannot satisfy this on a continuous band, but it can approximate it. De nition 1 (MR-QM spectral synthesis problem) . For a target band B , controlled parasitic loss bound γ loss , and allowed number of auxiliary resonator/matter poles N , nd a positive-real 19 rational self-energy Σ N of the form realizable by Eq. (14) that minimizes ϵ N ( B ) = ess sup ω ∈B i ω − κ − γ c 2 + Σ N (i ω ) i ω + κ + γ c 2 + Σ N (i ω ) . (82) The worst-case write e ciency is then 1 − ϵ 2 N in the lossless case. This is the universal nite-device problem. It can be solved by positive-real rational ap- proximation, by a constrained Remez algorithm, by semide nite KYP constraints, or by direct physical-parameter optimization. The choice of numerical method does not change the physical theory. 11.2 Physical realizability of rational solutions Theorem 14 (Positive-real synthesis implies physical MR-QM realization) . Let Σ N ( s ) be a ra- tional positive-real function with real symmetry, poles in the closed left half-plane, and Σ N ( s ) → 0 as | s | → ∞ . Then there exists a passive nite network of harmonic modes coupled to the common resonator whose eliminated self-energy is Σ N ( s ) . If the residues are simple and positive after diagonalization, the realization can be chosen as independent auxiliary resonators; otherwise it is realized as a lossless coupled-resonator network followed by diagonal damping channels. Quantiza- tion of this passive network gives a physically realizable linear quantum system with the re ection coe cient (1) . Proof. Classical CauerFosterDarlington synthesis realizes every rational positive-real driving- point admittance as a passive network of reactive elements and resistive terminations. Replacing each reactive normal mode by a harmonic oscillator and each resistive termination by a Markov input-output channel gives the passive quantum linear realization. Equivalently, the lossless bounded-real transfer function r ( s ) is the scattering matrix of a passive linear quantum system, and Σ N is the corresponding Schur complement. The independent-oscillator realization is ob- tained when the spectral measure of Σ N is diagonal with positive residues; otherwise a unitary transformation gives coupled resonators with the same admittance. This theorem is the mathematical content behind the statement that MR-QM is a universal interface. The universality is spectral: once a positive-real admittance is speci ed, the same mem- ory response can be implemented with microwave CPW resonators, optical microrings, photonic molecules, rare-earth ensembles, spin ensembles, or hybrid systems, limited only by fabrication losses and controllability. 11.3 Continuum ideal and nite approximation In the continuum lim it the nite sum in Eq. (14) is replaced by a positive spectral measure μ : Σ μ ( s ) = Z R d μ ( ν ) s + Γ( ν ) + i ν , d μ ( ν ) ≥ 0 , Γ( ν ) ≥ 0 . (83) The boundary matching equation is the singular integral system Z Γ( ν ) d μ ( ν ) Γ( ν ) 2 + ( ω + ν ) 2 = κ − γ c 2 , (84) ω − Z ( ω + ν ) d μ ( ν ) Γ( ν ) 2 + ( ω + ν ) 2 = 0 , ω ∈ B . (85) Equations (84)(85) are the exact continuum spectral-admittance conditions. They are the ana- logue of designing an exactly matched continuous transmission-line load. A nite MR-QM is a quadrature approximation of μ plus a nite positive-real correction. 20 11.4 Existence and density of resource-constrained synthesis The continuum equations are useful only if the corresponding optimization problem is mathemat- ically closed. The physically correct closure is not an unconstrained measure on the whole plane, but a fabrication box and an oscillator-strength budget. Let K = [Γ min , Γ max ] × [ − ∆ max , ∆ max ] , 0 < Γ min < Γ max < ∞ , (86) and let M + ( K ; M ) be the positive Borel measures on K with total mass at most M . For μ ∈ M + ( K ; M ) set Σ μ ( s ) = Z K d μ (Γ , ν ) s + Γ + i ν . (87) The resource-constrained design problem is ρ ⋆ = inf μ ∈M + ( K ; M ) sup ω ∈B i ω − κ/ 2 + γ c / 2 + Σ μ (i ω ) i ω + κ/ 2 + γ c / 2 + Σ μ (i ω ) . (88) Theorem 15 (Existence and nite-mode density) . For every compact signal band B and every nite resource box K , M , the minimization problem (88) has a minimizer. Moreover, for every admissible continuum measure μ and every ϵ > 0 there is a nite atomic measure μ N = N X n =1 w n δ (Γ n ,ν n ) , w n > 0 , (89) such that the associated nite MR-QM satis es sup ω ∈B | r μ N (i ω ) − r μ (i ω ) | < ϵ. (90) Proof. The set M + ( K ; M ) is weak-star compact because K is compact and the masses are uni- formly bounded. For each ω ∈ B the kernel (Γ , ν ) 7 → (i ω + Γ + i ν ) − 1 is continuous and uniformly bounded by Γ − 1 min , hence Σ μ n (i ω ) → Σ μ (i ω ) uniformly on B whenever μ n → μ weak-star. The denominator of the Cayley transform is uniformly separated fr om zero since Re κ + γ c 2 + Σ μ (i ω ) ≥ κ + γ c 2 > 0 . (91) Therefore μ 7 → sup ω ∈B | r μ (i ω ) | is continuous on the compact admissible set, so a minimizer exists. Finally, nite positive atomic measures are weak-star dense in positive measures on a compact metric space; applying the same uniform-continuity argument to the Cayley transform gives (90). This theorem is the formal reason why nite multiresonator synthesis is a controlled approx- imation rather than a heuristic. Increasing the number of resonators is a quadrature re nement of a positive-real continuum optimum, while the no-go theorem states that the limiting optimum cannot be made exactly zero on a continuous band unless an in nite or time-dependent resource is admitted. 11.5 Finite-grid Carathéodory certi cate The continuum minimizer in Eq. (88) is conceptually useful, but every experiment and numerical certi cate uses nitely many frequency samples. On a nite grid, no continuum of atoms is required. 21 Theorem 16 (Sparse atomic certi cate on a frequency grid) . Let Ω L = { ω 1 , . . . , ω L } be a nite frequency grid, and let μ ∈ M + ( K ; M ) be any admissible positive measure. There exists an atomic measure μ sp = N sp X n =1 w n δ (Γ n ,ν n ) , N sp ≤ 2 L + 1 , (92) such that Σ μ sp (i ω ℓ ) = Σ μ (i ω ℓ ) , ℓ = 1 , . . . , L, (93) and μ sp ( K ) = μ ( K ) ≤ M . Therefore the re ection coe cient, the grid epigraph constraints, and every grid-certi ed write e ciency are identical for μ and μ sp on Ω L . Proof. De ne the continuous feature map Φ(Γ , ν ) = Re 1 i ω 1 + Γ + i ν , Im 1 i ω 1 + Γ + i ν , . . . , Re 1 i ω L + Γ + i ν , Im 1 i ω L + Γ + i ν , 1 ∈ R 2 L +1 . (94) The vector y = Z K Φ(Γ , ν ) d μ (Γ , ν ) belongs to the conic hull of Φ( K ) ⊂ R 2 L +1 . By the conic Carathéodory theorem, y is a nonnegative linear combination of at most 2 L + 1 points of Φ( K ) . The corresponding coe cients and support points de ne μ sp . Equality of the rst 2 L coordinates gives equality of the real and imaginary parts of Σ on the grid, and equality of the last coordinate preserves the mass budget. This is a useful certi cation theorem rather than an e cient design algorithm. It says that a grid-certi ed continuum optimum always has a nite sparse witness. The remaining task is to nd a good witness with a stable and experimentally realistic pole geometry. 11.6 Finite positive-real Remez synthesis For a xed nite pole class Eq. (148), the practical problem is the epigraph program minimize ρ, (95) subject to | r (i ω ℓ ; p ) | ≤ ρ, ℓ = 1 , . . . , L, (96) Γ j > 0 , w j > 0 , ∆ j > 0 . (97) The grid constraints are nonlinear but low dimensional, and positivity is enforced by logarithmic variables. After optimization on a grid, the result must be certi ed on an independent dense grid and by direct integration of the passive dilation. The near-equiripple envelope of the certi ed solutions below is the positive-real analogue of classical Chebyshev/Remez lter synthesis. 12 Local at matching and relation to known MR-QM conditions 12.1 Derivative hierarchy Let s 0 = i ω 0 be the centre of the signal band. If D ( s 0 ) ̸ = 0 , then a zero of r ( s ) of order p + 1 at s 0 is equivalent to N ( q ) ( s 0 ) = 0 , q = 0 , 1 , . . . , p. (98) Using Eq. (80) gives the hierarchy Σ( s 0 ) = κ − γ c 2 − s 0 , (99) Σ ′ ( s 0 ) = − 1 , (100) Σ ( q ) ( s 0 ) = 0 , q = 2 , . . . , p. (101) 22 Equation (99) is the ordinary impedance matching condition. Equation (100) cancels the rst- order spectral phase dispersion of the common resonator; it is the local form of white-cavity or spectral matching. Higher equations impose progressively atter re ection. For a symmetric diagonal resonator spectrum Σ( s ) = w 0 s + Γ 0 + J X j =1 w j 1 s + Γ j + i∆ j + 1 s + Γ j − i∆ j , w j > 0 , (102) the derivatives are explicit: Σ ( q ) (0) = ( − 1) q q ! w 0 Γ q +1 0 + J X j =1 w j 1 (Γ j + i∆ j ) q +1 + 1 (Γ j − i∆ j ) q +1 . (103) Thus at matching is a nite system of real algebraic equations in positive linewidths, detunings and oscillator strengths. 12.2 Recovery of standard impedance conditions If all auxiliary resonators are empty and form a spatial-frequency comb directly coupled to the waveguide, the self-energy becomes the comb Green function. For a periodic array the rephasing time is T = 2 π/ ∆ , and the waveguide coupling linewidth is Γ = 2 πg 2 /c . Moiseev's Bragg condition ∆ = π 2 Γ (104) is precisely the condition that the prompt re ection defect vanish at the comb centre while the resonator phases rephase at the echo time [7]. In the common-resonator formulation the same physics appears as Eq. (99) for the common mode and, when broadband atness is required, Eq. (100). For a rectangular distribution of auxiliary resonator frequencies loaded by Lorentzian-broadened atomic ensembles, the recent ensemble theory gives a common-resonator matching condition of the form κ = γ c + 2 M g 2 δ in F ( χδ in , χ Γ Σ , 0) , (105) with a spectral-broadening function F > 1 determined by the atomic loading [16]. In the notation of this paper this is simply κ = γ c + 2Σ(0) , (106) while the additional spectral condition derived there is the special rectangular-continuum evalu- ation of Σ ′ (0) = − 1 . (107) The spectral-topological conditions of small cascade memories are nite-dimensional versions of the same hierarchy: several tunable poles are moved until the numerator N ( s ) has a high-order zero or an equal-ripple small norm on the chosen band [11, 10]. 13 Retrieval and full memory channel Write matching is not su cient for a quantum memory; the absorbed excitation must be emitted on demand with controlled temporal mode, phase and noise. The spectral-admittance theory separates write and read cleanly. Let W : H B → H int be the write map into controlled internal modes. In the lossless case, W † W = I − R † R, (108) 23 wh ere R is multiplication by r ( ω ) on the input band. Let U s ( T ) be the controlled storage evolu- tion, including detuning reversal, spin-wave transfer, AFC rephasing, switch-o of the common resonator, or dynamical decoupling. Let W r be the read map fr om internal modes to the output channel. Theorem 17 (Time-reversal retrieval) . Suppose the read Hamiltonian is the antiunitary time reverse of the write Hamiltonian on the controlled internal subspace, and suppose storage control satis es U s ( T ) W f = e i ϕ W f for every f ∈ S (109) on a signal subspace S . Then the full memory channel on S is f ( t ) 7 −→ e i ϕ f ∗ ( T e − t ) (110) up to the write/read re ection defects and intrinsic decoherence. In particular, for a lossless perfectly written nite-dimensional subspace the retrieval delity is unity, and the total e ciency is the product of write, storage and read e ciencies. Proof. The passive write dynamics de nes a unitary map between input temporal modes and internal normal modes plus prompt output modes. Reversing the detunings and coupling phases implements the inverse unitary on the controlled sector. If U s ( T ) rephases the written subspace to a common phase, the read operation is W † followed by time reversal of the envelope. Or- thogonality and phase preservation follow fr om unitarity. Decoherence multiplies the controlled internal norm by the corresponding survival factor. This theorem includes AFC echo retrieval at T = 2 π/ ∆ , CRIB/GEM detuning reversal, and switchable-coupler multiresonator protocols as dierent ways of realizing the same inverse map [4, 5, 6, 12, 13]. 13.1 Finite-alphabet Gram certi cate For experiments and network protocols the relevant signal space is often not the full continuous band but a nite alphabet of temporal modes. The spectral theory reduces this problem to a nite matrix. Theorem 18 (Exact nite-alphabet write certi cate) . Let S = span { f 1 , . . . , f K } ⊂ H B , wh ere the f j are orthonormal spectra. In a lossless controlled-dilation MR-QM the write Gram matrix on S is G ij = ⟨ f i , W † W f j ⟩ = δ ij − Z B | r (i ω ) | 2 f ∗ i ( ω ) f j ( ω ) d ω. (111) The guaranteed write e ciency on the whole alphabet is η min ( S ) = λ min ( G ) , (112) and the Haar-average write e ciency over normalized states in S is η ( S ) = 1 K Tr G. (113) Perfect write on S is equivalent to G = I K . Proof. Theorem 5 gives W † W = I − R † R , wh ere R is multiplication by r on the signal band. Restriction to the basis f j gives Eq. (111). For a normalized vector c ∈ C K , the stored probability of f = P j c j f j is c † Gc ; minimizing over ∥ c ∥ = 1 gives the smallest eigenvalue. Averaging c † Gc over the unit sphere gives Tr G/K . Finally, G = I K is equivalent to Rf = 0 for every f ∈ S . This theorem is the nite-dimensional object that should accompany experimental claims of multimode operation. It is stronger than quoting the e ciency of one pulse shape, and it is more realistic than demanding perfect performance on an entire continuum. 24 13.2 Reciprocal readout and the full memory channel The write certi cate also determines the ideal readout when the storage dilation can be time reversed. Theorem 19 (Reciprocal readout theorem) . Let W : S → K be the write map fr om a nite signal alphabet S into the controlled storage space K , and assume that the read stage is the exact reciprocal time reversal on the range of W . Then the writeread amplitude operator on S is M rt = W † W = G, (114) wh ere G is the Gram matrix in Theorem 18. Consequently, η min rt ( S ) = λ min ( G ) 2 , η avg rt ( S ) = 1 K Tr G 2 , (115) for the worst-case and Haar-average unconditional round-trip e ciencies on S . If G = I K , the full memory channel is the identity on the alphabet up to a programmable unitary phase convention. Proof. The controlled dilation is unitary on the direct sum of the signal and storage ports. The reciprocal read operation is obtained by reversing the write Hamiltonian phases and interchanging input and output temporal boundary conditions; on the stored subspace this is the adjoint map W † . Hence the round-trip amplitude is W † W . In the alphabet basis this is exactly the Gram matrix computed in Eq. (111). The singular values of the positive matrix G are its eigenvalues. Therefore the worst-case output probability is min ∥ c ∥ =1 ∥ Gc ∥ 2 = λ min ( G ) 2 , and Haar averaging gives Tr G 2 /K . This result separates two e ciencies that are often con ated. The write e ciency is governed by G , whereas the unconditional writeread e ciency is governed by G 2 for reciprocal retrieval. Conditional state delity after a successful retrieval may be high even when the unconditional probability is reduced; the Gram certi cate reports the probability loss without hiding it in postselection. 13.3 Noise-channel certi cate A quantum memory certi cate is incomplete unless it speci es noise. In the linear weak-excitation regime the same controlled dilation xes the noise channel. Theorem 20 (Noise form of reciprocal MR-QM) . Let the hypotheses of Theorem 19 hold and let G be the nite-alphabet write Gram matrix. In the Heisenberg picture the ideal reciprocal readout channel on the alphabet annihilation vector ˆ a has the form ˆ a out = G ˆ a in + E ˆ v , (116) wh ere ˆ v is an environmental bosonic vector and EE † = I − G 2 . (117) Consequently, in the eigenbasis of G the channel is a product of pure-loss channels with transmis- sivities τ α = g 2 α , 0 ≤ g α ≤ 1 , (118) wh ere g α are the eigenvalues of G . If the environmental modes have normally ordered covariance ¯ N E = ⟨ ˆ v † ˆ v ⟩ , the added output photon-noise matrix is N add = E ¯ N E E † . (119) 25 For a K -mode single-photon qudit encoded in S , the unconditional success probability is at least λ min ( G ) 2 . The conditional overlap with the intended qudit after a successful readout obeys F cond ≥ λ min ( G ) λ max ( G ) 2 . (120) In particular, if the only imperfection is the prompt re ection bound G ≥ η min I and λ max ( G ) ≤ 1 , then F cond ≥ η 2 min . Proof. The full signalstorageenvironment evolution is unitary, so the output annihilation op- erators must preserve canonical commutators. The reciprocal amplitude on the alphabet is G by Theorem 19; therefore the remaining environmental term must satisfy Eq. (117). Diagonal- izing the positive contraction G = U diag( g α ) U † reduces Eq. (116) to independent beamsplitter channels with transmissivities g 2 α . Equation (119) is the normally ordered covariance of the envi- ronmental contribution. For an input qudit | ψ ⟩ = P α c α | α ⟩ , the unnormalized retrieved single- photon component is G | ψ ⟩ . Its probability is ⟨ ψ | G 2 | ψ ⟩ ≥ λ min ( G ) 2 . The conditional overlap with the intended qudit is |⟨ ψ | G | ψ ⟩| 2 ⟨ ψ | G 2 | ψ ⟩ ≥ λ min ( G ) 2 λ max ( G ) 2 , which gives Eq. (120). This theorem turns the usual e ciency statement into a quantum-channel statement. Vacuum uncontrolled ports give quantum-limited loss. Thermally occupied microwave or mechanical ports add the normally ordered noise in Eq. (119), which must be reported separately fr om coherent e ciency. Theorem 21 (Operational distance to an ideal nite-alphabet memory) . Consider the vacuum- noise reciprocal channel of Theorem 20 on a K -dimensional single-photon alphabet, and let the positive round-trip amplitude be G with eigenvalues 0 ≤ g α ≤ 1 . Let I be the ideal memory channel on the same alphabet, with the loss output treated as an orthogonal erasure ag. Then P wc succ ≥ g 2 min , (121) F e = | Tr G | 2 K 2 , (122) F = Tr( G 2 ) + | Tr G | 2 K ( K + 1) , (123) ∥E G − I∥ ⋄ ≤ 2 p 2(1 − g min ) , (124) wh ere g min = λ min ( G ) . If storage decoherence or readout imbalance is represented by a known contraction D on the stored alphabet, the same formulas hold with G replaced by the singular-value contraction A = D 1 / 2 G , except that F e and F use A and Eq. (124) uses s min ( A ) . Proof. A Stinespring isometry for the vacuum pure-loss alphabet channel is V | ψ ⟩ = G | ψ ⟩ out | 0 ⟩ e + ( I − G 2 ) 1 / 2 | ψ ⟩ e | er ⟩ out . The worst-case success probability is ∥ G | ψ ⟩∥ 2 ≥ g 2 min . The entanglement delity follows fr om the standard Kraus formula; only the no-erasure Kraus operator G overlaps the ideal output, giving Eq. (122). Haar averaging R |⟨ ψ | G | ψ ⟩| 2 d ψ gives Eq. (123). For every normalized | ψ ⟩ , ∥ ( V − V 0 ) | ψ ⟩∥ 2 = ∥ ( G − I ) | ψ ⟩∥ 2 + ∥ ( I − G 2 ) 1 / 2 | ψ ⟩∥ 2 ≤ (1 − g min ) 2 + 1 − g 2 min = 2(1 − g min ) , wh ere V 0 | ψ ⟩ = | ψ ⟩ out | 0 ⟩ e . The diamond distance between channels is at most twice the operator- norm distance between these Stinespring isometries, which proves Eq. (124). A known storage contraction is absorbed into the no-erasure amplitude A , followed by the same singular-value argument. 26 This theorem gives an operational end point for the certi cate: the same Gram matrix that reports write and round-trip probabilities also gives entanglement delity, average unconditional qudit delity and a worst-case channel-norm distance to an ideal memory. It is intentionally conservative, but it is independent of a particular quantum-network protocol. 13.4 Finite-mode capacity of the write map The same operator formulation gives a sharp nite-dimensional obstruction that is independent of the details of the resonator topology. Theorem 22 (Rank capacity bound) . Assume that after the write stage the controllable long- lived memory sector is an M -dimensional Hilbert space and that the write operation is passive and lossless except for the prompt output channel. Let W : H B → C M be the write map. If K orthonormal input modes f 1 , . . . , f K are written with e ciencies η j = ∥ W f j ∥ 2 , then K X j =1 η j ≤ M. (125) Consequently, exact unit-e ciency storage of a K -dimensional temporal alphabet requires K ≤ M , and storage with average e ciency at least 1 − ϵ requires K (1 − ϵ ) ≤ M. (126) Proof. Passivity gives 0 ≤ W † W ≤ I and rank( W † W ) ≤ M . Therefore W † W has at most M nonzero eigenvalues, each no larger than one. For P K = P K j =1 | f j ⟩⟨ f j | , K X j =1 η j = Tr( P K W † W P K ) ≤ Tr( W † W ) ≤ M. (127) The two corollaries follow immediately. This bound is the nite-mode form of the time-bandwidth limitation. It does not forbid high e ciency on a prescribed nite alphabet, but it forbids interpreting a nite passive memory as an exact isometry on an unlimited continuum of orthogonal temporal modes. Together with Theorem 12, the rank bound gives the operational universality hierarchy. A known single mode can be captured exactly by scalar time-dependent matching. A K -dimensional known alphabet can be exactly stored by the matrix dynamic construction of Theorem 13 only when the controlled storage dimension is at least K . A continuous band contains in nitely many orthogonal modes and therefore cannot be exactly represented by a nite passive memory; it can only be approximated, with the stationary approximation quality governed by the positive-real minimax and BodeFano certi cates. 14 Fixed-pole convex synthesis and lower certi cates The nonconvex part of nite positive-real synthesis is the placement of passive poles. Once a pole support is xed, however, the oscillator-strength subproblem has a convex structure. This observation turns a proposed xed architecture into a globally checkable certi cate rather than a heuristic t. Let h n ( ω ) = 1 i ω + Γ n + i ν n , Γ n > 0 , (128) or a symmetry-constrained real combination such as h n ( ω ) = 1 i ω + Γ n + i∆ n + 1 i ω + Γ n − i∆ n . (129) 27 For nonnegative weights w n ≥ 0 , de ne Σ w (i ω ) = N X n =1 w n h n ( ω ) . (130) Theorem 23 (Fixed-pole convexity) . Fix 0 ≤ ρ < 1 and a nite grid Ω L = { ω ℓ } L ℓ =1 . The set of nonnegative oscillator-strength vectors w ∈ R N + satisfying | r w (i ω ℓ ) | ≤ ρ, ℓ = 1 , . . . , L, (131) is convex. More explicitly, Eq. (131) is equivalent to q ℓ ( w ; ρ ) ≤ 0 , ℓ = 1 , . . . , L, (132) wh ere q ℓ ( w ; ρ ) = | A ℓ + h T ℓ w | 2 − ρ 2 | B ℓ + h T ℓ w | 2 , (133) A ℓ = i ω ℓ − κ 2 + γ c 2 , B ℓ = i ω ℓ + κ 2 + γ c 2 , (134) and h ℓ = ( h 1 ( ω ℓ ) , . . . , h N ( ω ℓ )) T . Each q ℓ has Hessian ∇ 2 q ℓ = 2(1 − ρ 2 ) Re( h ℓ h † ℓ ) ⪰ 0 . (135) Consequently, for xed poles the minimum attainable grid re ection norm can be found by bisection in ρ with convex feasibility tests, and any infeasibility dual certi cate is a rigorous lower bound for that pole support. Proof. The denominator of the passive Cayley transform has positive real part on the imaginary axis because Re[ κ/ 2 + γ c / 2 + Σ w (i ω )] > 0 , so multiplying | r w | ≤ ρ by the denominator is nonsingular and gives | A ℓ + h T ℓ w | 2 ≤ ρ 2 | B ℓ + h T ℓ w | 2 . (136) Expanding the two squared moduli in the real variables w n yields q ℓ . The quadratic part is (1 − ρ 2 ) | h T ℓ w | 2 = w T (1 − ρ 2 ) Re( h ℓ h † ℓ ) w, (137) which is positive semide nite for ρ < 1 . Hence each sublevel set q ℓ ( w ; ρ ) ≤ 0 is convex, and the intersection with the positive orthant is convex. Feasibility is monotone in ρ , giving the bisection statement. This theorem separates discovery fr om certi cation. Nonlinear search may be used to nd useful pole locations, but once a xed support is proposed, the best nonnegative weights and the lower bound for that support are globally certi able. For implementation, write Q ℓ = (1 − ρ 2 ) Re( h ℓ h † ℓ ) , (138) b ℓ = Re { ( A ℓ − ρ 2 B ℓ ) ∗ h ℓ } , (139) c ℓ = | A ℓ | 2 − ρ 2 | B ℓ | 2 , (140) so that q ℓ ( w ; ρ ) = w T Q ℓ w + 2 b T ℓ w + c ℓ . (141) A large grid can be handled by cutting planes: solve on a coarse grid, add the dense-grid frequency with largest violation, and iterate until the independent dense-grid violation is below tolerance. 28 choose pole support solve convex weight feasibility apply sampling margin memory-channel certi cate dual infeasibility lower bound Figure 5: Fixed-support certi cation work ow. Pole discovery is geometric and may be noncon- vex, but the oscillator-strength certi cate for a proposed support is a convex feasibility problem with checkable lower bounds. 15 Deterministic robustness certi cate Monte-Carlo fabrication studies are useful but not su cient for certi cation. The spectral- admittance form gives a deterministic perturbation bound. Theorem 24 (Self-energy perturbation bound) . On a compact band B , de ne D ( ω ) = i ω + κ + γ c 2 + Σ(i ω ) , d = inf ω ∈B | D ( ω ) | . (142) Let a perturbed device have self-energy Σ + δ Σ with ϵ = sup ω ∈B | δ Σ(i ω ) | < d. (143) Then sup ω ∈B | r Σ+ δ Σ (i ω ) − r Σ (i ω ) | ≤ κϵ d ( d − ϵ ) . (144) If ρ = sup ω ∈B | r Σ (i ω ) | , the perturbed worst-case write e ciency satis es η min w (Σ + δ Σ) ≥ 1 − ρ + κϵ d ( d − ϵ ) 2 . (145) Proof. For xed ω write r Σ = N + Σ D , r Σ+ δ Σ = N + Σ + δ Σ D + δ Σ , wh ere N = i ω − κ/ 2 + γ c / 2 and D = i ω + κ/ 2 + γ c / 2 + Σ . A direct subtraction gives r Σ+ δ Σ − r Σ = κ δ Σ D ( D + δ Σ) . (146) Since | D | ≥ d and | D + δ Σ | ≥ d − ϵ , Eq. (144) follows. The e ciency bound is the triangle inequality applied to sup | r | . The bound is conservative but deterministic. It converts a calibrated admissible self-energy er- ror into a guaranteed lower e ciency without assuming a probability distribution over fabrication errors. 29 16 Numerical certi cation of the nite-mode synthesis This section gives a reproducible numerical check of the preceding theory. The purpose is not to claim a globally optimal experimental design, but to verify the certi cation chain: positive-real passivity on the boundary and in the open right half-plane, Schur contractivity, causal Schur- to-admittance inversion, minimum-phase/all-pass ambiguity, complex-response uncertainty prop- agation with full out-of-alphabet prompt leakage, operational nite-alphabet channel distance, the exact spectral energy identity, the BodeFano area constraint, the superiority of nite-band minimax matching over local Taylor cancellation, controlled-dilation consistency, deterministic self-energy robustness, Monte-Carlo robustness under fabrication errors, data-to-band measure- ment margins, capped dynamic-control leakage, and thermal-noise conversion into added output photons. We work in normalized units κ = 2 , γ c = 0 , r (i ω ) = i ω − 1 + Σ(i ω ) i ω + 1 + Σ(i ω ) . (147) The self-energy is chosen in the passive symmetric rational form Σ( s ) = w 0 s + Γ 0 + J X j =1 w j 1 s + Γ j + i∆ j + 1 s + Γ j − i∆ j , Γ j , w j , ∆ j > 0 . (148) This is a nite positive-real Stieltjes approximation to the ideal boundary admittance Σ id (i ω ) = 1 − i ω . 16.1 Benchmark and synthesized designs The single-pole locally matched admittance is Σ 1 ( s ) = 1 / ( s + 1) , which satis es Σ 1 (0) = 1 and Σ ′ 1 (0) = − 1 . The fourth-order local benchmark is Γ 0 = 0 . 517250 , w 0 = 0 . 371644 , (149) Γ 1 = 0 . 241311 , ∆ 1 = 0 . 341305 , w 1 = 0 . 101906 , (150) Γ 2 = 0 . 079200 , ∆ 2 = 4 . 938562 , w 2 = 0 . 001444 . (151) It cancels several derivatives at the carrier but is not optimized at the band edge. The eleven-internal-mode positive-real design used for the highest-performance static certi - cate is Γ 0 = 0 . 381536667 , w 0 = 0 . 247495288 , (152) Γ 1 = 0 . 299963214 , ∆ 1 = 0 . 376081898 , w 1 = 0 . 149484487 , (153) Γ 2 = 0 . 222611544 , ∆ 2 = 0 . 667360031 , w 2 = 0 . 100207794 , (154) Γ 3 = 0 . 139096754 , ∆ 3 = 0 . 879428329 , w 3 = 0 . 055251743 , (155) Γ 4 = 0 . 051704895 , ∆ 4 = 0 . 993077310 , w 4 = 0 . 016034014 , (156) Γ 5 = 0 . 104592206 , ∆ 5 = 1 . 067080282 , w 5 = 0 . 000275262 . (157) It is a certi ed positive-real design; global optimality over all pole locations is not claimed. 16.2 Spectral, time-domain, BodeFano and robustness checks The same rational admittance is realized as the controlled passive dilation of Theorem 2, ˙ a ( t ) = − a ( t ) − i X j g j b j ( t ) − √ 2 f in ( t ) , (158) ˙ b j ( t ) = − (Γ j + i∆ j ) b j ( t ) − i g j a ( t ) , (159) f out ( t ) = f in ( t ) + √ 2 a ( t ) , (160) 30 − 1 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 normalized detuning ω | r (i ω ) | single-pole local fourth-order local eleven-mode Figure 6: Re ection defect for positive-real nite-mode self-energies. Local Taylor cancellation is not a nite-band certi cate; minimax synthesis controls the largest prompt re ection on the target interval. 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 0 . 94 0 . 96 0 . 98 1 half-bandwidth B worst-case write e ciency eleven-mode design BodeFano ceiling Figure 7: Certi ed worst-case write e ciency η min w = 1 − max | ω |≤ B | r (i ω ) | 2 versus normalized half-bandwidth, compared with the one-port BodeFano ceiling. wh ere g j = √ w j and the symmetric detuned modes in Eq. (148) are included separately. For a normalized Gaussian spectrum with σ = 0 . 35 , | F σ ( ω ) | 2 = exp[ − ω 2 / (2 σ 2 )] √ 2 πσ , (161) the spectral prediction is η spec ( σ ) = 1 − Z ∞ −∞ | r (i ω ) | 2 | F σ ( ω ) | 2 d ω. (162) The controlled-dilation storage probability is η store = X j Z ∞ −∞ 2Γ j | b j ( t ) | 2 d t. (163) The spectral and time-domain calculations agree to the displayed precision. For reciprocal readout, Theorem 19 turns the write certi cate into a round-trip certi cate. If only the band-uniform bound η min w is used, the conservative worst-case reciprocal round-trip probability and the generic conditional qudit- delity lower bound are both ( η min w ) 2 . The BodeFano area A = Z ∞ −∞ log | r (i ω ) | − 1 d ω (164) 31 − 1 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 5 . 5 6 6 . 5 7 · 10 − 2 normalized detuning ω | r (i ω ) | dense grid ρ Figure 8: Dense-grid certi cation of the eleven-internal-mode design. The plotted envelope shows the nite-band minimax character and the certi ed re ection norm ρ = 0 . 069464144 . Table 1: Numerical certi cation. The columns η spec G and η ODE G compare spectral quadrature with direct time-domain controlled-dilation storage for a Gaussian signal with σ = 0 . 35 . design max | ω |≤ 1 | r | η min w min Re Σ min | D | η spec G η ODE G single-pole local 0 . 447214 0 . 800000 9 . 90 e − 03 1 . 581139 0 . 989878 0 . 989878 fourth-order local 0 . 707894 0 . 498886 2 . 42 e − 03 1 . 307557 0 . 972821 0 . 972821 three-mode minimax 0 . 109370 0 . 988038 7 . 34 e − 03 1 . 874755 0 . 990092 0 . 990092 ve-mode minimax 0 . 087250 0 . 992387 4 . 39 e − 03 1 . 852348 0 . 993161 0 . 993161 seven-mode minimax 0 . 075746 0 . 994263 3 . 17 e − 03 1 . 859257 0 . 994436 0 . 994436 eleven-mode certi ed 0 . 069464 0 . 995175 2 . 47 e − 03 1 . 882222 0 . 995167 0 . 995167 is numerically saturated at A = 2 π for the minimum-phase synthesized designs. The fourth-order local design leaves area unused because of a right-half-plane re ection zero. This con rms that nite-band performance is mainly a problem of redistributing a xed logarithmic matching area over frequency. For the eleven-mode design, the denominator margin is d = min | ω |≤ 1 | D ( ω ) | = 1 . 882222 . The deterministic perturbation certi cate of Theorem 24 then gives the following conservative guarantees for a bounded self-energy error ϵ = ∥ δ Σ ∥ L ∞ ( B ) . To quantify fabrication sensitivity beyond deterministic norm balls, all positive parameters in Eq. (157) were perturbed independently by log-normal factors exp( σ dis Z ) with Z ∼ N (0 , 1) , and η min w was recomputed on a dense grid for 1000 samples. The xed-pole convexity theorem adds a certi cation layer: for any proposed pole support, the weight-optimization subproblem is a globally checkable convex feasibility problem. The designs below were obtained by nonlinear pole exploration followed by independent dense-grid checks; a xed-architecture experimental claim should also report the convex bisection result and its dual gap. The numerical conclusion is sharper than the earlier local theory. A small passive array can be extremely good on a prescribed band, but only when optimized by the same norm that appears in the exact memory theorem. Derivative cancellation is a narrowband asymptotic tool; positive-real nite-band synthesis is the relevant certi cate. The BodeFano theorem adds a global limitation: even perfect synthesis cannot exceed the one-port logarithmic area budget, and deterministic robustness must be quoted as a margin on Σ rather than only as a best- t simulation. The dynamic-capture gures and capped-control table are not included as competing broad- band stationary designs. They are resource witnesses. They show that exact absorption is possible once the input temporal subspace is known, the storage dimension is su cient, and the coupling is allowed to vary in time, while a stationary universal interface must solve a harder approximation problem constrained by a xed BodeFano area. 32 Table 2: Derivative-aware data-to-band certi cate for the eleven-mode design on [ − 1 , 1] . Here h is the ll distance of a uniform sampling grid, L r = sup | d r (i ω ) / d ω | = 3 . 338616 , and the certi ed e ciency is 1 − ( ρ grid + L r h ) 2 . The continuum value is 0 . 995175 . grid points h ρ grid ρ grid + L r h certi ed η min w 401 2 . 500 e − 03 0 . 069464 0 . 077811 0 . 993945 1001 1 . 000 e − 03 0 . 069464 0 . 072803 0 . 994700 2001 5 . 000 e − 04 0 . 069464 0 . 071133 0 . 994940 4001 2 . 500 e − 04 0 . 069464 0 . 070299 0 . 995058 1 4 3 5 7 11 0 . 2 0 . 4 0 . 6 0 . 8 1 internal-mode design label certi ed probability write reciprocal round trip Figure 9: Write and reciprocal writeread probability certi cates obtained fr om the same re ec- tion norm. The labels denote the single-pole local benchmark, fourth-order local benchmark, and the three-, ve-, seven-, and eleven-mode nite-band minimax designs. 16.3 Reproducibility All numerical values in the certi cation tables are obtained fr om the same rational self-energy used in the analytic model. The independent checks are: dense-grid evaluation of max | r | and veri cation of min Re Σ ; spectral and time-domain controlled-dilation storage; the derivative- aware (Lipschitz) data-to-band certi cate; the write / reciprocal round-trip / conditional- delity channel table; BodeFano log-area integration; deterministic and Monte-Carlo log-normal ro- bustness; causal Schur-to-admittance inversion and all-pass diagnostics; nite-alphabet complex- response uncertainty propagation; the operational Legendre channel-distance certi cate; extended passivity, Schur and thermal-noise margins; data-to-band margins under amplitude-calibration uncertainty; capped dynamic-control leakage for a chirped Gaussian; a time-domain writestore read reciprocal-readout simulation; the de la Vallée-Poussin two-sided optimality bracket; and the physical-units translation of the certi ed recipe. The accompanying script numerical_certification.py writes numerical_certification_results.json and numerical_certification_results.txt , printing a pass/fail comparison against the value quoted in the manuscript for every headline number; it reproduces all checks without using tted data external to the paper. 17 Experimental translation The theory suggests a compact experimental reporting standard. A memory-grade multiresonator interface should report: 1. the measured or tted complex scalar self-energy Σ( s ) and re ection function r (i ω ) , includ- ing phase and pole locations, or, for vector interfaces, the full complex prompt scattering matrix R (i ω ) on the declared signal channels; 2. the physical storage dilation: which linewidths correspond to reversible storage ports, which 33 Table 3: Channel-level certi cate derived from the re ection norm. The conditional delity bound is conservative because it uses only λ min ( G ) and λ max ( G ) ≤ 1 . design η min w η min rt F cond lower bound single-pole local 0 . 800000 0 . 640000 0 . 640000 fourth-order local 0 . 498886 0 . 248887 0 . 248887 three-mode minimax 0 . 988038 0 . 976219 0 . 976219 ve-mode minimax 0 . 992387 0 . 984833 0 . 984833 seven-mode minimax 0 . 994263 0 . 988558 0 . 988558 eleven-mode certi ed 0 . 995175 0 . 990373 0 . 990373 1 4 3 5 7 11 0 . 96 0 . 98 1 internal-mode design label A/ (2 π ) Figure 10: BodeFano area utilization A/ (2 π ) for the normalized designs. correspond to uncontrolled loss, and which port-to-register trapping map implements The- orem 3; 3. the signal space: a continuous band B , a pulse family, or a nite alphabet S ; 4. the minimax re ection certi cate max ω ∈B | r (i ω ) | , or ess sup ω ∈B σ max R (i ω ) for vector in- terfaces, together with the nite-sampling derivative margin of Theorem 7, the causal- realizability check of Theorem 8, and the amplitude and phase calibration uncertainties when the certi cate is inferred from measured data; 5. the BodeFano area utilization A/ ( πκ ) , the xed-pole convex certi cate when a xed ar- chitecture is claimed, the complex-response uncertainty margin of Theorem 9, and the deterministic perturbation margin d with an admissible ∥ δ Σ ∥ ∞ ; 6. for nite alphabets, the write Gram matrix G , its smallest eigenvalue, the corresponding reciprocal round-trip certi cate G 2 , and the operational distance or delity certi cate of Theorem 21; 7. the readout operation and whether it implements the reciprocal adjoint map assumed in Theorem 19; 8. the measured output noise or an upper bound on the thermal occupation of uncontrolled ports entering Eq. (119); 9. for dynamically controlled memories, the full scalar or vector control waveform, its cal- ibration error, the storage dimension, the signal subspace for which it is valid, and the capped-control leakage integral of Proposition 3. This standard is compatible with broadband MR-QM-interface measurements, integrated resonator- array proposals, atomic-ensemble multiresonator memories, and pre-created macroscopic-coherence 34 1 4 3 5 7 11 0 0 . 2 0 . 4 internal-mode design label ceiling gap at B = 1 Figure 11: Gap between each design's certi ed worst-case write e ciency at B = 1 and the BodeFano ceiling 1 − e − 2 π . Finite-band minimax designs are close to the passive area lim it, while local Taylor matching can fail at the band edge. Table 4: BodeFano area utilization and remaining e ciency gap at B = 1 . The theoretical ceiling is A/ (2 π ) = 1 and η min w ≤ 1 − e − 2 π = 0 . 998133 . design A A/ (2 π ) ceiling gap single-pole local 6 . 283185 1 . 000000 0 . 198133 fourth-order local 6 . 207203 0 . 987907 0 . 499247 three-mode minimax 6 . 283185 1 . 000000 0 . 010094 ve-mode minimax 6 . 283185 1 . 000000 0 . 005745 seven-mode minimax 6 . 283185 1 . 000000 0 . 003870 eleven-mode certi ed 6 . 283185 1 . 000000 0 . 002958 memories. It prevents high absorption from being confused with high quantum-memory e ciency when the absorbed excitation is not stored in a reversible degree of freedom. Theorem 25 (Memory-grade reporting su ciency) . For a linear weak-excitation MR-QM on a declared signal space S , the following data are su cient to give a architecture-independent lower certi cate for the declared linear quantum memory channel: the prompt scattering operator R on S with a continuum or sampling margin, the map identifying controlled storage ports, the reciprocal readout error on the stored subspace, and the normally ordered covariance of uncontrolled noise ports. In the lossless reciprocal case the write, round-trip, and conditional single-photon qudit certi cates are η min w = λ min ( G ) , η min rt = λ min ( G ) 2 , F cond ≥ [ λ min ( G ) /λ max ( G )] 2 , wh ere G = I − R † R on S . Treating failed retrieval as an orthogonal erasure ag, the same data also give the unconditional entanglement- delity and diamond-distance certi cates F e = | Tr G | 2 /K 2 , F = Tr( G 2 ) + | Tr G | 2 K ( K + 1) , ∥E G − I∥ ⋄ ≤ 2 p 2(1 − λ min ( G )) . If only the band-uniform scalar bound ∥ r ∥ L ∞ ( B ) ≤ ρ is known, these reduce to η min w ≥ 1 − ρ 2 , η min rt ≥ (1 − ρ 2 ) 2 , F cond ≥ (1 − ρ 2 ) 2 , ∥E G − I∥ ⋄ ≤ min { 2 , 2 √ 2 ρ } . Proof. The statement is Theorems 5, 6, 7, 3, 19, and 20 restricted to the declared signal space. The scalar reduction follows fr om R † R ≤ ρ 2 I , hence G ≥ (1 − ρ 2 ) I and G ≤ I . The theory leads to a direct engineering work ow. 35 Table 5: Deterministic robustness bound for the eleven-mode certi ed design. ϵ ∆ r bound certi ed η min w 1 . 0 e − 04 5 . 646 e − 05 0 . 995167 1 . 0 e − 03 5 . 648 e − 04 0 . 995096 5 . 0 e − 03 2 . 830 e − 03 0 . 994774 1 . 0 e − 02 5 . 675 e − 03 0 . 994354 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 0 . 9 0 . 92 0 . 94 0 . 96 0 . 98 1 independent parameter disorder (%) worst-case write e ciency 5 th percentile median 95 th percentile Figure 12: Monte-Carlo robustness certi cate for the eleven-internal-mode design under indepen- dent log-normal perturbations of linewidths, detunings and oscillator strengths. Step 1: choose the signal space. Specify whether the memory must be approximately universal on a continuous band B , exactly preserve a nite alphabet of modes, or optimize a known wavepacket distribution. These are mathematically dierent problems. Step 2: choose the allowed positive-real class. For empty auxiliary resonators use Eq. (102) with xed fabrication linewidths. For atomic ensembles use Eq. (15) or the Lorentzian reduction (17). For coupled resonator molecules use a full matrix self-energy Σ( s ) = g † sI + Γ b 2 + iΩ b + Φ( s ) − 1 g . (165) Step 3: solve the spectral synthesis problem. Minimize Eq. (82), or impose the deriva- tive hierarchy (99)(101) when the signal bandwidth is narrow. The optimization variables are experimentally meaningful: detunings ∆ m , couplings g m , eective linewidths Γ m , atomic optical depths N m | f m | 2 , and external coupling κ . Step 4: enforce switchability and storage isolation. High write e ciency is not enough. After absorption, the common resonator or waveguide coupling should be switched o, or the excitation should be transferred to dark spin modes. This prevents the matched interface from becoming a loss channel during storage. Step 5: implement rephasing readout. Use AFC periodicity, CRIB/GEM detuning rever- sal, controlled auxiliary resonator frequency inversion, or a programmed inverse coupling sequence to implement the time-reversal theorem. Step 6: budget errors by norm. For any normalized input in H B , 1 − η total ≤ ϵ 2 write + ϵ 2 read + p decoh + p control + O ( ϵ 4 ) , (166) 36 Table 6: Robustness of the eleven-internal-mode certi ed design. Entries are percentiles of η min w over 1000 disorder samples. σ dis (%) 1% 5% 50% 95% 99% 0 . 5 0 . 989161 0 . 991602 0 . 994101 0 . 994904 0 . 995049 1 . 0 0 . 981817 0 . 986558 0 . 992903 0 . 994508 0 . 994791 2 . 0 0 . 963380 0 . 974040 0 . 989589 0 . 993570 0 . 994153 5 . 0 0 . 911003 0 . 936970 0 . 976034 0 . 990135 0 . 992069 Table 7: Causal-realizability and phase-ambiguity diagnostics for the eleven-mode design. The inverse-admittance reconstruction uses Eq. (47). The all-pass row multiplies the re ection coef- cient by B 0 . 5 ( s ) , which leaves | r | unchanged but changes the phase and group delay. diagnostic quantity value inverse consistency max | ω |≤ 1 | Σ r − Σ | 4 . 48 e − 16 passive inverse min | ω |≤ 1 Re Σ r (i ω ) 8 . 814 e − 01 all-pass ambiguity max || B 0 . 5 r | − | r || 2 . 78 e − 17 all-pass phase eect max | ω |≤ 1 τ 0 . 5 ( ω ) 4 . 000 all-pass phase eect min | ω |≤ 1 τ 0 . 5 ( ω ) 0 . 8003 wh ere ϵ write = ∥ r write ∥ L ∞ ( B ) and similarly for readout. This converts spectral plots into rigorous memory error budgets. 18 Optimal minimax synthesis, certi ed convergence, and 2026 state of the art The preceding sections establish that a passive multiresonator memory is an admittance synthe- sizer whose worst-case write e ciency is 1 − ess sup | ω |≤ 1 | r | 2 , that the achievable re ection on a nite band is oored by the BodeFano area law ρ ≥ e − πκ/ 2 B , and that for a xed pole support the oscillator-strength subproblem is convex. This section closes the loop left open in the preced- ing treatment, wh ere global optimality over all pole locations is not claimed: we (i) compute optimal passive minimax designs and exhibit a quantitatively improved eleven-mode design, (ii) certify them globally by equioscillation, the convex weight-optimum, and an independent global search, (iii) prove the optimum is minimum-phase and area-saturating and reduce the gap to the oor to a single tail log-area, (iv) sharpen the channel certi cate and prove a vector-port singular- value capacity oor, (v) position the certi ed numbers against the 20232026 experimental and theoretical state of the art, and (vi) give a buildable physical recipe and a fabrication-robust synthesis that trades little nominal e ciency for a much tighter worst-case tolerance. 18.1 State of the art and the role of a worst-case certi cate Table 13 collects the gures a 2026 memory paper must engage. The best demonstrated cavity- based e ciencies are now 80 . 3% for weak coherent pulses and 69 . 8% for telecom heralded single photons in an integrated rare-earth microcavity, with twenty temporal modes at 51 . 3% average e ciency [18]; the best microwave multiresonator storage is at the 60 73% level [30], with recent dynamically coupled devices emphasizing pulse-shape preservation [14, 12]. Against these, the multiresonator program predicts theoretical supere ciencies above 99 . 9% [11]. The contribution of the present framework is orthogonal to chasing a single high number: it certi es a worst-case guarantee over an entire continuous band and an entire signal alphabet, of the kind a quantum interconnect must assume. The certi ed worst-case write e ciency 0 . 995890 reported below is a uniform band guarantee, not a center-frequency or average value. 37 Table 8: Finite-alphabet complex-response uncertainty certi cate for the eleven-mode design. The alphabet is the orthonormal pair { 1 , ω } on [ − 1 , 1] . The Gram matrix is computed fr om the full prompt-output norm R | b r | 2 f ∗ i f j , not fr om the compressed same-alphabet leakage amplitude. A pointwise complex S-parameter error | δr | ≤ ϵ implies λ min ( G true ) ≥ λ min ( b G ) − ϵ (2 ∥ b R ∥ + ϵ ) . ϵ certi ed λ min ( G true ) observed minimum max observed | δr | 1 . 0 e − 04 0 . 995915 0 . 995927 9 . 999 e − 05 1 . 0 e − 03 0 . 995799 0 . 995925 1 . 000 e − 03 2 . 5 e − 03 0 . 995602 0 . 995919 2 . 500 e − 03 5 . 0 e − 03 0 . 995264 0 . 995903 4 . 999 e − 03 1 . 0 e − 02 0 . 994551 0 . 995866 9 . 999 e − 03 Table 9: Operational nite-alphabet channel-distance certi cate for the eleven-mode design. The Legendre alphabets are orthonormal on [ − 1 , 1] . The values use the full prompt-output Gram matrix, reciprocal vacuum readout, and Theorem 21. K g min P wc succ F e F diamond upper bound 2 0 . 995927 0 . 991871 0 . 992031 0 . 992031 0 . 180503 4 0 . 995826 0 . 991669 0 . 992180 0 . 992180 0 . 182737 8 0 . 995805 0 . 991628 0 . 992289 0 . 992289 0 . 183191 18.2 Optimal passive minimax designs We solve, over the passive symmetric positive-real class with N = 1 + 2 M internal modes, ρ N = min Σ N passive max | ω |≤ 1 | r (i ω ) | , (167) by a soft-maximum (log-sum-exp) homotopy with multistart and warm starts across N , polishing the true maximum and certifying it on a dense grid with the Lipschitz margin of Sec. 6. Two structural facts anchor the result. First, the single-pole optimum is analytic: the minimax single resonator gives ρ 1 = ( √ 2 − 1) 2 = 3 − 2 √ 2 ≈ 0 . 171573 , η 1 = 1 − ρ 2 1 ≈ 0 . 970563 , (168) which is far better than the locally at (center-matched) single pole with ρ = 0 . 447 ; this is the cleanest illustration that minimax matching is not local impedance matching. Second, as N grows the equiripple optimum descends monotonically toward the BodeFano oor e − π = 0 . 043214 . Table 14 and Figs. 1314 report the certi ed series. The eleven-mode design is improved from the benchmark value ρ = 0 . 069464 to ρ 11 = 0 . 064112 , η 11 = 0 . 995890 , round trip ≥ 0 . 991796 , (169) and the gap to the oor closes root-exponentially in the mode count. Over the equiripple-certi ed range the data are well described by ρ N − e − π ≈ C e − c √ N (3 ≤ N ≤ 19) , (170) with c ≈ 0 . 64 and coe cient of determination R 2 = 0 . 99 , a markedly better t than a geometric law q N ( R 2 = 0 . 95 ); the out-of-band area itself obeys A tail ( N ) ≈ 3 . 5 e − 0 . 50 √ N ( R 2 = 0 . 996 , Fig. 15). The √ N rate is exactly the GoncharRakhmanov Stahl rate of best rational approxi- mation in the presence of a boundary singularity [36, 35]here the unit-step transition of | r | at the band edge ω = ± 1 , wh ere the band and its complement meet with no spectral gap; this is the direct analogue of the classical e − π √ n rate for best rational approximation of | x | on a symmetric interval [34, 35]. Each optimized design is certi ed by its equioscillation count : | r (i ω ) | attains 38 Table 10: Extended passivity, measurement, and noise margins for the eleven-mode design. The passivity check is evaluated on [ − 20 , 20] ; the measurement rows use the 1001-point grid certi cate and add an absolute amplitude uncertainty u to the certi ed re ection norm. certi cate quantity value passivity min | ω |≤ 20 Re Σ(i ω ) 6 . 152 e − 04 Schur contractivity max | ω |≤ 20 max( | r | − 1 , 0) 0 . 000 e + 00 denominator margin min | ω |≤ 20 | D (i ω ) | 1 . 313346 thermal example n env = 0 . 1 ⇒ N add ≤ 9 . 627 e − 04 photons Table 11: Data-to-band margins under nite amplitude calibration uncertainty u . The baseline is the derivative-aware 1001-point grid certi cate for the eleven-mode design. u certi ed ρ certi ed η min w 0 . 0000 0 . 072803 0 . 994700 0 . 0010 0 . 073803 0 . 994553 0 . 0025 0 . 075303 0 . 994329 0 . 0050 0 . 077803 0 . 993947 0 . 0100 0 . 082803 0 . 993144 its maximum ρ N at exactly N + 2 points of the band (Table 14, column alt.), the Chebyshev alternation signature of a minimax optimum; by the de la Vallée-Poussin theorem an N + 2 -fold equioscillation both certi es the design as the best in its rational class and bounds the optimum fr om below by the smallest attained ripple [37, 38], here within 10 − 5 of ρ N for every certi ed design (the two-sided bracket is computed explicitly by the accompanying script). This is indepen- dently con rmed by the convex weight certi cate of Sec. 18.3. Together with the universal oor (Theorem 11) wh ere the area law gives 2 ln(1 /ρ ) ≤ R B ln(1 / | r | ) d ω ≤ R R ln(1 / | r | ) d ω ≤ πκ , so that any passive memory with max B | r | ≤ ρ obeys ρ ≥ e − π the certi ed series brack- ets the order- N optimum between a rigorous lower bound and an equioscillation-certi ed value, and exhibits e − π = 0 . 043214 as the exact in mum, proven as a lower bound and approached root-exponentially. 18.3 Support-optimality certi cate The search in (167) is nonconvex in the pole locations but convex in the oscillator strengths for a xed support (Sec. Fixed-pole convex synthesis and lower certi cates). We exploit this to certify each synthesized design: freezing the discovered poles, we solve the convex second-order- cone program ρ ⋆ supp = min w ≥ 0 max ω | r (i ω ; w ) | , | A ω + h ⊤ ω w | 2 ≤ ρ 2 | B ω + h ⊤ ω w | 2 , (171) with A ω = i ω − 1 , B ω = i ω + 1 , and h ω the vector of basis admittances; each constraint is convex because its Hessian (1 − ρ 2 ) Re( h ω h † ω ) ⪰ 0 . A cutting-plane re nement (adding the densest continuum violators) closes the grid-to-continuum gap. For the discovered supports the convex weight-optimum reproduces the synthesized value, N = 5 : ρ ⋆ supp = 0 . 087270 ( synth. 0 . 087250) , N = 11 : ρ ⋆ supp = 0 . 064132 ( synth. 0 . 064112) , (172) agreeing to four digits. This certi es that the oscillator strengths are optimal for the synthesized pole supports: any residual gap to the oor is a pole-placement gap, not a weight gap. Combined with the universal lower bound ρ N ≥ e − π , each design is thus bracketed between a rigorous oor and a support-certi ed optimum. 39 Table 12: Capped dynamic-control simulation for a chirped Gaussian temporal mode. The ideal dynamic law is replaced by κ c ( t ) = min[ κ ideal ( t ) , κ cap ] and the actual inputoutput ODE is integrated. κ cap output leak nal stored energy energy-balance error 3 . 0 4 . 578 e − 04 0 . 999542 1 . 25 e − 07 4 . 0 2 . 709 e − 05 0 . 999973 1 . 26 e − 07 5 . 0 1 . 152 e − 06 0 . 999999 1 . 26 e − 07 6 . 0 3 . 369 e − 08 1 . 000000 1 . 26 e − 07 8 . 0 8 . 437 e − 12 1 . 000000 1 . 26 e − 07 12 . 0 5 . 689 e − 15 1 . 000000 1 . 26 e − 07 Table 13: Representative quantum-memory performance, 20102026. Demonstrated values are experimental; predicted values are theoretical ceilings. The present work reports a certi ed worst- case band guarantee. Platform / work Type E ciency Modes Note Integrated RE microcavity [18] demonstrated 80 . 3% / 69 . 8% 20 weak coh./single phot. Microwave multiresonator [30] demonstrated 60 73% 2 single-/high-photon Microwave RF-SQUID [14] demonstrated 57 . 5% d. 1 pulse preservation Impedance-matched cavity [5] predicted → 100% 1 matching condition Spectral-topological MR-QM [11] predicted > 99 . 9% few optimized design This work (certi ed) certi ed 0 . 9959 w.c. 11 uniform on | ω | ≤ 1 18.4 Structure and global optimality of the optimum The certi ed designs share a sharp structural signature that both explains their optimality and quanti es their approach to the BodeFano oor. Proposition 4 (Minimum phase and area saturation) . The minimax-optimal passive re ection is minimum phase: all zeros of r ( s ) lie in the closed left half-plane, so r has no right half-plane zeros and the BodeFano log-area is saturated, R R ln(1 / | r (i ω ) | ) d ω = πκ . Proof sketch. By the area identity, R R ln(1 / | r | ) d ω = π κ − 2 P j Re z j over the right half-plane zeros z j . A right half-plane zero can be re ected to its mirror image by an inner (all-pass) Blaschke factor B with | B (i ω ) | = 1 : the magnitude | r | on the axis, hence max B | r | , is unchanged, while the log-area strictly increases (the term − 2 Re z j < 0 is removed). A larger log-area budget concentrated on the band can only lower the equiripple level; therefore an optimal design carries no right half-plane zero and saturates the area law. Passivity of the re ected realization follows because the spectral-factor (minimum-phase) completion of a positive-real self-energy is positive- real. This is con rmed to numerical precision for every certi ed design: all re ection zeros lie in the open left half-plane, all poles are stable, and the computed log-area equals 2 π (with κ = 2 ) to ve digits. The proposition turns the gap to the oor into a single transparent quantity. Splitting the saturated area into its band and out-of-band parts, Z B ln 1 | r | d ω | {z } A band ( N ) + Z | ω | > 1 ln 1 | r | d ω | {z } A tail ( N ) = πκ = 2 π, (173) and using A band ≥ 2 ln(1 /ρ N ) gives the exact statement ρ N ≥ exp − 1 2 A band ( N ) = exp − π + 1 2 A tail ( N ) , (174) 40 Table 14: Certi ed optimal minimax designs. ρ N is the dense-grid worst-case re ection; η N = 1 − ρ 2 N ; the gap is to the BodeFano oor e − π = 0 . 043214 . Every design equioscillates at exactly N +2 band points (column alt.), the Chebyshev signature of a minimax optimum, and is passive to machine precision with a saturated BodeFano log-area ( = 2 π ). The eleven-mode row improves the reference design ( ρ = 0 . 069464 , η = 0 . 995175 ). N ρ N η N ρ N − e − π alt. 1 0.171573 0.970563 0.128359 3 3 0.109370 0.988038 0.066157 5 5 0.087250 0.992387 0.044036 7 7 0.075746 0.994263 0.032532 9 9 0.068757 0.995273 0.025543 11 11 0.064112 0.995890 0.020898 13 13 0.060833 0.996299 0.017619 15 15 0.058376 0.996592 0.015162 17 17 0.056527 0.996805 0.013313 19 19 0.056028 0.996861 0.012814 21 2.5 5.0 7.5 10.0 12.5 15.0 17.5 internal modes N 0.04 0.06 0.08 0.10 0.12 0.14 0.16 max | | 1 | r | Optimal passive minimax synthesis equiripple-certified Bode--Fano floor e original 11-mode Figure 13: Optimal passive minimax re ection ρ N versus internal-mode count N , descending toward the BodeFano oor e − π (dashed). The reference eleven-mode design (open square) is improved by the optimized design. so the gap to the oor e − π is governed entirely by the out-of-band tail log-area A tail ( N ) the unavoidable cost of a nite-degree transition fr om | r | ≤ ρ N at the band edge to | r | → 1 at in nity. Numerically A tail / 2 π falls monotonically from 0 . 347 at N = 1 to 0 . 069 at N = 19 (Fig. 16), while A band / 2 ln(1 /ρ N ) → 1 : the band re ection becomes uniform and the design tends to the ideal brick wall as the tail vanishes, driving ρ N → e − π . Equation (174) is the mechanism behind the root-exponential convergence (170): an out-of-band area that decays like e − c √ N produces a gap to the oor that decays at the same root-exponential rate. This numerical law and its structural mechanism motivate a precise conjecture, which would close the one remaining analytic gap a rigorous N -dependent lower bound to complement the equioscillation and convex certi cates. Conjecture 1 (Root-exponential approach to the BodeFano oor) . The order- N passive mini- max optimum obeys ρ N − e − π = exp − ( c + o (1)) √ N as N → ∞ , for a constant c > 0 ; equivalently the saturated out-of-band log-area satis es A tail ( N ) = exp( − ( c ′ + o (1)) √ N ) . The rate is that of best rational approximation near a boundary singularity (GoncharRakhmanovStahl), the singu- larity being the unit-step transition of | r | at ω = ± 1 . 41 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 | r (i )| Equiripple reflection (optimized) N = 3 N = 7 N = 11 N = 15 e Figure 14: Equiripple re ection magnitude | r (i ω ) | of the optimized designs on the signal band. Increasing N lowers and equalizes the ripple toward the oor (dashed). 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N 4.5 4.0 3.5 3.0 2.5 2.0 ln( N e ) Root-exponential convergence to the floor certified designs root-exp fit R 2 = 0.987 Figure 15: Root-exponential convergence: ln( ρ N − e − π ) is essentially linear in √ N ( t R 2 = 0 . 99 ), the GoncharRakhmanovStahl signature of a boundary singularity, rather than linear in N (geometric, R 2 = 0 . 95 ). A proof would follow from a GoncharRakhmanovStahl analysis of the positive-real-constrained Chebyshev problem on the band [36, 35]: the minimum-phase and area-saturation structure of Proposition 4 reduces the problem to the boundary-value (magnitude) data, for which the √ N rate is the natural target. Combined with the rigorous identity (174), a lower bound A tail ( N ) ≥ e − c ′ √ N would upgrade the conjecture to a rigorous closed-form lower bound ρ N ≥ exp( − π + 1 2 e − c ′ √ N ) , matching the equioscillation certi cates analytically. This is the route we propose for the open problem. Finally, global optimality is corroborated by three independent certi cates that agree: the equioscillation count (Sec. 18.2), the convex weight-optimum (Sec. 18.3), and a from-scratch global search. A Sobol-initialized dierential-evolution optimizer with wide bounds, run independently of the homotopy synthesis, fails to beat the certi ed designs (e.g. at N = 5 it returns 0 . 087359 versus the certi ed 0 . 087250 ; at N = 7 , 0 . 082701 versus 0 . 075746 ). Together with Proposition 4 and the universal oor, the order- N optimum is thus pinned between a proven lower bound and three concordant optimality certi cates. The scalar certi cate generalizes to a vector port by replacing | r | with the largest singular 42 2.5 5.0 7.5 10.0 12.5 15.0 17.5 internal modes N 0.2 0.4 0.6 0.8 1.0 Bode--Fano log-area / 2 Area-saturation and the gap to the floor band A band /2 tail A tail /2 Figure 16: Saturated BodeFano log-area split between the band A band and the out-of-band tail A tail (each over 2 π ). The tail area the nite-degree transition cost and, by (174), the entire gap to the oor vanishes as N grows. value of the prompt scattering matrix R (i ω ) , the matrix Cayley transform of a matrix positive-real self-energy. We make the bound precise. Proposition 5 (Vector-port singular-value capacity oor) . Let a passive ( M + N ) -port couple M input channels to N internal storage modes, with prompt re ection matrix R (i ω ) the matrix Cayley transform of a matrix positive-real self-energy. Then R is contractive, I − R † R ⪰ 0 , the worst-input write e ciency is 1 − ess sup ω σ max ( R ) 2 , and when M > N , ess sup ω σ max R (i ω ) 2 ≥ 1 − N M . (175) Proof sketch. Contractivity is the matrix dilation of the scalar Schur property. The stored power in input direction u is ∥ u ∥ 2 − ∥ Ru ∥ 2 , maximized in the worst direction by σ max ( R ) , giving the e ciency. The oor is the memory specialization of the multiport power-loss-ratio bound of Nie and Hochwald [32]: the matrix I − R † R has rank at most N (only N storage modes can absorb), so at least M − N of its eigenvalues vanish; the corresponding input directions are unstored, forcing σ max ( R ) = 1 there and 1 M tr( I − R † R ) ≤ N/M . Specializing the multiport broadband-matching log-integral of Nie and Hochwald [32] to the memory self-energy class also yields a singular-value area law for P n ln(1 /σ n ( R )) . We verify the construction on a matched two-port built from the optimized scalar self-energy on the diagonal with a shared rank-one coupling mode (Fig. 17): the prompt matrix is contractive, I − R † R ⪰ 0 on the band (minimum eigenvalue ≥ 0 to numerical precision), the worst-case singular value is ess sup ω σ max ( R ) = 0 . 1139 (so η vec = 1 − σ 2 max = 0 . 9870 ), and crucially σ max ( R ) exceeds the diagonal magnitude | r 11 | by up to 0 . 025 : a single scalar re ection trace is insu cient to certify a coupled multiport memory, and the singular value is the correct gure of merit. The capacity oor is veri ed directly on an over-driven port ( M = 2 inputs, N = 1 storage mode, rank-one self-energy): ess sup ω σ max ( R ) = 1 , respecting p 1 − N/M = 0 . 7071 , i.e. the input direction orthogonal to the single storage mode is necessarily unstored. 18.5 Two-sided channel certi cate For a nite alphabet the write Gram matrix G = I − R † R on a K -mode Legendre basis yields a linear channel certi cate. For the optimized eleven-mode design we obtain, for K = 43 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 singular value / magnitude Vector-port: singular-value envelope max ( R ) | r 11 | Figure 17: Vector-port memory: the worst-case singular value σ max ( R ) (solid) controls the worst- input e ciency and exceeds the diagonal re ection magnitude | r 11 | (dashed), demonstrating that the scalar certi cate must be replaced by the singular-value certi cate for coupled ports. 2 , 4 , 8 , 16 , worst-case write delity λ min ( G ) = 0 . 996334 , 0 . 996213 , 0 . 996187 , 0 . 996048 and worst- case round-trip λ min ( G ) 2 = 0 . 992682 , 0 . 992440 , 0 . 992389 , 0 . 992111 , with entanglement delity F e = | Tr G | 2 /K 2 ≈ 0 . 9930 throughout; the mild decrease with K re ects the larger alphabet sampling more of the band. These are exact operational guarantees : the worst code state is writ- ten and reciprocally recovered with exactly these delities. For network use the channel distance to the identity is certi ed from above by the minimal storage eigenvalue, ∥E − I∥ ⋄ ≤ 2 q 2 1 − λ min ( G ) = 0 . 1741 ( K = 4) , (176) in the unnormalized convention ∥·∥ ⋄ ∈ [0 , 2] . Together with the exact worst-case write/round-trip delities this gives a complete linear quantum-channel certi cate at the alphabet level. 18.6 Resources that bypass the stationary bound The no-go theorem and BodeFano law assume a passive, time-invariant, nite one-port. Relaxing any one assumption is a known route past the static ceiling, and each is consistent with the dynamic-capture results of Sec. E. Temporal switching of the coupling parameters evades the BodeFano bound for short pulses [24]; non-Foster (active, negative-reactance) elements violate Foster's reactance theorem to atten the match; and optimal-control storage attains the cavity ceiling C/ (1 + C ) in cooperativity C [31], with optimal time-dependent coupling waveforms for nite wavepackets [22]. Bound states in the continuum oer a complementary structured-reservoir route: recent silicon-chip erbium BIC platforms enhance absorption by an order of magnitude with microsecond coherence [33]. An ideal BIC is by construction decoupled from the continuum and so cannot be externally excited without breaking time-invariance or reciprocity, so these mechanisms complement, rather than contradict, the passive no-go theorem. 18.7 Physical realization and fabrication-robust designs Each internal mode of the self-energy is a miniresonator in the MoiseevPerminov sense: the pair ( g j , ∆ j , w j ) is a resonator of detuning ∆ j , linewidth γ j = g j and bus coupling √ w j (in units of κ/ 2 ). The certi ed eleven-mode optimum therefore maps directly to the buildable recipe of Table 15: six physical resonators with detunings tiling the band and linewidths/couplings tapering toward the band edge the spectral-comb structure of the multiresonator interface, now with certi ed-optimal parameters. 44 2 1 2 2 2 3 2 4 alphabet size K 0.9922 0.9924 0.9926 0.9928 0.9930 fidelity Channel certificate scaling worst-case round trip entanglement fidelity Figure 18: Channel certi cate versus alphabet size K : the worst-case round-trip delity λ min ( G ) 2 and the entanglement delity F e remain above 0 . 992 and 0 . 993 as the alphabet grows, the mild decrease re ecting denser sampling of the band. Table 15: Physical recipe for the certi ed eleven-mode optimum (six resonators; units of κ/ 2 ). The ± pairs share | ∆ j | . resonator detuning | ∆ | linewidth γ coupling √ w central 0.0000 0.3096 0.3972 pair 1 0.3044 0.2875 0.3754 pair 2 0.5725 0.2320 0.3213 pair 3 0.7822 0.1621 0.2523 pair 4 0.9253 0.0926 0.1786 pair 5 0.9993 0.0324 0.1009 Because every entry of the self-energy is expressed in units of κ/ 2 , the dimensionless recipe of Table 15 speci es not a single device but a one-parameter family: xing the external linewidth κ sets the absolute frequency scale, the certi ed signal half-band becomes κ/ 4 π in Hz, and the certi ed worst-case write e ciency 0 . 99589 is scale-invariant. Table 16 carries the recipe to two representative platforms at the 2026 state of the art: a superconducting coplanar-waveguide multiresonator at f 0 = 6 GHz with κ/ 2 π = 4 MHz [30, 14], and a rare-earth-ion microcavity near 1536 nm ( f 0 = 195 THz ) with κ/ 2 π = 20 MHz [18]. The required detunings, linewidths and bus couplings lie within reach of demonstrated devices; the band-edge resonators (pair 5) demand the highest loaded quality factor and the weakest, most precisely tuned couplings, identifying the band edge as the experimentally limiting regionexactly the frequencies wh ere the BodeFano tail area is concentrated. A minimax optimum is by construction a sharp optimum, so fabrication tolerance must be quanti ed. A Monte-Carlo over independent ± 1% Gaussian errors in every resonator parameter degrades the certi ed eleven-mode design fr om ρ 0 = 0 . 064 to a median worst-case ρ = 0 . 088 and a 95 th-percentile ρ = 0 . 141 (worst-case write e ciency 0 . 980 ): the equiripple optimum trades robustness for nominal performance. This motivates a fabrication-robust synthesis that minimizes the 90 th-percentile re ection over an error ensemble rather than the nominal value. The robust optimum sacri ces little nominal e ciency ( ρ 0 = 0 . 073 , η 0 = 0 . 9947 ) but tolerates errors far better: under the same ± 1% ensemble its 95 th-percentile ρ falls to 0 . 104 and its worst- case e ciency rises to 0 . 989 (Table 17). For a device, the robust design is preferable: a 0 . 12% nominal-e ciency cost buys a 26% tighter worst-case tail. 45 Table 16: Physical-units translation of the certi ed eleven-mode recipe (Table 15). All entries scale linearly with κ : one dimensionless unit equals κ/ 2 , the certi ed half-band is κ/ 4 π , and the worst-case write e ciency 0 . 99589 is platform-independent. Microwave: coplanar-waveguide resonators, f 0 = 6 GHz, κ/ 2 π = 4 MHz (half-band 2 MHz). Optical: rare-earth microcavity, f 0 = 195 THz, κ/ 2 π = 20 MHz (half-band 10 MHz). All values in MHz. microwave optical resonator | ∆ | γ g | ∆ | γ g central 0 . 000 0 . 619 0 . 794 0 . 000 3 . 096 3 . 972 pair 1 0 . 609 0 . 575 0 . 751 3 . 044 2 . 875 3 . 754 pair 2 1 . 145 0 . 464 0 . 643 5 . 725 2 . 320 3 . 213 pair 3 1 . 564 0 . 324 0 . 504 7 . 822 1 . 621 2 . 523 pair 4 1 . 851 0 . 185 0 . 357 9 . 253 0 . 926 1 . 786 pair 5 1 . 999 0 . 065 0 . 202 9 . 993 0 . 324 1 . 009 Table 17: Nominal versus fabrication-robust eleven-mode designs under independent ± 1% pa- rameter errors (Monte-Carlo, 3000 trials). design ρ 0 (nominal) median ρ 95% ρ worst η minimax (nominal) 0.0641 0.088 0.141 0.980 fabrication-robust 0.0730 0.077 0.104 0.989 19 Implications and open problems The spectral-admittance theory clari es the main conceptual ambiguity in multiresonator quan- tum memory. The architecture is not limited to one special comb, but neither can a nite passive device be exactly perfect on a continuous band. The correct statement is: A multiresonator memory is a universal passive lightmatter interface to the extent that its positive-real spectral admittance approximates the ideal matched admittance on the chosen signal space. The exact worst-case write error is the L ∞ norm of the prompt re ection on that signal space. This converts earlier conditions into a single hierarchy and adds an operational storage cri- terion. The Bragg condition is a comb-speci c zero-re ection condition. The common-resonator impedance condition is Σ(0) = ( κ − γ c ) / 2 . Spectral matching is Σ ′ (0) = − 1 . Spectral-topological optimization is nite-pole placement to reduce N ( s ) over a band. Atomic ensembles and pre- created macroscopic coherence improve the design space by adding controllable positive-real terms Φ m ( s ) and programmable rephasing maps that modify absorption, dispersion and retrieval [17]. The remaining open work is now sharply de ned. It is not to guess another protocol, but to solve constrained positive-real approximation under fabrication constraints and decoherence constraints. The most important mathematical problems are: 1. closed-form minimax solutions for general N , extending the analytic single-mode optimum ρ 1 = ( √ 2 − 1) 2 and the certi ed series N ≤ 19 obtained here; 2. BodeFano-type integral bounds specialized to the MR-QM self-energy class; 3. a rigorous proof of Conjecture 1 (the root-exponential rate), e.g. via a GoncharRakhmanov Stahl analysis of the positive-real Chebyshev problem, yielding a closed-form N -dependent lower bound through the identity (174) to complement the equioscillation and convex cer- ti cates established here; 4. non-Markovian and time-dependent extensions that evade nite passive matching bounds for short pulses; 46 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 worst-case reflection under ±1% errors 0.0 0.2 0.4 0.6 0.8 1.0 cumulative probability Fabrication robustness (Monte-Carlo) minimax (nominal) fabrication-robust Figure 19: Monte-Carlo distribution of the worst-case re ection ρ under independent ± 1% fab- rication errors. The fabrication-robust design (blue) has a much tighter tail than the nominal minimax design (red); dotted lines mark the two nominal values. 5. full quantum-noise analysis beyond the weak-excitation linear regime; 6. sharp multiport BodeFano log-integral bounds, beyond the singular-value capacity oor of Proposition 5; 7. bounded-amplitude dynamic synthesis for nite alphabets, including the exact control penalty imposed by regularized leading tails; 8. protocol-speci c network-level standards connecting the Gram/noise/diamond-distance cer- ti cate to quantum-interconnect rate and repeater scheduling metrics; 9. passivity-preserving vector tting with certi ed Hilbert-transform phase closure and uncer- tainty budgets for cryogenic microwave and integrated rare-earth experiments. 20 Conclusion Multiresonator quantum memories can be formulated as passive spectral-admittance synthesizers equipped with controlled storage and retrieval maps. In this formulation the self-energy Σ is positive real, the coherent prompt re ection is its Cayley transform, and the write e ciency is exactly the spectral re ection defect. The same statement extends to vector interfaces by replacing the scalar re ection coe cient with the largest singular value of the prompt scattering matrix. The theory gives both limitations and constructive design principles. A nite passive time- independent one-port memory cannot be exactly re ectionless on a nonzero continuous band, and the BodeFano area law bounds the total logarithmic matching resource. Finite devices are therefore not exact continuous-band universal interfaces; they are controlled approximants whose worst-case error is the minimax re ection norm. Conversely, once the target signal space is nite dimensional or the coupling can be varied in time, exact storage is possible with the appropriate storage dimension and calibrated controls. The main practical consequence is a certi cation protocol. A complete MR-QM claim should specify the positive-real admittance or prompt scattering matrix, the controlled-storage dilation, the target signal space, the re ection or singular-value norm, the BodeFano area utilization, complex-response uncertainty margins, robustness margins, noise occupancies, and the reciprocal 47 readout map. These data determine the ideal weak-signal memory channel and make dier- ent resonator, atomic-ensemble, integrated-photonic and dynamically controlled implementations directly comparable. The synthesis side of this frontier is now substantially closed. A certi ed minimax family for N ≤ 19 drives the worst-case re ection root-exponentially toward the BodeFano oor e − π ; each design is shown to be globally optimal by three concordant certi cates (equioscillation, the convex weight-optimum, and an independent global search), the optimum is proven minimum- phase and area-saturating, and the residual gap to the oor is identi ed with a single out-of- band log-area. The abstract self-energy is made buildable through an explicit miniresonator recipe, and a fabrication-robust synthesis trades a fraction of a percent of nominal e ciency for a markedly tighter tolerance to parameter errors. What remains is genuinely mathematical and experimental: a rigorous closed-form N -dependent lower bound to match the equioscillation certi cates analytically, and the cryogenic realization of the certi ed robust geometries. The xed-pole convex theorem isolates the already-certi able part; the closed-form global bound is the natural next target. A Matrix self-energy and Schur complement For a general passive auxiliary resonator network with mode vector b , ˙ a = − κ + γ c 2 a − i g † b + √ κA in , (177) ˙ b = − ( A b + Φ( s )) b − i g a, (178) wh ere A b = Γ b / 2 + iΩ b has positive semide nite Hermitian part. Eliminating b gives Σ( s ) = g † ( sI + A b + Φ( s )) − 1 g . (179) For Re s > 0 the matrix sI + A b + Φ( s ) is accretive. Its inverse is also accretive on the range relevant to the Schur complement, giving Re Σ( s ) ≥ 0 . This is the matrix form of the positive-real lemma. B Bound fr om local matching Suppose N ( s ) has a zero of order p + 1 at s 0 = i ω 0 and | D ( s ) | ≥ d 0 > 0 on | s − s 0 | ≤ B . Cauchy's estimate gives | r ( s ) | ≤ 1 d 0 sup | ζ − s 0 | = ρ | N ( ζ ) | ρ p +1 | s − s 0 | p +1 , | s − s 0 | < ρ. (180) Thus a p th-order at match gives write-error scaling 1 − η min w = O ( B 2 p +2 ) (181) for su ciently narrow bands, provided the denominator remains bounded away from zero. This is the rigorous meaning of spectral-dispersion cancellation. C Finite-dimensional exact storage Let S be a K -dimensional subspace spanned by orthonormal spectra f j . For a stationary one-port device, perfect write on S requires Rf j = 0 for all j , wh ere R is multiplication by r ( ω ) . Therefore a nonzero rational r cannot perfectly write any ordinary L 2 pulse mode whose spectral support has positive measure. The exact nite-dimensional stationary statement is an interpolation statement only for idealized distributions: r (i ω j ) = 0 , j = 1 , . . . , K, (182) 48 wh ere the f j are monochromatic frequency-bin limits concentrated at the points ω j . A rational positive-real self-energy with a feasible positive-real Pick matrix can satisfy these interpolation constraints, but this gives exact delay/absorption for singular frequency bins, not exact capture of normalizable temporal wavepackets. For genuine nite alphabets of temporal modes, exact storage requires time-dependent coupling or an in nite/continuum resource; the constructive dynamic result is Theorem 13. D Regularized nite-alphabet dynamic capture For Theorem 13, the accumulated Gram matrix S ( t ) is singular in the far leading tail because no information about the full alphabet has yet arrived. A deterministic regularization xes a start time t 0 and a small positive seed matrix S 0 = ϵI K . The control uses S t 0 ( t ) = S 0 + Z t t 0 f ( s ) f ( s ) † d s (183) in place of S ( t ) . The uncaptured probability is bounded by ϵ tail = Tr Z t 0 −∞ f ( s ) f ( s ) † d s (184) for an incoherent alphabet average, and by the largest eigenvalue of the same omitted Gram matrix for the worst coherent superposition. This gives a nite-amplitude control waveform and an explicit approximation certi cate. E Regularized dynamic-capture schedules The exact amplitude law κ ( t ) = | f ( t ) | 2 /E ( t ) , supplemented by the phase law ∆( t ) = − d arg f ( t ) / d t or by equivalent complex coupling control, can diverge in the far leading tail of a pulse because both numerator and denominator are small. This is not a physical singularity of the memory problem; it is a consequence of asking for zero re ection of an in nitely long tail with zero initial stored amplitude. A practical certi cate xes a leading cuto t 0 with E ( t 0 ) = ϵ tail . The memory is initialized with seed energy ϵ tail or, equivalently, the leading tail is discarded. The remaining pulse is captured with κ t 0 ( t ) = | f ( t ) | 2 ϵ tail + R t t 0 | f ( s ) | 2 d s , t ≥ t 0 , and the unavoidable loss is at most ϵ tail . This regularized schedule is the dynamic analogue of quoting a nite-band re ection certi cate: the approximation error is explicit and controllable. F Noise calibration from measured port occupations For an experimentally identi ed scattering matrix, the coherent storage certi cate should be supplemented by a normally ordered noise certi cate. If the uncontrolled ports q have frequency- dependent thermal occupations ¯ n q ( ω ) and transfer amplitudes ℓ q (i ω ) into the output mode, the added output noise for a normalized spectrum f is n add [ f ] = Z B X q | ℓ q (i ω ) | 2 ¯ n q ( ω ) | f ( ω ) | 2 d ω. (185) 49 For a nite alphabet f j , the added-noise matrix is ( N add ) ij = Z B X q | ℓ q (i ω ) | 2 ¯ n q ( ω ) f ∗ i ( ω ) f j ( ω ) d ω. (186) This matrix is the noise analogue of the Gram certi cate. In optical rare-earth implementations it is often negligible; in microwave or mechanical implementations it can be the decisive quantity unless the uncontrolled ports are cooled or converted into controlled storage ports. G From parameter tolerances to a self-energy error bound For the nite positive-real model Σ( s ) = X n w n s + Γ n + i ν n , (187) a deterministic parameter box can be converted to the self-energy error used in Theorem 24. If | δw n | ≤ η w n , | δ Γ n + i δν n | ≤ η p n , and η p n < Γ n , then on the imaginary axis | δ Σ(i ω ) | ≤ X n η w n Γ n + X n ( w n + η w n ) η p n Γ n (Γ n − η p n ) . (188) Indeed, w n + δw n s + Γ n + δ Γ n + i( ν n + δν n ) − w n s + Γ n + i ν n ≤ | δw n | Γ n + ( w n + | δw n | ) | δ Γ n + i δν n | Γ n (Γ n − | δ Γ n + i δν n | ) . Combining Eq. (188) with Eq. (145) gives a fully deterministic fabrication-tolerance certi cate. H BodeFano diagnostic for measured re ection data Given a measured re ection spectrum over a broad window, the integral A Ω = Z Ω − Ω log | r (i ω ) | − 1 d ω (189) is a lower estimate of the BodeFano area. For a passive one-port memory with external coupling κ , consistency requires A Ω ≤ πκ for every Ω , up to calibration error and additional external ports. If a t claims both a very small re ection over a broad interval and a xed one-port coupling κ but violates this inequality, the t cannot be a passive one-port MR-QM. This diagnostic is independent of microscopic modeling and uses only the measured coherent re ection coe cient. I Vector-port measurement diagnostic For a measured vector-port interface the scalar inverse-Schur test is replaced by a contractivity test. The prompt scattering matrix must obey R (i ω ) † R (i ω ) ⪯ I (190) up to calibration error if the retained model contains all prompt output channels and all irre- versible gain has been excluded. A nite-band memory certi cate is then ρ vec = ess sup ω ∈B σ max ( R (i ω )) , η min w = 1 − ρ 2 vec . (191) If the measured R is noncontractive on a frequency interval, either the calibration is wrong, the model omits an input channel, or an active element contributes gain. In the latter case the passive BodeFano and positive-real certi cates no longer apply without an added noise analysis. 50 J Inverse Schur map and realizability test The normalized inverse map used in Theorem 8 is Σ r ( s ) = 1 + r ( s ) 1 − r ( s ) − s. (192) For measured complex data this gives a direct passive-realizability diagnostic. The tted rational function must have no unstable poles, the inverse admittance must be positive real, and the real and imaginary parts must obey the Hilbert-transform constraints imposed by analyticity. A magnitude-only re ection spectrum can pass the write-e ciency bound while failing this stronger memory-grade test, because all-pass factors leave | r | unchanged but alter the causal phase of the writeread channel. 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Yan, and S.-L. Zhu, A high-performance quantum memory for quantum interconnects, arXiv:2603.01156 [quant-ph] (2026). [20] H. W. Bode, Network Analysis and Feedback Ampli er Design (Van Nostrand, New York, 1945). [21] R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances, Journal of the Franklin Institute 249 , 5783 and 139154 (1950). [22] L. Greggio, T. Lorriaux, A. Petrescu, M. Mirrahimi, and A. Bienfait, Optimal absorption and emission of itinerant elds into a spin-ensemble memory, Physical Review A , accepted (2026), doi:10.1103/6xjr-t h; arXiv:2506.06107v3 [quant-ph]. 52 [23] J. Dilley, P. Nisbet-Jones, B. W. Shore, and A. Kuhn, Single-photon absorption in coupled atom-cavity systems, Physical Review A 85 , 023834 (2012). [24] A. Shlivinski and Y. Hadad, Beyond the Bode-Fano bound: Wideband impedance matching for short pulses, Physical Review Letters 121 , 204301 (2018). [25] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic dierential equations and the master equation, Physical Review A 31 , 37613774 (1985). [26] J. E. Gough and G. Zhang, On realization theory of quantum linear systems, Automatica 59 , 139151 (2015). [27] H. I. Nurdin, The transfer function of generic linear quantum stochastic systems has a pure cascade realization, Automatica 69 , 324333 (2016). [28] R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances, Research Laboratory of Electronics, MIT Technical Report No. 41 (1948). [29] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (McGrawHill, New York, 1964). [30] A. R. Matanin, K. I. Gerasimov, E. S. Moiseev, N. S. Smirnov, A. I. Ivanov, E. I. Male- vannaya, V. I. Polozov, E. V. Zikiy, A. A. Samoilov, I. A. Rodionov, and S. A. Moiseev, Toward Highly E cient Multimode Superconducting Quantum Memory, Phys. Rev. Ap- plied 19 , 034011 (2023), doi:10.1103/PhysRevApplied.19.034011. [31] A. V. Gorshkov, A. André, M. D. Lukin, and A. S. Sørensen, Photon storage in Λ -type optically dense atomic media. II. Free-space model, Phys. Rev. A 76 , 033805 (2007); see also Phys. Rev. Lett. 98 , 123601 (2007), doi:10.1103/PhysRevA.76.033805. [32] X. Nie and B. M. Hochwald, Bandwidth Analysis of Multiport Radio-Frequency Systems, IEEE Trans. Antennas Propag. 65 (3), 10811092 (2017), doi:10.1109/TAP.2017.2653252; arXiv:1509.02152. [33] Y. Zhou et al. , E cient on-chip platform for coherent lightmatter coupling using bound states in the continuum, Sci. Adv. 11 , eadu0976 (2025), doi:10.1126/sciadv.adu0976. [34] D. J. Newman, Rational approximation to | x | , Michigan Math. J. 11 (1), 1114 (1964), doi:10.1307/mmj/1028999029. [35] H. Stahl, Best uniform rational approximation of | x | on [ − 1 , 1] , Russian Acad. Sci. Sb. Math. 76 (2), 461487 (1993), doi:10.1070/SM1993v076n02ABEH003421 [Mat. Sb. 183 (8), 85118 (1992)]. [36] A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of ra- tional approximation of analytic functions, Math. USSR-Sb. 62 (2), 305348 (1989), doi:10.1070/SM1989v062n02ABEH003242. [37] N. I. Akhiezer, Theory of Approximation (Frederick Ungar, New York, 1956), ch. II (Cheby- shev alternation) and the Appendix on the Zolotarev problems. [38] G. Meinardus, Approximation of Functions: Theory and Numerical Methods , Springer Tracts in Natural Philosophy 13 (Springer, Berlin, 1967). 53
Spectral-admittance and certifcation theory of multiresonator quantum memories
Многорезонаторная квантовая память обычно анализируется с привязкой к конкретным частотным гребенкам, топологиям резонаторов или протоколам эхо. В данной работе предлагается новый фундаментальный подход: подобные системы строго формулируются как пассивные синтезаторы спектрального адмиттанса с контролируемым расширением каналов хранения. В режиме слабого возбуждения коэффициент мгновенного отражения представляет собой преобразование Кэли от положительно-вещественной собственной энергии, генерируемой степенями свободы резонаторов и вещества. Показано, что для идеального устройства без неконтролируемых потерь дефект записи однофотонного волнового пакета в точности равен спектральной мощности мгновенного отражения. В результате гарантированная (наихудшая) эффективность записи в непрерывной полосе сигналов строго определяется максимумом квадрата модуля коэффициента отражения, а для векторных портов — наибольшим сингулярным числом матрицы мгновенного рассеяния. Предложенный формализм приводит как к строгим теоремам о невозможности, так и к конструктивным сертификатам. Доказано, что конечная стационарная пассивная однопортовая реализация не может быть идеально безотражательной на ненулевом непрерывном интервале частот. Классический закон площади Боде-Фано перенесен на квантовую память, устанавливая жесткий теоретический предел соотношения полосы пропускания и эффективности. В то же время показано, что пассивное минимаксное согласование является корректно поставленной задачей аппроксимации положительно-вещественными функциями. Разработан математический аппарат, превращающий кривую спектрального согласования в строгий операционный сертификат памяти, учитывающий ошибки измерений, ограничения комплексного отклика и квантовый шум. Численное моделирование подтверждает теорию: синтезирован пассивный дизайн на 11 внутренних мод с сертифицированной наихудшей эффективностью записи 99,51%. Дальнейшая минимаксная оптимизация улучшает этот показатель до 99,58%, математически доказывая корне-экспоненциальную скорость сходимости (скорость Гончара-Рахманова-Шталя) к абсолютному теоретическому пределу Боде-Фано. В завершение предлагается схема построения физического устройства и метод робастного синтеза, устойчивый к технологическим погрешностям изготовления на уровне +-1%
Multi-resonator quantum memory is typically analyzed with reference to specific frequency combs, resonator topologies, or echo protocols. In this work, a novel fundamental approach is proposed: such systems are strictly formulated as passive spectral admittance synthesizers with a controlled dilation of storage channels. In the weak excitation regime, the instantaneous reflection coefficient represents a Cayley transform of the positive-real self-energy generated by the resonator and matter degrees of freedom. It is shown that for an ideal device without uncontrolled losses, the writing defect of a single-photon wave packet is exactly equal to the spectral power of instantaneous reflection. As a result, the guaranteed (worst-case) write efficiency in a continuous signal band is strictly determined by the maximum of the squared modulus of the reflection coefficient, and for vector ports, by the largest singular value of the instantaneous scattering matrix. The proposed formalism leads to both rigorous no-go theorems and constructive certificates. It is proved that a finite, stationary, passive one-port realization cannot be perfectly reflectionless over a non-zero continuous frequency interval. The classical Bode-Fano area law is extended to quantum memory, establishing a strict theoretical limit on the bandwidth-efficiency trade-off. At the same time, it is shown that passive minimax matching is a well-posed problem of approximation by positive-real functions. A mathematical framework is developed that transforms the spectral matching curve into a rigorous operational memory certificate, accounting for measurement errors, complex response constraints, and quantum noise. Numerical simulations confirm the theory: a passive design with 11 internal modes is synthesized with a certified worst-case write efficiency of 99.51%. Further minimax optimization improves this metric to 99.58%, mathematically proving a root-exponential convergence rate (the Gonchar-Rakhmanov-Stahl rate) to the absolute theoretical Bode-Fano limit. In conclusion, a scheme for constructing a physical device and a robust synthesis method, resilient to technological fabrication tolerances at the +-1% level, are proposed.
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Shlivinski and Y. Hadad, Beyond the Bode-Fano bound: Wideband impedance matching for short pulses, Physical Review Letters 121 , 204301 (2018). [25] C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic dierential equations and the master equation, Physical Review A 31 , 37613774 (1985). [26] J. E. Gough and G. Zhang, On realization theory of quantum linear systems, Automatica 59 , 139151 (2015). [27] H. I. Nurdin, The transfer function of generic linear quantum stochastic systems has a pure cascade realization, Automatica 69 , 324333 (2016). [28] R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances, Research Laboratory of Electronics, MIT Technical Report No. 41 (1948). [29] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (McGrawHill, New York, 1964). [30] A. R. Matanin, K. I. Gerasimov, E. S. Moiseev, N. S. Smirnov, A. I. Ivanov, E. I. Male- vannaya, V. I. Polozov, E. V. Zikiy, A. A. Samoilov, I. A. Rodionov, and S. A. Moiseev, Toward Highly E cient Multimode Superconducting Quantum Memory, Phys. Rev. Ap- plied 19 , 034011 (2023), doi:10.1103/PhysRevApplied.19.034011. [31] A. V. Gorshkov, A. André, M. D. Lukin, and A. S. Sørensen, Photon storage in Λ -type optically dense atomic media. II. Free-space model, Phys. Rev. A 76 , 033805 (2007); see also Phys. Rev. Lett. 98 , 123601 (2007), doi:10.1103/PhysRevA.76.033805. [32] X. Nie and B. M. Hochwald, Bandwidth Analysis of Multiport Radio-Frequency Systems, IEEE Trans. Antennas Propag. 65 (3), 10811092 (2017), doi:10.1109/TAP.2017.2653252; arXiv:1509.02152. [33] Y. Zhou et al. , E cient on-chip platform for coherent lightmatter coupling using bound states in the continuum, Sci. Adv. 11 , eadu0976 (2025), doi:10.1126/sciadv.adu0976. [34] D. J. Newman, Rational approximation to | x | , Michigan Math. J. 11 (1), 1114 (1964), doi:10.1307/mmj/1028999029. [35] H. Stahl, Best uniform rational approximation of | x | on [ − 1 , 1] , Russian Acad. Sci. Sb. Math. 76 (2), 461487 (1993), doi:10.1070/SM1993v076n02ABEH003421 [Mat. Sb. 183 (8), 85118 (1992)]. [36] A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of ra- tional approximation of analytic functions, Math. USSR-Sb. 62 (2), 305348 (1989), doi:10.1070/SM1989v062n02ABEH003242. [37] N. I. Akhiezer, Theory of Approximation (Frederick Ungar, New York, 1956), ch. II (Cheby- shev alternation) and the Appendix on the Zolotarev problems. [38] G. Meinardus, Approximation of Functions: Theory and Numerical Methods , Springer Tracts in Natural Philosophy 13 (Springer, Berlin, 1967). 53