Unrefereed research candidate. When m identical copies of a quantum state of light are measured with Gaussian measurements (linear optics, Gaussian auxiliary modes, homodyne detection), measuring the copies jointly can give more Fisher information than measuring them one by one (Navarro, Morelli, Sanz and Mehboudi, arXiv:2512.20534), who left finite-copy optimality open. For every smooth family of pure single-mode Gaussian states and every m, this package proves the exact maximum of the Fisher information over all fixed (non-adaptive) Gaussian measurements, including ancilla-assisted schemes: a one-angle formula attained by concentrating the mean into one bright output and homodyne detection of every output. It also gives the exact optimum when measurement-seed squeezing is capped (in normalised coordinates), the exact shortfall from the quantum Fisher information (nondecreasing in m, at most one copy's variance information λ2/2, with an explicit limit), and a necessary and sufficient condition for strict collective advantage. The archive contains the 9-page manuscript (PDF and LaTeX), numerical checks with semantic negative controls, CI and SHA-256 sums. Assurance: written proofs; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded prior-art search. Mixed states, multimode tangents and adaptive strategies are not covered. Creator: Anonymous. Text CC0-1.0; code MIT.
Unrefereed research candidate. When m identical copies of a quantum state of light are measured with Gaussian measurements (linear optics, Gaussian auxiliary modes, homodyne detection), measuring the copies jointly can give more Fisher information than measuring them one by one (Navarro, Morelli, Sanz and Mehboudi, arXiv:2512.20534), who left finite-copy optimality open. For every smooth family of pure single-mode Gaussian states and every m, this package proves the exact maximum of the Fisher information over all fixed (non-adaptive) Gaussian measurements, including ancilla-assisted schemes: a one-angle formula attained by concentrating the mean into one bright output and homodyne detection of every output. It also gives the exact optimum when measurement-seed squeezing is capped (in normalised coordinates), the exact shortfall from the quantum Fisher information (nondecreasing in m, at most one copy's variance information λ2/2, with an explicit limit), and a necessary and sufficient condition for strict collective advantage. The archive contains the 9-page manuscript (PDF and LaTeX), numerical checks with semantic negative controls, CI and SHA-256 sums. Assurance: written proofs; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded prior-art search. Mixed states, multimode tangents and adaptive strategies are not covered. Creator: Anonymous. Text CC0-1.0; code MIT.
Famously, the quantum Fisher information -- the maximum Fisher information over all physical measurements -- is additive for independent copies of a system and the optimal measurement acts locally. We are left to wonder: does the same hold when the set of accessible measurements is constrained? Such constraints are necessary to account for realistic experimental restrictions. Here, we consider a fully Gaussian scenario focusing on only Gaussian measurements. We prove that the optimal Gaussian measurement protocol remains local, if the information is encoded in either the displacement or the covariance matrix. However, when the information is imprinted on both, this no longer holds true: we construct a simple global Gaussian measurement where the Fisher information becomes super additive. These results can improve parameter estimation tasks via feasible tools. Namely, in quantum optical platforms our proposed global operation requires only passive global operations and single mode Gaussian measurements. We demonstrate this in two examples where we estimate squeezing and losses. While in the former case there is a significant gap between the Fisher information of the optimal Gaussian measurement and the quantum Fisher information for a single copy, this gap can be reduced with joint Gaussian measurements and closed in the asymptotic limit of many copies.
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 ε away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget Es. 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 ∑t=1TEK∥αt∥2≤TEp under every channel in the testing promise. The probe budget Ep is distinct from the signal budget Es defining the certification guarantee. Optimal mean-energy query laws. For fixed positive Es,Ep, fixed finite mode number m, and both testing errors at most δ, the minimax number of calls satisfies Tmean(1)(ε,δ)=ΘEs,Ep(ε−4logε1logδ1),Tmean(m)(ε,δ)=Θm,Es,Ep(ε−(2m+2)logδ1),m≥2. The bounds hold for sufficiently small ε, uniformly over 0<δ≤1/4. 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 tlogt for one mode and as tm for fixed m≥2. 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 logTG=Θ(ε−2) at fixed confidence. A fixed brightness cap gives logTcap=Θ(ε−2log(1/ε)). 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.
An unknown optical phase can hide quantum non-Gaussianity in every individual mode while leaving it certifiable in the joint light. This paper shows when Gaussian measurements can exclude the complete Gaussian convex hull without knowing the source's common phase, and gives explicit finite-sample tests. The results guide whether to preserve phase relations between emissions and how to process the resulting measurement records. 1. Certification against the full Gaussian model. Let Gn be the trace-norm-closed convex hull of all physical n-mode Gaussian states. For a fixed known ρ∈/G1, form the common-phase state Ωn=∫02π(UϕρUϕ†)⊗n2πdϕ,Uϕ=e−iϕN. Finite Gaussian measurements with a laboratory phase reference and bounded classical scores achieve arbitrarily small false-positive and missed-detection probabilities. The null includes arbitrary Gaussian entanglement and classical correlations, with unrestricted displacement, squeezing and energy. In particular, infσn∈Gn∥Ωn−σn∥1⟶2. The norm is the full, unhalved trace norm. An unavailable common phase therefore differs operationally from independently randomising the phase of each emission. 2. An exact coherence boundary and a joint-only resource. For the displaced two-Gaussian-packet family, centred by its actual mean before pure loss, the Gaussian-hull boundary is ρη,ν∈G1⟺ν≤νc,νc=exp[−2(1−η)(v−1/2)2v2R]. Here v>1/2 is the packet quadrature variance, R the momentum displacement, 0<τ,η<1 the packet weight and transmissivity, and 0≤ν≤1 the coherence visibility. The same membership boundary holds after common-phase averaging for every n≥3. At (v,R,τ,ν,η)=(20,2/9,9/50,1,1/2), each phase-averaged single mode is a Gaussian mixture, while the joint common-phase state lies outside Gn for n≥3. Independent phase averaging instead gives a Gaussian mixture at every block size. These states are separable and Wigner-positive: their difference is carried by phase relations across emissions. 3. The same two-packet source has an explicit finite certificate. Heterodyne phase training, a finite homodyne score and an eight-direction bounded tail test provide a complete certificate for the preceding parameter choice. With ideal Gaussian readout, the sufficient raw acquisition budget and certified errors are nraw=1015+2(4×1015)=9×1015,PFP<3.86×10−22,PFN<0.002068. This conservative budget establishes finite-sample certification of the joint-only resource; it is not an optimised experimental acquisition requirement. The PDF contains the measurement coefficients and complete proof. 4. A complementary two-million-trial implementation. For a displaced lossy single-photon source with unknown common phase, the target conditional state is ωφ=D(eiφ)2∣0⟩⟨0∣+∣1⟩⟨1∣D†(eiφ). A bounded homodyne score uses the full-Gaussian one-photon benchmark 33/(4e), 128 phase candidates and decision threshold 0.490. Including the specified source and readout tolerances, n=2×106,PFP<1.86×10−4,PFN<7.908×10−3. All vacuum emissions, loss events and measurement outcomes count towards the raw budget. This is a separate source family from the two-packet example. Together, the results distinguish unknown common phase from destructive phase randomisation and connect full-model exclusion to explicit Gaussian-measurement scores, error bounds and acquisition costs. The 15-page PDF includes the manuscript, two figures and complete supplementary proofs.
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 CQ(R)=supN:I(A:M)≤RI(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<η<1, write the output occupation as ηn+b, where b≥0 is the environment's added thermal occupation. With g(x)=(x+1)log(x+1)−xlogx, the exact optimum over all normal classical records is Ccl(n,R)=g(ηn+b)−g(ηuR+b),uR=g−1(max{g(n)−R,0}). A legal encoder applies vacuum loss with transmissivity λ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed b>0 and 0<λ<1/2, at the common budget Rn=g(n)+g(λn)−g((1−λ)n), its actual relevance satisfies v2=1+bηλnlog(1/n)+O(n),Ccl(n,Rn)=2ληlog(1+1/b)n+O(log(1/n)n). The full optical tail is included in the resource accounting. At n=0.01, η=b=0.5, λ=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 dimKk≤2k and I(Ak:Kk)≤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 αk be the missed-correlation probability and βk the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is αk≤ϵ⟹βk≥exp[−1−ϵkcnR+log2],cn=ηlog(1+1/n)log(1+1/(ηn+b)). At the finite parameters above and ϵ=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.
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 ka(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=2q. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is Θ(m). Along arbitrary growing-mode sequences, the precise criterion for improving that order is ka(D,m)=o(m)⟺am→∞andmaD→∞. 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,m2/D}. At the fixed energy and separation above, constants c,C>0, independent of m,D,a, give cmin{m,A/a}≤ka(D,m)≤Cmin{m,A/a+min(a−2,m2/3)},0<a≤1. At a=0, the sample cost is Θ(m). For ideal memory the complete rate is k1(D,m)=Θ(min{m,max(1,m2/D)}). With loss, the bounds match whenever A≥min{a−1,am2/3}. For example, a=m−1/6 and D=⌈m11/6⌉ give Θ(m1/3) preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for D≥m2, a=m−1/2, the stated bounds are Ω(m1/2) and O(m2/3). 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 m2 labels, only a positive register of dimension m+1 is retained. The three-visit route requires D≥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 N=aΨ+(1−a)Rτ, where Ψ is CPTP and Rτ(X)=Tr(X)τ, an exact flagged-erasure simulation gives kN(D,m)≥ka(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 Θ(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.
Quantum Fisher information adds over independent probes, so a single parameter is best measured probe by probe. We show that Gaussian measurements, the linear optics and homodyne detection of optical and microwave experiments, break this rule: two independent modes are better measured together. The uncertainty principle leaves a linear detector half of phase space, and for two modes the choice of half is a resource. Bell homodyne, a balanced beam splitter followed by two homodyne detectors, beats the best separate readout when the modes differ in quadrature width and in how the width responds to the parameter. Heterodyne detection splits a mode on a beam splitter to read both quadratures and pays one unit of vacuum noise for the empty port. Bell homodyne fills that port with the second mode, so the noise turns into signal. We prove that the gain stays below (2−1)2=17.157% for every parameter carried by widths. For thermal modes at one temperature, the canonical case, a gain was conjectured impossible. We prove that any frequency difference opens a temperature window and that the gain peaks at 12.699% at frequency ratio 3.318, where Bell homodyne is the optimal Gaussian measurement. Standard hardware reaches the gain: two coupled resonators with two homodyne detectors, or a phase-preserving amplifier whose idler band is fed by the same thermal source.