Quantum Boltzmann Brain

Hutan Ashrafian

Published Nov 1, 2026• 0 citations

Abstract

A quantum Boltzmann brain would arise if a fluctuation assembled an observer whose information processing used quantum switches, in which the order of physical operations is coherently controlled. A comparison with conventional substrates must account for both the assigned computation and the probability that the assembled apparatus can sustain it. I develop this comparison for a network distinguishing unknown commuting from anticommuting operations. With one invocation of each operation, an exact certificate bounds single-task fixed-order success by 11/12 for a specified finite ensemble. With an answer committed before each fresh task, the bound extends to networks retaining arbitrary quantum memory and identifies reliability requirements that the switch can satisfy but the matched fixed-order architecture cannot. For a shared fluctuating phase, a finite execution schedule also gives a lower bound on completed-batch success, including the probability that the reliable episode persists; an explicit example exceeds the causal upper bound. The network’s reliability region links residual order information, tolerated errors and phase correlations to both occupation and duration. With a fixed error budget, a fixed number r of phase coordinates and inverse-size visibility loss below the feasibility threshold, occupation scales as N^{-r/2}, while Brownian noise imposes an exponential penalty for sustaining a fixed duration as the number of outputs N grows. For one common phase with finite frequency-correlation time, fixed survival quantiles instead scale as N^{-1/2}. Incorporating these operational probabilities into a common thermal formation model yields a crossover criterion for the assembly saving required to offset the loss of reliable operation. The resulting theory makes a coherent-order contribution to fluctuation-observer comparisons calculable, conditional on the complete apparatus, its noise law and the functional requirements assigned to an observer.

AI Summary

The paper asks how coherent control of operation order could affect comparisons between fluctuation-assembled observers, once both apparatus formation and reliable computation are included. It models a “quantum Boltzmann brain” using quantum switches to distinguish unknown commuting from anticommuting operations, rather than specifying a general model of cognition. For a specified finite ensemble and one invocation of each operation, an exact certificate bounds fixed-order single-task success by $11/12$. Requiring an answer before each fresh task extends the bound to networks with arbitrary retained quantum memory. This yields reliability requirements achievable by the switch but not by the matched fixed-order architecture. For a shared fluctuating phase, a finite execution schedule supplies a lower bound on completed-batch success that includes the probability that the reliable episode persists; an explicit example exceeds the causal upper bound. The noise analysis relates residual order information, error tolerance and phase correlations to the frequency and duration of reliable operation. With fixed error budget, $r$ phase coordinates and inverse-size visibility loss below a feasibility threshold, occupation scales as $N^{-r/2}$ for $N$ outputs. Brownian phase noise produces an exponential penalty for maintaining a fixed duration as $N$ grows; for one common phase with finite frequency-correlation time, fixed survival quantiles instead scale as $N^{-1/2}$. A common thermal formation model gives a crossover criterion for assembly savings to compensate for unreliable operation. These conclusions are conditional on the complete apparatus, noise law and assigned observer requirements. Only the abstract was available for this summary.

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DOI: 10.5281/zenodo.23862736

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Quantum Boltzmann Brain

Hutan Ashrafian

Published Nov 1, 2026• 0 citations

Publisher

Abstract

AI Summary

For a specified finite ensemble and one invocation of each operation, an exact certificate bounds fixed-order single-task success by

11/12

. Requiring an answer before each fresh task extends the bound to networks with arbitrary retained quantum memory. This yields reliability requirements achievable by the switch but not by the matched fixed-order architecture. For a shared fluctuating phase, a finite execution schedule supplies a lower bound on completed-batch success that includes the probability that the reliable episode persists; an explicit example exceeds the causal upper bound.

The noise analysis relates residual order information, error tolerance and phase correlations to the frequency and duration of reliable operation. With fixed error budget,

r

phase coordinates and inverse-size visibility loss below a feasibility threshold, occupation scales as

N^{-r/2}

for

N

outputs. Brownian phase noise produces an exponential penalty for maintaining a fixed duration as

N

grows; for one common phase with finite frequency-correlation time, fixed survival quantiles instead scale as

N^{-1/2}

A common thermal formation model gives a crossover criterion for assembly savings to compensate for unreliable operation. These conclusions are conditional on the complete apparatus, noise law and assigned observer requirements. Only the abstract was available for this summary.

No sign of AI-generated textUnclear noveltyStrong assumptions

Written by gpt-6.1-sol from the abstract only. Automated: it can be wrong.

DOI: 10.5281/zenodo.23862736

Related papers

Sufficient condition on noise correlations for scalable quantum computing

PreskillJohn

Quantum Information and ComputationMar 1, 20130 citations

Distinct stability laws for Wigner entropy and non-Gaussian structure

Zixuan He

Loss can restore the vacuum entropy bound without erasing non-Gaussian structure. This paper determines when an entropy statement justifies a coherent-state approximation, when non-Gaussian structure survives optical loss, and what mode and source-request budgets are needed to use that structure. The results give quantum-optics researchers separate criteria for entropy recovery, state approximation, Gaussian concentration and conversion to a specified output. Version 1.5 — revised 10 October 2026. The 96-page PDF contains the main article, Methods, Supplementary Information and references. This revision connects the exact loss boundary to a mode-uniform near-equality certificate, finite-data validation, a conversion-cost lower bound for the same thermally repaired source, and an exact equal-entropy comparison of sources with different conversion-cost exponents. The central figures are updated accordingly. 1. Near-vacuum states can have an unbounded entropy deficit Write $\nu_m$ for the $m$ -mode vacuum and define $\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)$ , using natural logarithms; $H_1$ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, $E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.$ Here $\mathcal F_m$ is the trace-norm-closed convex hull of Gaussian states and $d_{\mathcal F_m}$ uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives $-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0$ , so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity $\eta$ and any fixed positive total input-energy budget $E$ , the sharp threshold is $\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}$ With $\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}$ , $\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}$ The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For $\alpha\ge2$ , established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input $\rho$ , define the half-loss output Shannon excess $s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.$ The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields $\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).$ The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most $\epsilon$ is guaranteed by $s\le(\ln2)\epsilon^2/16$ . Conversely, a surviving output Gaussian-hull gap enforces $s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.$ Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation $c/m$ has total input energy tending to $3c$ . Along the recovery boundary, its attenuated state obeys $\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.$ At half transmission this lower bound is $g(c)=(c/8)e^{-7c/8}$ . Starting from one complete block, Gaussian processing requires $\Theta(m)$ output modes to retain any fixed positive gap below this bound at success probability at least a fixed $p_0>0$ . Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target $\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25$ within half trace distance $10^{-6}$ . Let $C_m$ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed $0<c\le1$ , every achieving Gaussian protocol obeys the asymptotic lower bound $\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).$ This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become $h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).$ The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy $E_A,E_B\sim s/2$ . At exactly equal energy, the two excesses instead share $s_i(E)=2E-(35/8)E^2+O(E^3)$ and differ by $s_A(E)-s_B(E)=15E^3+O(E^4)$ . Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound $\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K$ , trusted half loss and all-mode on/off detection with efficiency at least $\xi_{\min}$ , the click probability $q$ gives $s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.$ A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses $K=4$ , $\tau_K=0$ , $\xi=0.8$ and $N=3{,}517{,}336$ independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most $0.05$ with false-acceptance probability at most $0.001$ and missed acceptance at most $0.001$ on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at $E=1$ , $\alpha=1.9$ , $\eta=0.9505$ and $m=10^5$ , the certified bound $\Delta_\alpha<0.06942$ rules out a target deficit of $0.1$ nat. A separate finite single-mode preparation model predicts positive Shannon deficit near $\eta=0.97$ , negative deficit near $\eta=0.85$ , and a Gaussian-hull witness above $4.4\times10^{-3}$ at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put $\alpha=2-\delta$ and $L=\ln(1/\delta)$ . For the maximum deficit $D_{m,E}$ at fixed total energy, $D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),$ where $T_h$ is the first mode number allowing a fixed positive deficit $h$ . Beyond a sufficiently large fixed multiple of $L/\delta$ , $D_{m,E}(2-\delta)=\Theta_E(m\delta)$ . The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds: $h(W_\rho)\ge m\ln(2\pi)+\varepsilon_m$ with $\varepsilon_m>0$ for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With $\mathcal C_{r,k}$ denoting the ordering-weighted norm of photon-number-difference block $k$ , $\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.$ For vacuum– $n$ -photon coherence and $1<\alpha<2$ , $\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.$ This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.Loss can restore the vacuum entropy bound without erasing non-Gaussian structure. This paper determines when an entropy statement justifies a coherent-state approximation, when non-Gaussian structure survives optical loss, and what mode and source-request budgets are needed to use that structure. The results give quantum-optics researchers separate criteria for entropy recovery, state approximation, Gaussian concentration and conversion to a specified output. Version 1.5 — revised 10 October 2026. The 96-page PDF contains the main article, Methods, Supplementary Information and references. This revision connects the exact loss boundary to a mode-uniform near-equality certificate, finite-data validation, a conversion-cost lower bound for the same thermally repaired source, and an exact equal-entropy comparison of sources with different conversion-cost exponents. The central figures are updated accordingly. 1. Near-vacuum states can have an unbounded entropy deficit Write $\nu_m$ for the $m$-mode vacuum and define $\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)$, using natural logarithms; $H_1$ is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, $$E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.$$ Here $\mathcal F_m$ is the trace-norm-closed convex hull of Gaussian states and $d_{\mathcal F_m}$ uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives $-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0$, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity $\eta$ and any fixed positive total input-energy budget $E$, the sharp threshold is $$\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}$$ With $\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}$, $$\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}$$ The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For $\alpha\ge2$, established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input $\rho$, define the half-loss output Shannon excess $$s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.$$ The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields $$\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).$$ The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most $\epsilon$ is guaranteed by $s\le(\ln2)\epsilon^2/16$. Conversely, a surviving output Gaussian-hull gap enforces $$s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.$$ Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation $c/m$ has total input energy tending to $3c$. Along the recovery boundary, its attenuated state obeys $$\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.$$ At half transmission this lower bound is $g(c)=(c/8)e^{-7c/8}$. Starting from one complete block, Gaussian processing requires $\Theta(m)$ output modes to retain any fixed positive gap below this bound at success probability at least a fixed $p_0>0$. Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target $\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25$ within half trace distance $10^{-6}$. Let $C_m$ be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed $0<c\le1$, every achieving Gaussian protocol obeys the asymptotic lower bound $$\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).$$ This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become $$h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).$$ The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy $E_A,E_B\sim s/2$. At exactly equal energy, the two excesses instead share $s_i(E)=2E-(35/8)E^2+O(E^3)$ and differ by $s_A(E)-s_B(E)=15E^3+O(E^4)$. Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound $\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K$, trusted half loss and all-mode on/off detection with efficiency at least $\xi_{\min}$, the click probability $q$ gives $$s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.$$ A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses $K=4$, $\tau_K=0$, $\xi=0.8$ and $N=3{,}517{,}336$ independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most $0.05$ with false-acceptance probability at most $0.001$ and missed acceptance at most $0.001$ on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at $E=1$, $\alpha=1.9$, $\eta=0.9505$ and $m=10^5$, the certified bound $\Delta_\alpha<0.06942$ rules out a target deficit of $0.1$ nat. A separate finite single-mode preparation model predicts positive Shannon deficit near $\eta=0.97$, negative deficit near $\eta=0.85$, and a Gaussian-hull witness above $4.4\times10^{-3}$ at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put $\alpha=2-\delta$ and $L=\ln(1/\delta)$. For the maximum deficit $D_{m,E}$ at fixed total energy, $$D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),$$ where $T_h$ is the first mode number allowing a fixed positive deficit $h$. Beyond a sufficiently large fixed multiple of $L/\delta$, $D_{m,E}(2-\delta)=\Theta_E(m\delta)$. The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds: $h(W_\rho)\ge m\ln(2\pi)+\varepsilon_m$ with $\varepsilon_m>0$ for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With $\mathcal C_{r,k}$ denoting the ordering-weighted norm of photon-number-difference block $k$, $$\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.$$ For vacuum–$n$-photon coherence and $1<\alpha<2$, $$\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.$$ This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.

Sep 23, 20260 citations

Feedback and Robust Local Readout of a Fixed Four-State Quantum Source

Jeremy Rodgers

This paper develops local quantum-reader decision problems within Shadow Theory’s source–readout programme. It asks when feedback improves access to a target distinction in a fixed quantum source, and how reliably that distinction can be read under detector noise and calibration uncertainty. The source supplies independent copies conditional on one hidden label, held fixed throughout the experiment, from four equatorial qubit preparations. The task is to identify their common positive or negative sign, rather than reconstruct the full preparation label. Although all four preparations have identical separate-mode intensities, phase-sensitive local measurements provide access to the target. The analysis compares adaptive readers, whose measurement settings depend on earlier records, with randomized nonadaptive readers, whose complete schedules are selected before outcomes are observed. Each measurement consumes one fresh copy. Every protocol must return a sign at a fixed deadline, retaining all settings, outcomes, nulls and specified timing records. Neither class uses collective measurements or coherent quantum memory between specimens. The main results establish equal optimal worst-case error for both reader classes through three copies across the shared efficiency and visibility model. At four copies, explicit rational feedback policies outperform every randomized nonadaptive schedule over the full continuous equatorial measurement menu, in both the ideal model and a specified noisy model. The comparison is supported by continuous-menu lower certificates rather than measurement-grid searches alone. For a one-percent worst-case error target, the minimum copy count is narrowed to either 13 or 14 for each reader class in the nominal model, and either 16 or 17 under the specified asymmetric gain and phase uncertainty. These adjacent thresholds remain unresolved. Further results give an exact robust repeated-angle frontier, extend the comparison to randomized choices of a fixed repeated angle, and identify convex-source conditions under which one reader is globally sufficient. The accompanying technical supplement and reproducibility materials provide measurement-allocation benchmarks, expanded finite-menu comparisons, compact feedback policies, exact optimization certificates, and timing and calibration interfaces. The results quantify target access under explicitly stated preparation, measurement and record-access conditions. They use standard quantum probabilities within Shadow Theory’s research programme and distinguish copy savings from total apparatus, control or calibration savings.This paper develops local quantum-reader decision problems within Shadow Theory’s source–readout programme. It asks when feedback improves access to a target distinction in a fixed quantum source, and how reliably that distinction can be read under detector noise and calibration uncertainty. The source supplies independent copies conditional on one hidden label, held fixed throughout the experiment, from four equatorial qubit preparations. The task is to identify their common positive or negative sign, rather than reconstruct the full preparation label. Although all four preparations have identical separate-mode intensities, phase-sensitive local measurements provide access to the target. The analysis compares adaptive readers, whose measurement settings depend on earlier records, with randomized nonadaptive readers, whose complete schedules are selected before outcomes are observed. Each measurement consumes one fresh copy. Every protocol must return a sign at a fixed deadline, retaining all settings, outcomes, nulls and specified timing records. Neither class uses collective measurements or coherent quantum memory between specimens. The main results establish equal optimal worst-case error for both reader classes through three copies across the shared efficiency and visibility model. At four copies, explicit rational feedback policies outperform every randomized nonadaptive schedule over the full continuous equatorial measurement menu, in both the ideal model and a specified noisy model. The comparison is supported by continuous-menu lower certificates rather than measurement-grid searches alone. For a one-percent worst-case error target, the minimum copy count is narrowed to either 13 or 14 for each reader class in the nominal model, and either 16 or 17 under the specified asymmetric gain and phase uncertainty. These adjacent thresholds remain unresolved. Further results give an exact robust repeated-angle frontier, extend the comparison to randomized choices of a fixed repeated angle, and identify convex-source conditions under which one reader is globally sufficient. The accompanying technical supplement and reproducibility materials provide measurement-allocation benchmarks, expanded finite-menu comparisons, compact feedback policies, exact optimization certificates, and timing and calibration interfaces. The results quantify target access under explicitly stated preparation, measurement and record-access conditions. They use standard quantum probabilities within Shadow Theory’s research programme and distinguish copy savings from total apparatus, control or calibration savings.

Oct 28, 20260 citations

On the Emergence of Quantum Boltzmann Fluctuation Dynamics near a Bose–Einstein Condensate

Thomas Chen, Michael Hott

arXiv:2106.03825Journal of Statistical PhysicsApr 4, 20236 citations

A Hybrid Quantum-Classical Algorithm for Training Quantum Boltzmann Machines

Christa Zoufal, Aurélien Lucchi, Stefan Woerner

Bulletin of the American Physical SocietyMar 2, 20200 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

For a specified finite ensemble and one invocation of each operation, an exact certificate bounds fixed-order single-task success by

11/12

The noise analysis relates residual order information, error tolerance and phase correlations to the frequency and duration of reliable operation. With fixed error budget,

r

phase coordinates and inverse-size visibility loss below a feasibility threshold, occupation scales as

N^{-r/2}

for

N

outputs. Brownian phase noise produces an exponential penalty for maintaining a fixed duration as

N

grows; for one common phase with finite frequency-correlation time, fixed survival quantiles instead scale as

N^{-1/2}

\nu_m

for the

m

-mode vacuum and define

\Delta_{\alpha,m}(\rho)=H_\alpha(W_{\nu_m})-H_\alpha(W_\rho)

, using natural logarithms;

H_1

is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously,

E_m=\operatorname{Tr}(N_m\rho_m)\to0,\qquad \tfrac12\|\rho_m-\nu_m\|_1\to0,\qquad d_{\mathcal F_m}(\rho_m)\to0,\qquad \Delta_{1,m}(\rho_m)\to+\infty.

Here

\mathcal F_m

is the trace-norm-closed convex hull of Gaussian states and

d_{\mathcal F_m}

uses half the trace norm. Thus proximity to the entire Gaussian model does not control Wigner entropy uniformly in mode number. Independent weak modes amplify a one-mode entropy effect while their total excitation vanishes. For the same sequence, half loss instead gives

-E_m\le\Delta_{1,m}(\mathcal L_{1/2}^{\otimes m}\rho_m)\le0

, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity

\eta

and any fixed positive total input-energy budget

E

, the sharp threshold is

\eta_c(\alpha)=\begin{cases}1/2,&0<\alpha\le1,\\ \alpha/2,&1<\alpha<2.\end{cases}

With

\mathcal P_{m,E}=\{\rho:W_\rho\ge0,\ \operatorname{Tr}(N_m\rho)\le E\}

\sup_{m\ge1}\sup_{\rho\in\mathcal P_{m,E}}\Delta_{\alpha,m}(\mathcal L_\eta^{\otimes m}\rho)=\begin{cases}0,&\eta\le\eta_c(\alpha),\\+\infty,&\eta>\eta_c(\alpha).\end{cases}

The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For

\alpha\ge2

, established Wigner-norm bounds give the vacuum entropy bound at every transmissivity for Wigner-positive outputs. These statements distinguish a universal guarantee from the earlier sign crossing that a particular source may exhibit. 3. Near equality gives a certificate for the entire input For an arbitrary input

\rho

, define the half-loss output Shannon excess

s(\rho)=h(W_{\mathcal L_{1/2}^{\otimes m}\rho})-m(1+\ln\pi)\ge0.

The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields

\inf_a\|\rho-|a\rangle\langle a|\|_1^2\le\frac{64}{\ln2}\,s(\rho).

The bound controls the full input state, with no energy or photon-support restriction. One coherent reference then controls every subsequent quantum channel and every unconditional measurement event. In particular, an event-probability error at most

\epsilon

is guaranteed by

s\le(\ln2)\epsilon^2/16

. Conversely, a surviving output Gaussian-hull gap enforces

s(\rho)\ge\frac{\ln2}{16}\,d_{\mathcal F_m}(\mathcal L_{1/2}^{\otimes m}\rho)^2.

Recovery of the inequality and proximity to equality therefore support different physical inferences: the former permits residual non-Gaussian structure, while the latter certifies a coherent-state approximation. 4. Surviving structure has mode and conversion costs A thermally repaired vacuum–three-photon source with per-mode occupation

c/m

has total input energy tending to

3c

. Along the recovery boundary, its attenuated state obeys

\liminf_{m\to\infty}d_{\mathcal F_m}(\widetilde\rho_{m,\eta_c})\ge c\eta_c^3\exp\{-c[1-(1-\eta_c)^3]\}>0.

At half transmission this lower bound is

g(c)=(c/8)e^{-7c/8}

. Starting from one complete block, Gaussian processing requires

\Theta(m)

output modes to retain any fixed positive gap below this bound at success probability at least a fixed

p_0>0

. Gaussian ancillary states, adaptive Gaussian measurements, feedback, discarding and postselection are included. A distinct task requests fresh independent blocks of this same half-loss source to produce the one-mode target

\tau_*=(18|0\rangle\langle0|+6|1\rangle\langle1|+|2\rangle\langle2|)/25

within half trace distance

10^{-6}

. Let

C_m

be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed

0<c\le1

, every achieving Gaussian protocol obeys the asymptotic lower bound

\log C_m\ge\frac1{128}(m/c)^{1/3}-\log m-O(1).

This is a source-request lower bound; the optimal achievable cost for this source remains undetermined. The proof evaluates a curvature-controlled input certificate for the actual thermally repaired state and applies the Gaussian conversion theorem of the companion paper cited below. 5. Exactly equal entropy does not fix conversion cost For two separate two-mode quartic source families, their mixture probabilities can be adjusted so that their half-loss Shannon entropies are exactly equal. Under a common target, tolerance and failure-inclusive accounting, the inherited matching conversion laws become

h(W_{\omega_A})=h(W_{\omega_B})=2(1+\ln\pi)+s,\qquad \log C_A(s)=\Theta(s^{-1/3}),\qquad \log C_B(s)=\Theta(s^{-1/2}).

The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy

E_A,E_B\sim s/2

. At exactly equal energy, the two excesses instead share

s_i(E)=2E-(35/8)E^2+O(E^3)

and differ by

s_A(E)-s_B(E)=15E^3+O(E^4)

. Entropy alone therefore cannot select the cheaper source for a fixed output task. 6. Finite data and finite mode budgets Under a declared displaced-input photon-tail bound

\operatorname{Tr}[N\mathbf1_{N>K}\sigma_a]\le\tau_K

, trusted half loss and all-mode on/off detection with efficiency at least

\xi_{\min}

, the click probability

q

gives

s(\sigma)\le A_Kq+\tau_K,\qquad A_K=\frac{K}{1-(1-\xi_{\min}/2)^K}.

A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses

K=4

\tau_K=0

\xi=0.8

and

N=3{,}517{,}336

independent validation blocks, accepting at most 41 clicks, to certify unconditional event error at most

0.05

with false-acceptance probability at most

0.001

and missed acceptance at most

0.001

on the stated low-click alternative class. All vacuum results count; training, calibration and source heralding are separate costs. When coherent-state approximation alone is the task, the same click measurement supplies a stronger direct fidelity test. The paper also gives all-input finite-budget exclusions above the loss-recovery line. For example, at

E=1

\alpha=1.9

\eta=0.9505

and

m=10^5

, the certified bound

\Delta_\alpha<0.06942

rules out a target deficit of

0.1

nat. A separate finite single-mode preparation model predicts positive Shannon deficit near

\eta=0.97

, negative deficit near

\eta=0.85

, and a Gaussian-hull witness above

4.4\times10^{-3}

at half transmission, with explicit preparation, measurement-error and raw-trial budgets in the PDF. 7. Dimensional scales, a strict entropy floor and ORCS Near order two, put

\alpha=2-\delta

and

L=\ln(1/\delta)

. For the maximum deficit

D_{m,E}

at fixed total energy,

D_{m,E}(2-\delta)=\Theta_{m,E}(\delta/L)\quad\text{for fixed }m,\qquad T_h(\delta,E)=\Theta_{E,h}(L/\delta),

where

T_h

is the first mode number allowing a fixed positive deficit

h

. Beyond a sufficiently large fixed multiple of

L/\delta

D_{m,E}(2-\delta)=\Theta_E(m\delta)

. The manuscript separates proved sufficient fronts from necessary exit bounds; the sharp first-crossing coefficient remains open. At every fixed mode number, an energy-unrestricted strict Shannon floor also holds:

h(W_\rho)\ge m\ln(2\pi)+\varepsilon_m

with

\varepsilon_m>0

for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With

\mathcal C_{r,k}

denoting the ordering-weighted norm of photon-number-difference block

k

\mathcal C_{r,k}(\mathcal L_\eta\rho)=\mathcal C_{T_\eta(r),k}(\rho),\qquad T_\eta(r)=1-\frac{1-r}{\eta}.

For vacuum–

n

-photon coherence and

1<\alpha<2

\mathcal C_{1-\alpha/2}(\mathcal L_\eta|n\rangle\langle0|)=\left(\frac{2\eta}{\alpha}\right)^n.

This formula exposes the high-order coherence mechanism at the loss threshold; the universal recovery result is proved through the full nonlinear Wigner-norm inequality. Related papers and history The counterexample paper Wigner entropy below vacuum: physical counterexamples and stability limits expands arXiv:2609.13312v1, submitted on 10 September 2026. This stability paper develops the dimensional, loss-dependent and operational consequences. Its state-approximation and conversion interfaces use the companion results From phase-space entropy to coherent-state error: mode-uniform Rényi–Wehrl stability and Phase records control the conversion cost of non-Gaussian light. The mode-retention comparison connects to Limits to Gaussian concentration of non-Gaussian light.

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.