The memory cost of quantum illumination

Zixuan He

Published Oct 20, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.23485507

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The memory cost of quantum illumination

Zixuan He

Published Oct 20, 2026• 0 citations

Publisher

Abstract

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.

DOI: 10.5281/zenodo.23485507

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Quantum limits and one-qubit advantage in optical predictive compression

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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

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

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

The quantum memory cost of optical correlations

Zixuan He

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.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.

Oct 29, 20260 citations

Brightness limits in quantum channel certification with coherent light

Zixuan He

How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least $\varepsilon$ away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget $E_s$ . This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys $\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p$ under every channel in the testing promise. The probe budget $E_p$ is distinct from the signal budget $E_s$ defining the certification guarantee. Optimal mean-energy query laws. For fixed positive $E_s,E_p$ , fixed finite mode number $m$ , and both testing errors at most $\delta$ , the minimax number of calls satisfies $T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),$ $T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.$ The bounds hold for sufficiently small $\varepsilon$ , uniformly over $0<\delta\le1/4$ . Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as $t\log t$ for one mode and as $t^m$ for fixed $m\ge2$ . A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which $\log T_{\mathrm G}=\Theta(\varepsilon^{-2})$ at fixed confidence. A fixed brightness cap gives $\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))$ . The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least $\varepsilon$ away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget $E_s$. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys \[\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p\] under every channel in the testing promise. The probe budget $E_p$ is distinct from the signal budget $E_s$ defining the certification guarantee. Optimal mean-energy query laws. For fixed positive $E_s,E_p$, fixed finite mode number $m$, and both testing errors at most $\delta$, the minimax number of calls satisfies \[T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),\] \[T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.\] The bounds hold for sufficiently small $\varepsilon$, uniformly over $0<\delta\le1/4$. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as $t\log t$ for one mode and as $t^m$ for fixed $m\ge2$. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which $\log T_{\mathrm G}=\Theta(\varepsilon^{-2})$ at fixed confidence. A fixed brightness cap gives $\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))$. The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.

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

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.

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

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.

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

, 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.

\varepsilon

away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget

E_s

. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys

\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p

under every channel in the testing promise. The probe budget

E_p

is distinct from the signal budget

E_s

defining the certification guarantee. Optimal mean-energy query laws. For fixed positive

E_s,E_p

, fixed finite mode number

m

, and both testing errors at most

\delta

, the minimax number of calls satisfies

T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),

T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.

The bounds hold for sufficiently small

\varepsilon

, uniformly over

0<\delta\le1/4

. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as

t\log t

for one mode and as

t^m

for fixed

m\ge2

. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which

\log T_{\mathrm G}=\Theta(\varepsilon^{-2})

at fixed confidence. A fixed brightness cap gives

\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))

. The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.

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.