A closed finite quantum system can return arbitrarily close to its initial state, but how far can a region's entropy depart from its initial value during the evolution? For finite quadratic fermionic systems, we reduce the maximum over all initial states and all times to a static spectral calculation: universal two-sided bounds identify it with B = ‖P − P̄‖1, where P selects the region and P̄ is its spectral dephasing. The criterion includes mixed, non-Gaussian and initially entangled states. In an explicit random Kitaev chain of 2N sites, the disorder-averaged maximum stays bounded uniformly in length away from statistical self-duality. At self-duality it equals (ln 2/6) ln N + o(ln N), including after optimally freezing any fixed number of spectral modes. The same leading term holds throughout each fixed finite-size window |χ|√N ≤ M. A single preparation, vacuum on one half and maximally mixed on the other, already produces logarithmic growth in its long-time averaged local entropy. Cross-boundary spectral channels thus determine both the existence of an entropy budget and its exact critical growth, even when the many-body preparation is unknown.
A closed finite quantum system can return arbitrarily close to its initial state, but how far can a region's entropy depart from its initial value during the evolution? For finite quadratic fermionic systems, we reduce the maximum over all initial states and all times to a static spectral calculation: universal two-sided bounds identify it with B = ‖P − P̄‖1, where P selects the region and P̄ is its spectral dephasing. The criterion includes mixed, non-Gaussian and initially entangled states. In an explicit random Kitaev chain of 2N sites, the disorder-averaged maximum stays bounded uniformly in length away from statistical self-duality. At self-duality it equals (ln 2/6) ln N + o(ln N), including after optimally freezing any fixed number of spectral modes. The same leading term holds throughout each fixed finite-size window |χ|√N ≤ M. A single preparation, vacuum on one half and maximally mixed on the other, already produces logarithmic growth in its long-time averaged local entropy. Cross-boundary spectral channels thus determine both the existence of an entropy budget and its exact critical growth, even when the many-body preparation is unknown.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.
When the two largest eigenvalues of an entanglement or negativity spectrum meet, the gap of the corresponding entanglement or negativity Hamiltonian closes while entropies stay smooth, an event we call a temporal criticality. We ask when such criticalities occur in open fermionic systems, in which order, and whether they precede the geometric decoherence time, at which the negativity starts to fall. We compare loss and gain, which respect fermion parity only on average (weak symmetry) and keep the state Gaussian, with density dephasing, which respects it in every jump (strong symmetry) and makes the state non-Gaussian. We find that Gaussianity, rather than the symmetry, fixes the criterion for a criticality. In a Gaussian state the entanglement gap of a subsystem closes exactly when its two parity sectors carry equal weight, and the negativity gap closes exactly when this happens for the subsystem or its complement, so the negativity criticality never comes after the entanglement one. Uniform balanced loss and gain only shrink the two parities and leave both criticalities in place at every rate, for any quadratic Hamiltonian. In an open Kitaev chain, weak imbalanced loss and gain shift the criticalities by amounts set by the occupations at the cut, and at strong rates they allow a closing near the relaxation time only when an odd number of neighboring sites pass half filling. For dephasing we solve the non-Gaussian dynamics exactly at the point where the bond terms of the chain commute. Weak dephasing delays both criticalities and, away from the chain ends, moves them ahead of the parity balance, where the parity rules of Gaussian states would place them. As the dephasing rate approaches critical damping, both criticalities recede to infinite time, and at strong dephasing the negativity starts to fall long before either gap begins to shrink.
In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size.We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally.We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions.Specifically, we prove that a translationally invariant marginal obeying three nonlinear constraints-all of which follow from the entropy scaling law straightforwardlymust be consistent with some global state on an infinite lattice.Moreover, we derive a closed-form expression for the maximum-entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy.The main assumptions of our construction are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.
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 νm for the m-mode vacuum and define Δα,m(ρ)=Hα(Wνm)−Hα(Wρ), using natural logarithms; H1 is Shannon entropy. Positive deficit means entropy below vacuum. We construct physical Wigner-positive states satisfying, simultaneously, Em=Tr(Nmρm)→0,21∥ρm−νm∥1→0,dFm(ρm)→0,Δ1,m(ρm)→+∞. Here Fm is the trace-norm-closed convex hull of Gaussian states and dFm 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 −Em≤Δ1,m(L1/2⊗mρm)≤0, so the entropy difference from vacuum tends to zero. 2. The exact pure-loss recovery boundary For Wigner-positive inputs, intensity transmissivity η and any fixed positive total input-energy budget E, the sharp threshold is ηc(α)={1/2,α/2,0<α≤1,1<α<2. With Pm,E={ρ:Wρ≥0,Tr(Nmρ)≤E}, supm≥1supρ∈Pm,EΔα,m(Lη⊗mρ)={0,+∞,η≤ηc(α),η>ηc(α). The recovery statement includes the boundary itself and is independent of mode number. Half loss also restores vacuum majorization for arbitrary density operators. For α≥2, 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 ρ, define the half-loss output Shannon excess s(ρ)=h(WL1/2⊗mρ)−m(1+lnπ)≥0. The half-loss identity identifies this quantity with the input's total Wehrl entropy excess. Applying mode-uniform Wehrl stability yields infa∥ρ−∣a⟩⟨a∣∥12≤ln264s(ρ). 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 ϵ is guaranteed by s≤(ln2)ϵ2/16. Conversely, a surviving output Gaussian-hull gap enforces s(ρ)≥16ln2dFm(L1/2⊗mρ)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 liminfm→∞dFm(ρm,ηc)≥cηc3exp{−c[1−(1−ηc)3]}>0. At half transmission this lower bound is g(c)=(c/8)e−7c/8. Starting from one complete block, Gaussian processing requires Θ(m) output modes to retain any fixed positive gap below this bound at success probability at least a fixed p0>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 τ∗=(18∣0⟩⟨0∣+6∣1⟩⟨1∣+∣2⟩⟨2∣)/25 within half trace distance 10−6. Let Cm be the infimum of expected complete-block requests divided by success probability, including all failed requests. For fixed 0<c≤1, every achieving Gaussian protocol obeys the asymptotic lower bound logCm≥1281(m/c)1/3−logm−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(WωA)=h(WωB)=2(1+lnπ)+s,logCA(s)=Θ(s−1/3),logCB(s)=Θ(s−1/2). The target tolerance is held fixed in these asymptotics. The delivered energies after entropy matching satisfy EA,EB∼s/2. At exactly equal energy, the two excesses instead share si(E)=2E−(35/8)E2+O(E3) and differ by sA(E)−sB(E)=15E3+O(E4). 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 Tr[N1N>Kσa]≤τK, trusted half loss and all-mode on/off detection with efficiency at least ξmin, the click probability q gives s(σ)≤AKq+τK,AK=1−(1−ξmin/2)KK. A one-sided binomial bound turns this into an entropy confidence certificate. An explicit undisplaced four-photon-cutoff example uses K=4, τK=0, ξ=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, α=1.9, η=0.9505 and m=105, the certified bound Δα<0.06942 rules out a target deficit of 0.1 nat. A separate finite single-mode preparation model predicts positive Shannon deficit near η=0.97, negative deficit near η=0.85, and a Gaussian-hull witness above 4.4×10−3 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 α=2−δ and L=ln(1/δ). For the maximum deficit Dm,E at fixed total energy, Dm,E(2−δ)=Θm,E(δ/L)for fixed m,Th(δ,E)=ΘE,h(L/δ), where Th is the first mode number allowing a fixed positive deficit h. Beyond a sufficiently large fixed multiple of L/δ, Dm,E(2−δ)=ΘE(mδ). 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(Wρ)≥mln(2π)+εm with εm>0 for all Wigner-positive states. The ordering-resolved coherence spectrum (ORCS) supplies an exact single-mode loss coordinate. With Cr,k denoting the ordering-weighted norm of photon-number-difference block k, Cr,k(Lηρ)=CTη(r),k(ρ),Tη(r)=1−η1−r. For vacuum–n-photon coherence and 1<α<2, C1−α/2(Lη∣n⟩⟨0∣)=(α2η)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.
In a spin chain governed by a local Hamiltonian, we consider a microcanonical ensemble in the middle of the energy spectrum and a contiguous subsystem whose length is a constant fraction of the system size. We prove that if the bandwidth of the ensemble is greater than a certain constant, then the average entanglement entropy (between the subsystem and the rest of the system) of eigenstates in the ensemble deviates from the maximum entropy by at least a positive constant. This result highlights the difference between the entanglement entropy of mid-spectrum eigenstates of (chaotic) local Hamiltonians and that of random states. We also prove that the former deviates from the thermodynamic entropy at the same energy by at least a positive constant.