The quantum memory cost of optical correlations

Zixuan He

Published Oct 29, 2026• 0 citations

Abstract

How much quantum memory is needed to restore an unknown optical pulse together with its correlations? This paper gives matching storage laws showing that the answer depends on the actual joint preparation, even when the optical marginal is a positive mixture of coherent states. Fix an optical mode number $m$, a known real $k$-dimensional displacement subspace with $1\le k\le2m$, and a positive mean photon budget $U_0$. The source is any probability mixture of coherent states in that subspace with actual mean photons at most $U_0$. One normal encoder and decoder must work for the entire unknown source class, and every transmitted message or flag counts toward the same register dimension. The reference is inaccessible to both maps. Let $\delta$ be the required half trace-distance error and $q_{\min}(\delta)$ the least encoded qubit count. As $\delta\downarrow0$, \[q_{\min}(\delta)=\begin{cases}\dfrac{k}{2}\log_2(1/\delta)+O(1),&\text{optical marginal or joint coherent preparation},\\k\log_2(1/\delta)+O(1),&\text{arbitrary reference extension or classical-bit reference}.\end{cases}\] A single inaccessible classical bit already witnesses the larger leading cost. To recover the smaller cost for correlations, the joint source must actually be prepared as $\omega_{AB}=\int |Jx\rangle\langle Jx|\otimes\tau_B(x)\,\mathrm d\mu(x)$, with $\int|x|^2\mathrm d\mu\le U_0$. Here $J$ is the known displacement embedding and $\tau_B(x)$ may be noncommuting normal reference states. The constants are uniform in the reference dimension and the orientation of $J$; the codec may depend on the known $J$. The mechanism is the difference between the probability of a rare bright component and the square-root amplitude of its retained correlations. Exact optical Gram matrices and a reference-coherence witness establish the converse for arbitrary normal decoders, including infinite-rank outputs. A common finite optical subspace supplies the attaining compression schemes while retaining the full photon tails. These are single-input universal memory-dimension laws at fixed $m,k,U_0$, with every communication resource counted. They complement established known-source and many-copy compression theory. The integrated 10-page PDF contains two figures, complete proofs and references.

DOI: 10.5281/zenodo.23778178

Related papers

Quantum limits and one-qubit advantage in optical predictive compression

Zixuan He

Oct 25, 20260 citations

Photon-number hierarchies in the quantum memory cost of heterodyne uncertainty

Zixuan He

Oct 25, 20260 citations

Memory capacity and lifetime thresholds for testing pure Gaussianity

Zixuan He

Oct 25, 20260 citations

Size and reliability limits of quantum memory for heterodyne uncertainty

Zixuan He

Oct 26, 20260 citations

The memory cost of quantum illumination

Zixuan He

Oct 20, 20260 citations

Optimal entanglement weights for heterodyne uncertainty with finite quantum memory

Zixuan He

Oct 25, 20260 citations

The quantum memory cost of optical correlations

Zixuan He

Published Oct 29, 2026• 0 citations

Publisher

Abstract

How much quantum memory is needed to restore an unknown optical pulse together with its correlations? This paper gives matching storage laws showing that the answer depends on the actual joint preparation, even when the optical marginal is a positive mixture of coherent states. Fix an optical mode number (m), a known real (k)-dimensional displacement subspace with (1\le k\le2m), and a positive mean photon budget (U_0). The source is any probability mixture of coherent states in that subspace with actual mean photons at most (U_0). One normal encoder and decoder must work for the entire unknown source class, and every transmitted message or flag counts toward the same register dimension. The reference is inaccessible to both maps. Let (\delta) be the required half trace-distance error and (q_{\min}(\delta)) the least encoded qubit count. As (\delta\downarrow0), [q_{\min}(\delta)=\begin{cases}\dfrac{k}{2}\log_2(1/\delta)+O(1),&\text{optical marginal or joint coherent preparation},\k\log_2(1/\delta)+O(1),&\text{arbitrary reference extension or classical-bit reference}.\end{cases}] A single inaccessible classical bit already witnesses the larger leading cost. To recover the smaller cost for correlations, the joint source must actually be prepared as (\omega_{AB}=\int |Jx\rangle\langle Jx|\otimes\tau_B(x),\mathrm d\mu(x)), with (\int|x|^2\mathrm d\mu\le U_0). Here (J) is the known displacement embedding and (\tau_B(x)) may be noncommuting normal reference states. The constants are uniform in the reference dimension and the orientation of (J); the codec may depend on the known (J). The mechanism is the difference between the probability of a rare bright component and the square-root amplitude of its retained correlations. Exact optical Gram matrices and a reference-coherence witness establish the converse for arbitrary normal decoders, including infinite-rank outputs. A common finite optical subspace supplies the attaining compression schemes while retaining the full photon tails. These are single-input universal memory-dimension laws at fixed (m,k,U_0), with every communication resource counted. They complement established known-source and many-copy compression theory. The integrated 10-page PDF contains two figures, complete proofs and references.

DOI: 10.5281/zenodo.23778178

Related papers

Quantum limits and one-qubit advantage in optical predictive compression

Zixuan He

What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on $A$ is prepared with its actual purifier $C$ . A normal encoder acts on $C$ to produce memory $M$ , while a prescribed Gaussian channel maps $A$ to the relevant optical output $B$ . The source-information cost and useful correlation are $r=I(A:M)$ and $v=I(B:M)$ , respectively. All logarithms are natural. The paper determines $C_{\mathrm Q}(R)=\sup_{\mathcal N:\,I(A:M)\le R}I(B:M).$ For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number $n$ and a thermal attenuator with transmissivity $0<\eta<1$ , write the output occupation as $\eta n+b$ , where $b\ge0$ is the environment's added thermal occupation. With $g(x)=(x+1)\log(x+1)-x\log x$ , the exact optimum over all normal classical records is $C_{\mathrm{cl}}(n,R)=g(\eta n+b)-g(\eta u_R+b),\qquad u_R=g^{-1}\!\bigl(\max\{g(n)-R,0\}\bigr).$ A legal encoder applies vacuum loss with transmissivity $\lambda$ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed $b>0$ and $0<\lambda<1/2$ , at the common budget $R_n=g(n)+g(\lambda n)-g((1-\lambda)n)$ , its actual relevance satisfies $v_2=\frac{\eta\lambda}{1+b}\,n\log(1/n)+O(n),\qquad C_{\mathrm{cl}}(n,R_n)=2\lambda\eta\log(1+1/b)\,n+O\!\left(\frac{n}{\log(1/n)}\right).$ The full optical tail is included in the resource accounting. At $n=0.01$ , $\eta=b=0.5$ , $\lambda=0.2$ and $R=0.023875$ , the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For $k$ uses, the retained predictive memory obeys $\dim K_k\le2^k$ and $I(A^k:K_k)\le kR$ , including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let $\alpha_k$ be the missed-correlation probability and $\beta_k$ the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is $\alpha_k\le\epsilon\quad\Longrightarrow\quad \beta_k\ge\exp\!\left[-\frac{k c_nR+\log2}{1-\epsilon}\right],\qquad c_n=\eta\frac{\log(1+1/(\eta n+b))}{\log(1+1/n)}.$ At the finite parameters above and $\epsilon=0.1$ , the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on $A$ is prepared with its actual purifier $C$. A normal encoder acts on $C$ to produce memory $M$, while a prescribed Gaussian channel maps $A$ to the relevant optical output $B$. The source-information cost and useful correlation are $r=I(A:M)$ and $v=I(B:M)$, respectively. All logarithms are natural. The paper determines \[ C_{\mathrm Q}(R)=\sup_{\mathcal N:\,I(A:M)\le R}I(B:M). \] For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number $n$ and a thermal attenuator with transmissivity $0<\eta<1$, write the output occupation as $\eta n+b$, where $b\ge0$ is the environment's added thermal occupation. With $g(x)=(x+1)\log(x+1)-x\log x$, the exact optimum over all normal classical records is \[ C_{\mathrm{cl}}(n,R)=g(\eta n+b)-g(\eta u_R+b),\qquad u_R=g^{-1}\!\bigl(\max\{g(n)-R,0\}\bigr). \] A legal encoder applies vacuum loss with transmissivity $\lambda$ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed $b>0$ and $0<\lambda<1/2$, at the common budget $R_n=g(n)+g(\lambda n)-g((1-\lambda)n)$, its actual relevance satisfies \[ v_2=\frac{\eta\lambda}{1+b}\,n\log(1/n)+O(n),\qquad C_{\mathrm{cl}}(n,R_n)=2\lambda\eta\log(1+1/b)\,n+O\!\left(\frac{n}{\log(1/n)}\right). \] The full optical tail is included in the resource accounting. At $n=0.01$, $\eta=b=0.5$, $\lambda=0.2$ and $R=0.023875$, the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For $k$ uses, the retained predictive memory obeys $\dim K_k\le2^k$ and $I(A^k:K_k)\le kR$, including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let $\alpha_k$ be the missed-correlation probability and $\beta_k$ the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is \[ \alpha_k\le\epsilon\quad\Longrightarrow\quad \beta_k\ge\exp\!\left[-\frac{k c_nR+\log2}{1-\epsilon}\right],\qquad c_n=\eta\frac{\log(1+1/(\eta n+b))}{\log(1+1/n)}. \] At the finite parameters above and $\epsilon=0.1$, the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.

Oct 25, 20260 citations

Photon-number hierarchies in the quantum memory cost of heterodyne uncertainty

Zixuan He

How much quantum memory is needed to retain the contributions of rare photon events to a measurement record? This paper determines the memory scales required to approach the best heterodyne conditional entropy when the total optical energy tends to zero. Its resource laws distinguish precisions for which counting photon configurations suffices from critical precisions at which energy-calibrated entropy corrections determine additional memory costs. Full preparation model and reference value. The optical system has $m$ bosonic modes, total mean photon number at most $E$ , and an entire accessible quantum side system $B$ of dimension at most $D$ , including retained preparation labels. The optical source and its correlations with the memory are optimized over all normal mixed states; arbitrary coherences and infinite photon tails are allowed. With natural logarithms and heterodyne measure $d\mu=\pi^{-m}d^{2m}z$ , define $V_{m,D}(E)=\min_{\rho_{AB}}H_\mu(Z|B),\qquad H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B).$ Here $Z$ is the heterodyne outcome and $\chi$ is Holevo information. The unlimited-memory optimum at the same energy is $B_m(E)=m-E\ln(1+m/E),\qquad D_{\min}(m,E,\varepsilon)=\min\{D:V_{m,D}(E)-B_m(E)\le\varepsilon\}.$ The tolerance $\varepsilon$ is a total conditional-entropy excess in nats. It is not a measurement-error probability. Classical side information satisfies $H_\mu(Z|Y)\ge m$ ; entangled quantum side information can go below this benchmark. The resource $D$ is a Hilbert-space dimension, corresponding to $\lceil\log_2D\rceil$ qubits. Precision selects a photon-number hierarchy. Along arbitrary joint limits $m\to\infty$ , $E\to0$ , with no prescribed relation between $E$ and $1/m$ , $\frac{\varepsilon}{E}\to c>0\quad\Longrightarrow\quad D_{\min}\sim e^{-c}m.$ For every fixed integer $K\ge2$ , $\frac{\varepsilon}{E^K}\to c\in(0,1/K!)\quad\Longrightarrow\quad D_{\min}\sim(1-K!c)\frac{m^K}{K!}.$ Although higher-photon events become rarer, they occupy increasingly many orthogonal mode configurations. Resolving their entropy contribution therefore requires a larger memory dimension. In qubits, the fixed- $K$ law is $K\log_2m+O(1)$ . Critical precisions resolve costs beyond sector counting. At the exact tolerance $E^2/2$ , the complete vacuum–one-photon head is not sufficient at leading dimension: $D_{\min}(m,E,E^2/2)=m+1+(1+o(1))\left(\frac m2+\frac{Em^2}{6}\right).$ The error is relative to the additional dimension in parentheses. Both terms come from the entropy of the mean-calibrated finite head, together with an all-state output-entropy comparison. At the next critical tolerance $E^3/6$ , the paper determines an added-to-removed-level transition near the complete two-photon head $\binom{m+2}{2}$ , including the boundary near $mE=4$ and precisely qualified integer conclusions. An exact finite-energy source. For every finite $m$ , $2\le D\le m+1$ , and $0<E\le10^{-4}$ , let $r=D-1$ and choose orthonormal one-photon modes $u_1,\ldots,u_r$ . The preparation $|\Psi\rangle=\sqrt{1-E}|0\rangle_A|0\rangle_B+\sqrt{E/r}\sum_{j=1}^{r}|u_j\rangle_A|j\rangle_B$ attains the original global minimum over the full preparation class. Its vacuum–one-photon support is a conclusion of the optimization, rather than an imposed photon cutoff. The paper also characterizes exact small-energy optima for every fixed complete photon head. A record-reconstruction interpretation with explicit resource accounting. A common finite heterodyne digitizer, including a genuine overflow symbol, approximates the conditional entropy uniformly over all legal preparations. For independent repetitions of a known preparation, classical–quantum Slepian–Wolf coding identifies the asymptotic reconstruction rate with $H(X|B)$ . The uniform comparison transfers the hierarchy to the excess optimized communication rate relative to unlimited finite memory using the same digitizer. Preserving the first critical formula requires digitization error $\xi=o(E^2/m+E^3)$ ; the corresponding error scales for the separated second-critical regimes are also given. The decoder may act collectively after receiving the classical message. For $n$ independent preparations, the supplied side-system dimension is at most $D^n$ and total mean optical energy is at most $nE$ . This is a supplied-memory resource law, not an optimization over arbitrary correlated block preparations, a minimum Schumacher-compressed storage rate, or a bound on peak decoder workspace. Coding limits are taken at fixed finite parameters before the low-energy, growing-mode limit. The work develops the low-photon, growing-mode regime of the preparation problem studied in Optimal entanglement weights for heterodyne uncertainty with finite quantum memory (doi:10.5281/zenodo.23647071). The integrated 29-page paper contains the complete proofs, three figures and two tables.How much quantum memory is needed to retain the contributions of rare photon events to a measurement record? This paper determines the memory scales required to approach the best heterodyne conditional entropy when the total optical energy tends to zero. Its resource laws distinguish precisions for which counting photon configurations suffices from critical precisions at which energy-calibrated entropy corrections determine additional memory costs. Full preparation model and reference value. The optical system has $m$ bosonic modes, total mean photon number at most $E$, and an entire accessible quantum side system $B$ of dimension at most $D$, including retained preparation labels. The optical source and its correlations with the memory are optimized over all normal mixed states; arbitrary coherences and infinite photon tails are allowed. With natural logarithms and heterodyne measure $d\mu=\pi^{-m}d^{2m}z$, define \[V_{m,D}(E)=\min_{\rho_{AB}}H_\mu(Z|B),\qquad H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B).\] Here $Z$ is the heterodyne outcome and $\chi$ is Holevo information. The unlimited-memory optimum at the same energy is \[B_m(E)=m-E\ln(1+m/E),\qquad D_{\min}(m,E,\varepsilon)=\min\{D:V_{m,D}(E)-B_m(E)\le\varepsilon\}.\] The tolerance $\varepsilon$ is a total conditional-entropy excess in nats. It is not a measurement-error probability. Classical side information satisfies $H_\mu(Z|Y)\ge m$; entangled quantum side information can go below this benchmark. The resource $D$ is a Hilbert-space dimension, corresponding to $\lceil\log_2D\rceil$ qubits. Precision selects a photon-number hierarchy. Along arbitrary joint limits $m\to\infty$, $E\to0$, with no prescribed relation between $E$ and $1/m$, \[\frac{\varepsilon}{E}\to c>0\quad\Longrightarrow\quad D_{\min}\sim e^{-c}m.\] For every fixed integer $K\ge2$, \[\frac{\varepsilon}{E^K}\to c\in(0,1/K!)\quad\Longrightarrow\quad D_{\min}\sim(1-K!c)\frac{m^K}{K!}.\] Although higher-photon events become rarer, they occupy increasingly many orthogonal mode configurations. Resolving their entropy contribution therefore requires a larger memory dimension. In qubits, the fixed-$K$ law is $K\log_2m+O(1)$. Critical precisions resolve costs beyond sector counting. At the exact tolerance $E^2/2$, the complete vacuum–one-photon head is not sufficient at leading dimension: \[D_{\min}(m,E,E^2/2)=m+1+(1+o(1))\left(\frac m2+\frac{Em^2}{6}\right).\] The error is relative to the additional dimension in parentheses. Both terms come from the entropy of the mean-calibrated finite head, together with an all-state output-entropy comparison. At the next critical tolerance $E^3/6$, the paper determines an added-to-removed-level transition near the complete two-photon head $\binom{m+2}{2}$, including the boundary near $mE=4$ and precisely qualified integer conclusions. An exact finite-energy source. For every finite $m$, $2\le D\le m+1$, and $0<E\le10^{-4}$, let $r=D-1$ and choose orthonormal one-photon modes $u_1,\ldots,u_r$. The preparation \[|\Psi\rangle=\sqrt{1-E}|0\rangle_A|0\rangle_B+\sqrt{E/r}\sum_{j=1}^{r}|u_j\rangle_A|j\rangle_B\] attains the original global minimum over the full preparation class. Its vacuum–one-photon support is a conclusion of the optimization, rather than an imposed photon cutoff. The paper also characterizes exact small-energy optima for every fixed complete photon head. A record-reconstruction interpretation with explicit resource accounting. A common finite heterodyne digitizer, including a genuine overflow symbol, approximates the conditional entropy uniformly over all legal preparations. For independent repetitions of a known preparation, classical–quantum Slepian–Wolf coding identifies the asymptotic reconstruction rate with $H(X|B)$. The uniform comparison transfers the hierarchy to the excess optimized communication rate relative to unlimited finite memory using the same digitizer. Preserving the first critical formula requires digitization error $\xi=o(E^2/m+E^3)$; the corresponding error scales for the separated second-critical regimes are also given. The decoder may act collectively after receiving the classical message. For $n$ independent preparations, the supplied side-system dimension is at most $D^n$ and total mean optical energy is at most $nE$. This is a supplied-memory resource law, not an optimization over arbitrary correlated block preparations, a minimum Schumacher-compressed storage rate, or a bound on peak decoder workspace. Coding limits are taken at fixed finite parameters before the low-energy, growing-mode limit. The work develops the low-photon, growing-mode regime of the preparation problem studied in Optimal entanglement weights for heterodyne uncertainty with finite quantum memory (doi:10.5281/zenodo.23647071). The integrated 29-page paper contains the complete proofs, three figures and two tables.

Oct 25, 20260 citations

Memory capacity and lifetime thresholds for testing pure Gaussianity

Zixuan He

How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on $m$ bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least $1/32$ from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most $D$ , while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability $a$ , or is erased with a classical flag. The quantity $k_a(D,m)$ is the smallest deterministic original-preparation cap achieving worst-case error at most $1/3$ . Failed preparations, discarded trials and reserved positions are all charged. A retained register of $q$ qubits has $D=2^q$ . An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is $\Theta(m)$ . Along arbitrary growing-mode sequences, the precise criterion for improving that order is $k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty.$ The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when $a=O(1/m)$ , yet a survival probability tending to zero can still permit an advantage. Let $A=\max\{1,m^2/D\}$ . At the fixed energy and separation above, constants $c,C>0$ , independent of $m,D,a$ , give $c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1.$ At $a=0$ , the sample cost is $\Theta(m)$ . For ideal memory the complete rate is $k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right).$ With loss, the bounds match whenever $A\ge\min\{a^{-1},am^{2/3}\}$ . For example, $a=m^{-1/6}$ and $D=\lceil m^{11/6}\rceil$ give $\Theta(m^{1/3})$ preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for $D\ge m^2$ , $a=m^{-1/2}$ , the stated bounds are $\Omega(m^{1/2})$ and $O(m^{2/3})$ . Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order $m^2$ labels, only a positive register of dimension $m+1$ is retained. The three-visit route requires $D\ge m+1$ and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition $\mathcal N=a\Psi+(1-a)\mathcal R_\tau$ , where $\Psi$ is CPTP and $\mathcal R_\tau(X)=\operatorname{Tr}(X)\tau$ , an exact flagged-erasure simulation gives $k_{\mathcal N}(D,m)\ge k_a(D,m)$ . The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction $1-q=O(1/m)$ therefore forces $\Theta(m)$ samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on $m$ bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least $1/32$ from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most $D$, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability $a$, or is erased with a classical flag. The quantity $k_a(D,m)$ is the smallest deterministic original-preparation cap achieving worst-case error at most $1/3$. Failed preparations, discarded trials and reserved positions are all charged. A retained register of $q$ qubits has $D=2^q$. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is $\Theta(m)$. Along arbitrary growing-mode sequences, the precise criterion for improving that order is \[ k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty. \] The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when $a=O(1/m)$, yet a survival probability tending to zero can still permit an advantage. Let $A=\max\{1,m^2/D\}$. At the fixed energy and separation above, constants $c,C>0$, independent of $m,D,a$, give \[ c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1. \] At $a=0$, the sample cost is $\Theta(m)$. For ideal memory the complete rate is \[ k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right). \] With loss, the bounds match whenever $A\ge\min\{a^{-1},am^{2/3}\}$. For example, $a=m^{-1/6}$ and $D=\lceil m^{11/6}\rceil$ give $\Theta(m^{1/3})$ preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for $D\ge m^2$, $a=m^{-1/2}$, the stated bounds are $\Omega(m^{1/2})$ and $O(m^{2/3})$. Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order $m^2$ labels, only a positive register of dimension $m+1$ is retained. The three-visit route requires $D\ge m+1$ and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition $\mathcal N=a\Psi+(1-a)\mathcal R_\tau$, where $\Psi$ is CPTP and $\mathcal R_\tau(X)=\operatorname{Tr}(X)\tau$, an exact flagged-erasure simulation gives $k_{\mathcal N}(D,m)\ge k_a(D,m)$. The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction $1-q=O(1/m)$ therefore forces $\Theta(m)$ samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.

Oct 25, 20260 citations

Size and reliability limits of quantum memory for heterodyne uncertainty

Zixuan He

Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has $m$ modes and total mean photon number at most $E$ . Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most $d$ . Independently, the entire register survives with probability $1-p$ or is erased into an orthogonal readable failure state. The physical output therefore has $d+1$ levels; a total $q$ -qubit register permits $d\le2^q-1$ . Every event is retained, with no postselection. For heterodyne measure $d^{2m}z/\pi^m$ and natural logarithms, the exact all-preparation reduction is $V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.$ Here $W$ is Wehrl entropy, $S$ is von Neumann entropy and $N$ is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define $g(x)=(1+x)\ln(1+x)-x\ln x$ , $S_*=m g(E/m)$ , the ideal unlimited-memory value $B_m(E)=m-E\ln(1+m/E)$ , and $a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}$ . For any target excess satisfying $0<\varepsilon<a_ES_*$ , $p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.$ A finite register reaches $V_{\rm er}\le B_m(E)+\varepsilon$ if and only if $p<p_{\rm crit}$ . At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let $S_d$ be the largest entropy of any rank-at-most- $d$ optical state within the same photon budget, and $\Delta_d=S_*-S_d$ . For $0\le p<a_E$ , $pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.$ The first contribution survives at unlimited dimension, whereas $\Delta_d$ tends to zero as storage grows. An exact finite Gibbs program computes $S_d$ , including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At $0<E\le10^{-4}$ , $2\le d\le m+1$ and $p<1/8$ , an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has $m$ modes and total mean photon number at most $E$. Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most $d$. Independently, the entire register survives with probability $1-p$ or is erased into an orthogonal readable failure state. The physical output therefore has $d+1$ levels; a total $q$-qubit register permits $d\le2^q-1$. Every event is retained, with no postselection. For heterodyne measure $d^{2m}z/\pi^m$ and natural logarithms, the exact all-preparation reduction is \[V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.\] Here $W$ is Wehrl entropy, $S$ is von Neumann entropy and $N$ is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define $g(x)=(1+x)\ln(1+x)-x\ln x$, $S_*=m g(E/m)$, the ideal unlimited-memory value $B_m(E)=m-E\ln(1+m/E)$, and $a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}$. For any target excess satisfying $0<\varepsilon<a_ES_*$, \[p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.\] A finite register reaches $V_{\rm er}\le B_m(E)+\varepsilon$ if and only if $p<p_{\rm crit}$. At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let $S_d$ be the largest entropy of any rank-at-most-$d$ optical state within the same photon budget, and $\Delta_d=S_*-S_d$. For $0\le p<a_E$, \[pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.\] The first contribution survives at unlimited dimension, whereas $\Delta_d$ tends to zero as storage grows. An exact finite Gibbs program computes $S_d$, including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At $0<E\le10^{-4}$, $2\le d\le m+1$ and $p<1/8$, an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.

Oct 26, 20260 citations

The memory cost of quantum illumination

Zixuan He

Quantum illumination uses a retained idler to improve detection of a weak optical return in thermal noise. For a source designer, the practical resource question is how much ideal quantum storage is needed for a specified improvement. This work gives the exact asymptotic answer when the signal-photon budget and the total retained-register dimension are constrained independently. Exact finite-memory frontier Consider a thermal attenuator with transmissivity $0<\eta<1$ and vacuum-output background occupation $b>0$ , tested against the replacer that returns the same thermal state. For $n$ parallel uses, every admissible probe obeys a mean total signal-photon ceiling $nN$ and a strict bound $\dim R\le\lfloor e^{nP}\rfloor$ on the entire retained register, including retained source records. The memory rate is $P/\ln2$ ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is $\Xi(N,P)=\eta\ln(1+1/b)N+G\!\left(\min\{P,g(N)\}\right).$ Here all logarithms are natural, $g(x)=(x+1)\ln(x+1)-x\ln x$ , with $g(0)=0$ , and $G(p)=g(b)+p-F\!\left(g^{-1}(p)\right).$ The function $F(x)$ is the entropy of the full complementary output for a thermal input of mean occupation $x$ ; the complement includes the purification of the thermal environment. The first term in $\Xi$ is the optimum without retained memory, and $G$ is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are $P_{\mathrm{sat}}(N)=g(N),\qquad G'(0^+)=\frac{\eta}{1+b}.$ Because $G$ is strictly increasing and strictly concave, at a positive signal budget $N>0$ the minimum rate retaining a fixed fraction $0<r<1$ of the full additive memory benefit is $P_r(N)=G^{-1}\!\left(rG(g(N))\right)<r\,g(N).$ At low brightness, $P_r(N)\sim r g(N)$ . At high brightness, this fixed-fraction storage cost remains bounded while the exact-saturation cost diverges. The relative detection benefit simultaneously tends to zero: saving a large fraction of the saturation storage does not imply a large improvement over the total no-memory exponent. Connection to optical communication For the same thermal channel, let $C(N,P)$ denote the classical capacity under the corresponding photon budget and a strict initial maximally entangled Schmidt-rank supply. The two separately defined tasks satisfy $\Xi(N,P)-\Xi(N,0)=C(N,P)-C(N,0).$ This identity equates optimized incremental values, not the physical resources or protocols. The manuscript explicitly attributes the shared Gaussian-reference contraction and entropy-support tools to the capacity analysis. Its detection-specific contribution is the original-register mixed-probe converse, strict finite-register attainment and operational testing limit. Scope. The result concerns parallel, phase-known thermal attenuation, ideal retained storage and the vanishing-miss Stein criterion. It is an asymptotic resource benchmark rather than a finite-sample receiver or a noisy-memory optimum. The integrated article includes the complete proofs and two figure groups showing both storage cost and the value obtained from it.Quantum illumination uses a retained idler to improve detection of a weak optical return in thermal noise. For a source designer, the practical resource question is how much ideal quantum storage is needed for a specified improvement. This work gives the exact asymptotic answer when the signal-photon budget and the total retained-register dimension are constrained independently. Exact finite-memory frontier Consider a thermal attenuator with transmissivity $0<\eta<1$ and vacuum-output background occupation $b>0$, tested against the replacer that returns the same thermal state. For $n$ parallel uses, every admissible probe obeys a mean total signal-photon ceiling $nN$ and a strict bound $\dim R\le\lfloor e^{nP}\rfloor$ on the entire retained register, including retained source records. The memory rate is $P/\ln2$ ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is \[\Xi(N,P)=\eta\ln(1+1/b)N+G\!\left(\min\{P,g(N)\}\right).\] Here all logarithms are natural, $g(x)=(x+1)\ln(x+1)-x\ln x$, with $g(0)=0$, and \[G(p)=g(b)+p-F\!\left(g^{-1}(p)\right).\] The function $F(x)$ is the entropy of the full complementary output for a thermal input of mean occupation $x$; the complement includes the purification of the thermal environment. The first term in $\Xi$ is the optimum without retained memory, and $G$ is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are \[P_{\mathrm{sat}}(N)=g(N),\qquad G'(0^+)=\frac{\eta}{1+b}.\] Because $G$ is strictly increasing and strictly concave, at a positive signal budget $N>0$ the minimum rate retaining a fixed fraction $0<r<1$ of the full additive memory benefit is \[P_r(N)=G^{-1}\!\left(rG(g(N))\right)<r\,g(N).\] At low brightness, $P_r(N)\sim r g(N)$. At high brightness, this fixed-fraction storage cost remains bounded while the exact-saturation cost diverges. The relative detection benefit simultaneously tends to zero: saving a large fraction of the saturation storage does not imply a large improvement over the total no-memory exponent. Connection to optical communication For the same thermal channel, let $C(N,P)$ denote the classical capacity under the corresponding photon budget and a strict initial maximally entangled Schmidt-rank supply. The two separately defined tasks satisfy \[\Xi(N,P)-\Xi(N,0)=C(N,P)-C(N,0).\] This identity equates optimized incremental values, not the physical resources or protocols. The manuscript explicitly attributes the shared Gaussian-reference contraction and entropy-support tools to the capacity analysis. Its detection-specific contribution is the original-register mixed-probe converse, strict finite-register attainment and operational testing limit. Scope. The result concerns parallel, phase-known thermal attenuation, ideal retained storage and the vanishing-miss Stein criterion. It is an asymptotic resource benchmark rather than a finite-sample receiver or a noisy-memory optimum. The integrated article includes the complete proofs and two figure groups showing both storage cost and the value obtained from it.

Oct 20, 20260 citations

Optimal entanglement weights for heterodyne uncertainty with finite quantum memory

Zixuan He

A finite quantum memory is specified by more than its entanglement entropy. This paper determines how to shape its Schmidt weights and allocate optical photons to minimize the quantum conditional uncertainty of a prescribed heterodyne measurement. The resulting resource laws let researchers compare a uniformly weighted entangled source with a shaped source, and assess when sharing one memory across optical modes is more efficient than independently preparing mode–memory pairs. Complete preparation model and sharp leading resource law. Let $V_{m,D}(E)$ be the minimum of $H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B)$ over all normal, possibly mixed optical–memory states with total mean photon number at most $E$ and total accessible memory dimension at most $D$ . Here $Z$ is the outcome of an $m$ -mode heterodyne measurement, $B$ is the retained quantum memory, $\chi$ is Holevo information, and $d\mu=\pi^{-m}d^{2m}z$ . Accessible flags count toward the same memory budget. For every fixed $m$ and every joint limit $D,E\to\infty$ , $V_{m,D}(E)\sim\frac{\Lambda_m(E/T_D)}{T_D},\qquad T_D=(m!D)^{1/m}.$ The function $\Lambda_m$ is an explicitly characterized constrained radial ground-state energy. Its Kummer and Bessel profiles give legal attaining preparations. The converse retains arbitrary coherent inputs and their photon tails, rather than optimizing only a diagonal ansatz. Why shaping changes the resource exponent. At the common photon ceiling $E=T_D$ , optimally shaped sources have uncertainty of order $D^{-1/m}$ , whereas final flat Schmidt spectra have optimal order $D^{-1/(2m)}$ , even after optimizing the occupied optical subspace. Flat weights create a sharp support boundary; tapering the weights replaces its boundary-layer cost by a smaller gradient cost. To reach a small target uncertainty $\varepsilon$ , this changes the leading memory requirement from $2m\log_2(1/\varepsilon)$ to $m\log_2(1/\varepsilon)$ , and the attainable photon scaling from quadratic to linear in $1/\varepsilon$ . This improvement is compatible with nearly maximal entanglement entropy. For the explicit canonical saturated construction, $S(\rho_{B,D})=\ln D-c_m+o(1),\qquad \|\rho_{B,D}-I_D/D\|_1\longrightarrow L_m>0.$ The full trace-norm limits are approximately $L_1=0.8771$ and $L_2=1.1170$ ; the half-trace-distance convention halves these numbers. Thus a vanishing relative entropy deficit does not make the optimized spectrum uniform. Collective preparation at fully counted resources. For two modes in the energy-sufficient asymptotic regime, sharing quantum memory reduces the leading conditional entropy by about $10.24\%$ relative to the optimized independent mode–memory preparation class. That comparator includes local state optimization, integer memory and energy allocations, mixtures and charged accessible flags; it is not the class of all separable joint states. A separate finite construction uses seven coherent optical inputs, three retained qubits and $0.21$ input photons. Under the specified loss, phase and stochastic memory-fault contract, its retained photon budget is at most $0.200781$ , and its conditional entropy is strictly lower than that of the same independent comparator by more than $0.00547535$ nats. The integrated article gives the preparation circuit, one-sided numerical bounds and complete same-source error accounting. Interpretation. The objective uses quantum side information. A classical measurement of the memory generally has accessible information below $\chi(Z:B)$ , so the conditional-entropy advantages do not by themselves assert the same improvement in estimation precision or classical readout performance. The finite circuit is a rigorously specified source model, not a qualification of an implemented laboratory apparatus. The single PDF contains the article, five figures, full proofs and references.A finite quantum memory is specified by more than its entanglement entropy. This paper determines how to shape its Schmidt weights and allocate optical photons to minimize the quantum conditional uncertainty of a prescribed heterodyne measurement. The resulting resource laws let researchers compare a uniformly weighted entangled source with a shaped source, and assess when sharing one memory across optical modes is more efficient than independently preparing mode–memory pairs. Complete preparation model and sharp leading resource law. Let $V_{m,D}(E)$ be the minimum of $H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B)$ over all normal, possibly mixed optical–memory states with total mean photon number at most $E$ and total accessible memory dimension at most $D$. Here $Z$ is the outcome of an $m$-mode heterodyne measurement, $B$ is the retained quantum memory, $\chi$ is Holevo information, and $d\mu=\pi^{-m}d^{2m}z$. Accessible flags count toward the same memory budget. For every fixed $m$ and every joint limit $D,E\to\infty$, \[V_{m,D}(E)\sim\frac{\Lambda_m(E/T_D)}{T_D},\qquad T_D=(m!D)^{1/m}.\] The function $\Lambda_m$ is an explicitly characterized constrained radial ground-state energy. Its Kummer and Bessel profiles give legal attaining preparations. The converse retains arbitrary coherent inputs and their photon tails, rather than optimizing only a diagonal ansatz. Why shaping changes the resource exponent. At the common photon ceiling $E=T_D$, optimally shaped sources have uncertainty of order $D^{-1/m}$, whereas final flat Schmidt spectra have optimal order $D^{-1/(2m)}$, even after optimizing the occupied optical subspace. Flat weights create a sharp support boundary; tapering the weights replaces its boundary-layer cost by a smaller gradient cost. To reach a small target uncertainty $\varepsilon$, this changes the leading memory requirement from $2m\log_2(1/\varepsilon)$ to $m\log_2(1/\varepsilon)$, and the attainable photon scaling from quadratic to linear in $1/\varepsilon$. This improvement is compatible with nearly maximal entanglement entropy. For the explicit canonical saturated construction, \[S(\rho_{B,D})=\ln D-c_m+o(1),\qquad \|\rho_{B,D}-I_D/D\|_1\longrightarrow L_m>0.\] The full trace-norm limits are approximately $L_1=0.8771$ and $L_2=1.1170$; the half-trace-distance convention halves these numbers. Thus a vanishing relative entropy deficit does not make the optimized spectrum uniform. Collective preparation at fully counted resources. For two modes in the energy-sufficient asymptotic regime, sharing quantum memory reduces the leading conditional entropy by about $10.24\%$ relative to the optimized independent mode–memory preparation class. That comparator includes local state optimization, integer memory and energy allocations, mixtures and charged accessible flags; it is not the class of all separable joint states. A separate finite construction uses seven coherent optical inputs, three retained qubits and $0.21$ input photons. Under the specified loss, phase and stochastic memory-fault contract, its retained photon budget is at most $0.200781$, and its conditional entropy is strictly lower than that of the same independent comparator by more than $0.00547535$ nats. The integrated article gives the preparation circuit, one-sided numerical bounds and complete same-source error accounting. Interpretation. The objective uses quantum side information. A classical measurement of the memory generally has accessible information below $\chi(Z:B)$, so the conditional-entropy advantages do not by themselves assert the same improvement in estimation precision or classical readout performance. The finite circuit is a rigorously specified source model, not a qualification of an implemented laboratory apparatus. The single PDF contains the article, five figures, full proofs and references.

Oct 25, 20260 citations

How much quantum memory is needed to restore an unknown optical pulse together with its correlations? This paper gives matching storage laws showing that the answer depends on the actual joint preparation, even when the optical marginal is a positive mixture of coherent states. Fix an optical mode number (m), a known real (k)-dimensional displacement subspace with (1\le k\le2m), and a positive mean photon budget (U_0). The source is any probability mixture of coherent states in that subspace with actual mean photons at most (U_0). One normal encoder and decoder must work for the entire unknown source class, and every transmitted message or flag counts toward the same register dimension. The reference is inaccessible to both maps. Let (\delta) be the required half trace-distance error and (q_{\min}(\delta)) the least encoded qubit count. As (\delta\downarrow0), [q_{\min}(\delta)=\begin{cases}\dfrac{k}{2}\log_2(1/\delta)+O(1),&\text{optical marginal or joint coherent preparation},\k\log_2(1/\delta)+O(1),&\text{arbitrary reference extension or classical-bit reference}.\end{cases}] A single inaccessible classical bit already witnesses the larger leading cost. To recover the smaller cost for correlations, the joint source must actually be prepared as (\omega_{AB}=\int |Jx\rangle\langle Jx|\otimes\tau_B(x),\mathrm d\mu(x)), with (\int|x|^2\mathrm d\mu\le U_0). Here (J) is the known displacement embedding and (\tau_B(x)) may be noncommuting normal reference states. The constants are uniform in the reference dimension and the orientation of (J); the codec may depend on the known (J). The mechanism is the difference between the probability of a rare bright component and the square-root amplitude of its retained correlations. Exact optical Gram matrices and a reference-coherence witness establish the converse for arbitrary normal decoders, including infinite-rank outputs. A common finite optical subspace supplies the attaining compression schemes while retaining the full photon tails. These are single-input universal memory-dimension laws at fixed (m,k,U_0), with every communication resource counted. They complement established known-source and many-copy compression theory. The integrated 10-page PDF contains two figures, complete proofs and references.

A

is prepared with its actual purifier

C

. A normal encoder acts on

C

to produce memory

M

, while a prescribed Gaussian channel maps

A

to the relevant optical output

B

. The source-information cost and useful correlation are

r=I(A:M)

and

v=I(B:M)

, respectively. All logarithms are natural. The paper determines

C_{\mathrm Q}(R)=\sup_{\mathcal N:\,I(A:M)\le R}I(B:M).

For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number

n

and a thermal attenuator with transmissivity

0<\eta<1

, write the output occupation as

\eta n+b

, where

b\ge0

is the environment's added thermal occupation. With

g(x)=(x+1)\log(x+1)-x\log x

, the exact optimum over all normal classical records is

C_{\mathrm{cl}}(n,R)=g(\eta n+b)-g(\eta u_R+b),\qquad u_R=g^{-1}\!\bigl(\max\{g(n)-R,0\}\bigr).

A legal encoder applies vacuum loss with transmissivity

\lambda

to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed

b>0

and

0<\lambda<1/2

, at the common budget

R_n=g(n)+g(\lambda n)-g((1-\lambda)n)

, its actual relevance satisfies

v_2=\frac{\eta\lambda}{1+b}\,n\log(1/n)+O(n),\qquad C_{\mathrm{cl}}(n,R_n)=2\lambda\eta\log(1+1/b)\,n+O\!\left(\frac{n}{\log(1/n)}\right).

The full optical tail is included in the resource accounting. At

n=0.01

\eta=b=0.5

\lambda=0.2

and

R=0.023875

, the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For

k

uses, the retained predictive memory obeys

\dim K_k\le2^k

and

I(A^k:K_k)\le kR

, including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let

\alpha_k

be the missed-correlation probability and

\beta_k

the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is

\alpha_k\le\epsilon\quad\Longrightarrow\quad \beta_k\ge\exp\!\left[-\frac{k c_nR+\log2}{1-\epsilon}\right],\qquad c_n=\eta\frac{\log(1+1/(\eta n+b))}{\log(1+1/n)}.

At the finite parameters above and

\epsilon=0.1

, the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.

m

bosonic modes, total mean photon number at most

E

, and an entire accessible quantum side system

B

of dimension at most

D

, including retained preparation labels. The optical source and its correlations with the memory are optimized over all normal mixed states; arbitrary coherences and infinite photon tails are allowed. With natural logarithms and heterodyne measure

d\mu=\pi^{-m}d^{2m}z

, define

V_{m,D}(E)=\min_{\rho_{AB}}H_\mu(Z|B),\qquad H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B).

Here

Z

is the heterodyne outcome and

\chi

is Holevo information. The unlimited-memory optimum at the same energy is

B_m(E)=m-E\ln(1+m/E),\qquad D_{\min}(m,E,\varepsilon)=\min\{D:V_{m,D}(E)-B_m(E)\le\varepsilon\}.

The tolerance

\varepsilon

is a total conditional-entropy excess in nats. It is not a measurement-error probability. Classical side information satisfies

H_\mu(Z|Y)\ge m

; entangled quantum side information can go below this benchmark. The resource

D

is a Hilbert-space dimension, corresponding to

\lceil\log_2D\rceil

qubits. Precision selects a photon-number hierarchy. Along arbitrary joint limits

m\to\infty

E\to0

, with no prescribed relation between

E

and

1/m

\frac{\varepsilon}{E}\to c>0\quad\Longrightarrow\quad D_{\min}\sim e^{-c}m.

For every fixed integer

K\ge2

\frac{\varepsilon}{E^K}\to c\in(0,1/K!)\quad\Longrightarrow\quad D_{\min}\sim(1-K!c)\frac{m^K}{K!}.

Although higher-photon events become rarer, they occupy increasingly many orthogonal mode configurations. Resolving their entropy contribution therefore requires a larger memory dimension. In qubits, the fixed-

K

law is

K\log_2m+O(1)

. Critical precisions resolve costs beyond sector counting. At the exact tolerance

E^2/2

, the complete vacuum–one-photon head is not sufficient at leading dimension:

D_{\min}(m,E,E^2/2)=m+1+(1+o(1))\left(\frac m2+\frac{Em^2}{6}\right).

The error is relative to the additional dimension in parentheses. Both terms come from the entropy of the mean-calibrated finite head, together with an all-state output-entropy comparison. At the next critical tolerance

E^3/6

, the paper determines an added-to-removed-level transition near the complete two-photon head

\binom{m+2}{2}

, including the boundary near

mE=4

and precisely qualified integer conclusions. An exact finite-energy source. For every finite

m

2\le D\le m+1

, and

0<E\le10^{-4}

, let

r=D-1

and choose orthonormal one-photon modes

u_1,\ldots,u_r

. The preparation

|\Psi\rangle=\sqrt{1-E}|0\rangle_A|0\rangle_B+\sqrt{E/r}\sum_{j=1}^{r}|u_j\rangle_A|j\rangle_B

attains the original global minimum over the full preparation class. Its vacuum–one-photon support is a conclusion of the optimization, rather than an imposed photon cutoff. The paper also characterizes exact small-energy optima for every fixed complete photon head. A record-reconstruction interpretation with explicit resource accounting. A common finite heterodyne digitizer, including a genuine overflow symbol, approximates the conditional entropy uniformly over all legal preparations. For independent repetitions of a known preparation, classical–quantum Slepian–Wolf coding identifies the asymptotic reconstruction rate with

H(X|B)

. The uniform comparison transfers the hierarchy to the excess optimized communication rate relative to unlimited finite memory using the same digitizer. Preserving the first critical formula requires digitization error

\xi=o(E^2/m+E^3)

; the corresponding error scales for the separated second-critical regimes are also given. The decoder may act collectively after receiving the classical message. For

n

independent preparations, the supplied side-system dimension is at most

D^n

and total mean optical energy is at most

nE

. This is a supplied-memory resource law, not an optimization over arbitrary correlated block preparations, a minimum Schumacher-compressed storage rate, or a bound on peak decoder workspace. Coding limits are taken at fixed finite parameters before the low-energy, growing-mode limit. The work develops the low-photon, growing-mode regime of the preparation problem studied in Optimal entanglement weights for heterodyne uncertainty with finite quantum memory (doi:10.5281/zenodo.23647071). The integrated 29-page paper contains the complete proofs, three figures and two tables.

m

bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least

1/32

from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most

D

, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability

a

, or is erased with a classical flag. The quantity

k_a(D,m)

is the smallest deterministic original-preparation cap achieving worst-case error at most

1/3

. Failed preparations, discarded trials and reserved positions are all charged. A retained register of

q

qubits has

D=2^q

. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is

\Theta(m)

. Along arbitrary growing-mode sequences, the precise criterion for improving that order is

k_a(D,m)=o(m)\quad\Longleftrightarrow\quad am\to\infty\ \text{and}\ \frac{aD}{m}\to\infty.

The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when

a=O(1/m)

, yet a survival probability tending to zero can still permit an advantage. Let

A=\max\{1,m^2/D\}

. At the fixed energy and separation above, constants

c,C>0

, independent of

m,D,a

, give

c\min\{m,A/a\}\le k_a(D,m)\le C\min\{m,A/a+\min(a^{-2},m^{2/3})\},\qquad 0<a\le1.

a=0

, the sample cost is

\Theta(m)

. For ideal memory the complete rate is

k_1(D,m)=\Theta\!\left(\min\{m,\max(1,m^2/D)\}\right).

With loss, the bounds match whenever

A\ge\min\{a^{-1},am^{2/3}\}

. For example,

a=m^{-1/6}

and

D=\lceil m^{11/6}\rceil

give

\Theta(m^{1/3})

preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for

D\ge m^2

a=m^{-1/2}

, the stated bounds are

\Omega(m^{1/2})

and

O(m^{2/3})

. Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order

m^2

labels, only a positive register of dimension

m+1

is retained. The three-visit route requires

D\ge m+1

and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition

\mathcal N=a\Psi+(1-a)\mathcal R_\tau

, where

\Psi

is CPTP and

\mathcal R_\tau(X)=\operatorname{Tr}(X)\tau

, an exact flagged-erasure simulation gives

k_{\mathcal N}(D,m)\ge k_a(D,m)

. The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction

1-q=O(1/m)

therefore forces

\Theta(m)

samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.

m

modes and total mean photon number at most

E

. Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most

d

. Independently, the entire register survives with probability

1-p

or is erased into an orthogonal readable failure state. The physical output therefore has

d+1

levels; a total

q

-qubit register permits

d\le2^q-1

. Every event is retained, with no postselection. For heterodyne measure

d^{2m}z/\pi^m

and natural logarithms, the exact all-preparation reduction is

V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.

Here

W

is Wehrl entropy,

S

is von Neumann entropy and

N

is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define

g(x)=(1+x)\ln(1+x)-x\ln x

S_*=m g(E/m)

, the ideal unlimited-memory value

B_m(E)=m-E\ln(1+m/E)

, and

a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}

. For any target excess satisfying

0<\varepsilon<a_ES_*

p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.

A finite register reaches

V_{\rm er}\le B_m(E)+\varepsilon

if and only if

p<p_{\rm crit}

. At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let

S_d

be the largest entropy of any rank-at-most-

d

optical state within the same photon budget, and

\Delta_d=S_*-S_d

. For

0\le p<a_E

pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.

The first contribution survives at unlimited dimension, whereas

\Delta_d

tends to zero as storage grows. An exact finite Gibbs program computes

S_d

, including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At

0<E\le10^{-4}

2\le d\le m+1

and

p<1/8

, an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.

0<\eta<1

and vacuum-output background occupation

b>0

, tested against the replacer that returns the same thermal state. For

n

parallel uses, every admissible probe obeys a mean total signal-photon ceiling

nN

and a strict bound

\dim R\le\lfloor e^{nP}\rfloor

on the entire retained register, including retained source records. The memory rate is

P/\ln2

ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is

\Xi(N,P)=\eta\ln(1+1/b)N+G\!\left(\min\{P,g(N)\}\right).

Here all logarithms are natural,

g(x)=(x+1)\ln(x+1)-x\ln x

, with

g(0)=0

, and

G(p)=g(b)+p-F\!\left(g^{-1}(p)\right).

The function

F(x)

is the entropy of the full complementary output for a thermal input of mean occupation

x

; the complement includes the purification of the thermal environment. The first term in

\Xi

is the optimum without retained memory, and

G

is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are

P_{\mathrm{sat}}(N)=g(N),\qquad G'(0^+)=\frac{\eta}{1+b}.

Because

G

is strictly increasing and strictly concave, at a positive signal budget

N>0

the minimum rate retaining a fixed fraction

0<r<1

of the full additive memory benefit is

P_r(N)=G^{-1}\!\left(rG(g(N))\right)<r\,g(N).

At low brightness,

P_r(N)\sim r g(N)

. At high brightness, this fixed-fraction storage cost remains bounded while the exact-saturation cost diverges. The relative detection benefit simultaneously tends to zero: saving a large fraction of the saturation storage does not imply a large improvement over the total no-memory exponent. Connection to optical communication For the same thermal channel, let

C(N,P)

denote the classical capacity under the corresponding photon budget and a strict initial maximally entangled Schmidt-rank supply. The two separately defined tasks satisfy

\Xi(N,P)-\Xi(N,0)=C(N,P)-C(N,0).

This identity equates optimized incremental values, not the physical resources or protocols. The manuscript explicitly attributes the shared Gaussian-reference contraction and entropy-support tools to the capacity analysis. Its detection-specific contribution is the original-register mixed-probe converse, strict finite-register attainment and operational testing limit. Scope. The result concerns parallel, phase-known thermal attenuation, ideal retained storage and the vanishing-miss Stein criterion. It is an asymptotic resource benchmark rather than a finite-sample receiver or a noisy-memory optimum. The integrated article includes the complete proofs and two figure groups showing both storage cost and the value obtained from it.

V_{m,D}(E)

be the minimum of

H_\mu(Z|B)=h_\mu(Z)-\chi(Z:B)

over all normal, possibly mixed optical–memory states with total mean photon number at most

E

and total accessible memory dimension at most

D

. Here

Z

is the outcome of an

m

-mode heterodyne measurement,

B

is the retained quantum memory,

\chi

is Holevo information, and

d\mu=\pi^{-m}d^{2m}z

. Accessible flags count toward the same memory budget. For every fixed

m

and every joint limit

D,E\to\infty

V_{m,D}(E)\sim\frac{\Lambda_m(E/T_D)}{T_D},\qquad T_D=(m!D)^{1/m}.

The function

\Lambda_m

is an explicitly characterized constrained radial ground-state energy. Its Kummer and Bessel profiles give legal attaining preparations. The converse retains arbitrary coherent inputs and their photon tails, rather than optimizing only a diagonal ansatz. Why shaping changes the resource exponent. At the common photon ceiling

E=T_D

, optimally shaped sources have uncertainty of order

D^{-1/m}

, whereas final flat Schmidt spectra have optimal order

D^{-1/(2m)}

, even after optimizing the occupied optical subspace. Flat weights create a sharp support boundary; tapering the weights replaces its boundary-layer cost by a smaller gradient cost. To reach a small target uncertainty

\varepsilon

, this changes the leading memory requirement from

2m\log_2(1/\varepsilon)

m\log_2(1/\varepsilon)

, and the attainable photon scaling from quadratic to linear in

1/\varepsilon

. This improvement is compatible with nearly maximal entanglement entropy. For the explicit canonical saturated construction,

S(\rho_{B,D})=\ln D-c_m+o(1),\qquad \|\rho_{B,D}-I_D/D\|_1\longrightarrow L_m>0.

The full trace-norm limits are approximately

L_1=0.8771

and

L_2=1.1170

; the half-trace-distance convention halves these numbers. Thus a vanishing relative entropy deficit does not make the optimized spectrum uniform. Collective preparation at fully counted resources. For two modes in the energy-sufficient asymptotic regime, sharing quantum memory reduces the leading conditional entropy by about

10.24\%

relative to the optimized independent mode–memory preparation class. That comparator includes local state optimization, integer memory and energy allocations, mixtures and charged accessible flags; it is not the class of all separable joint states. A separate finite construction uses seven coherent optical inputs, three retained qubits and

0.21

input photons. Under the specified loss, phase and stochastic memory-fault contract, its retained photon budget is at most

0.200781

, and its conditional entropy is strictly lower than that of the same independent comparator by more than

0.00547535

nats. The integrated article gives the preparation circuit, one-sided numerical bounds and complete same-source error accounting. Interpretation. The objective uses quantum side information. A classical measurement of the memory generally has accessible information below

\chi(Z:B)

, so the conditional-entropy advantages do not by themselves assert the same improvement in estimation precision or classical readout performance. The finite circuit is a rigorously specified source model, not a qualification of an implemented laboratory apparatus. The single PDF contains the article, five figures, full proofs and references.