bibkey: shirokov2012entropycontinuity
authors: M. E. Shirokov
year: 2012
title: "On a conjectured property of the von Neumann entropy valid in the commutative case"
doi: null
url: https://arxiv.org/abs/1201.1093v1
claim: "Entropy continuity on a closed convex family is conjectured to imply an all-temperature Hamiltonian energy bound and a common continuous bounded relative entropy reference."
strata_touched:
  - D5/S3/Quantum/Analysis/OrliczCommonEnergy
  - D5/S3/Quantum/Analysis/TraceClass/EntropyMaximizer
  - D5/S3/Quantum/Analysis/TraceClass/ContinuousEnergyHamiltonian
  - D5/S3/Quantum/Analysis/TraceClass/FiniteEntropyRelativeEntropy
license: citation-only
triage: anchor

Entropy continuity and the scalar Orlicz sufficient condition

Verified locator

Original conjectures: https://arxiv.org/abs/1201.1093v1

Original entropy-continuity premises

Shirokov, arXiv:1201.1093v1, Section 3, Conjectures 1 and 2, asks about arbitrary trace-norm closed convex families of quantum states with finite continuous von Neumann entropy. Conjecture 1 asks for containment in an energy ball of a positive Hamiltonian with finite partition function at every positive inverse temperature. Conjecture 2 asks for a single state whose relative entropy is continuous and bounded on the family. Neither conjecture assumes commutation of the states.

The related equivalence between continuous logarithmic reference energy and an all-temperature energy bound is in Shirokov, Entropic characteristics of subsets of states, arXiv:quant-ph/0510073v2, Proposition 4. This is a source for the mathematical objective, not a claim that the scalar theorem below is stated in either paper.

For a positive normalized reference sequence s and nonnegative normalized masses d(x,n), the module uses the actual functions

$${\Phi }_p(t)=t(\log \max (1,t){)}^p,{\Psi }_p(u)=\underset{t\ge 0}{\sup }\{ut-{\Phi }_p(t)\}.$$

A finite uniform extended Phi moment and finiteness of the actual extended dual series at every positive scale imply uniform tails of (-log s(n))*d(x,n). One common monotone natural weight, tending to infinity, preserves these tails with arbitrarily small additive energy cost. For coordinatewise continuous d on any topological space, the resulting scalar energy is finite and continuous, and its Boltzmann series is summable at every positive inverse temperature. The scalar implication uses p>0; nonvacuous faithful infinite-reference applications of the every-scale condition use p>1.

This conditional sufficient criterion has stronger premises than entropy continuity. No universal implication from the original entropy premise to a fixed logarithmic-power moment is claimed. The spectral, noncommuting trace, self-adjoint domain and actual heat-trace joins are separate mathematical obligations. The scalar theorem does not resolve either original conjecture or establish historical originality.

Global entropy maximum from the original family premises

D5/S3/Quantum/Analysis/TraceClass/EntropyMaximizer works on the actual trace-class Banach space of any complete complex Hilbert space. Its nonempty, closed real-convex family consists of positive trace-one operators with finite actual singular spectral entropy and relatively continuous actual real entropy. It assumes no energy condition, compactness or entropy bound.

Every state has trace norm one. At a fixed state x, continuity of entropy relative to the family gives a neighborhood where entropy is below S(x)+1. A single sufficiently small positive t puts every mixture (1-t)x+ty in that neighborhood, since the state family has diameter at most two. Nonnegative concavity gives t*S(y)<=S(x)+1 and hence a finite global upper bound. Strong entropy concavity with modulus one half, completeness of the closed family and upper semicontinuity then give an actual unique global maximizer gamma and

norm(rho-gamma)^2/8 <= S(gamma)-S(rho).

The quantitative concavity supplier is attributed in Library/QuantumBounds/androulakis2023nussbaumszkola.md; attainment uses the standard maximizing-sequence argument in StrongConcavityAttainment. This is a consequence used along the original entropy-continuity route, not a theorem attributed verbatim to the conjecture paper. No originality, common-reference energy conclusion or all-temperature Hamiltonian conclusion is asserted. Separability, commutation and absence of kernel fibers are not premises of this intermediate result.

Thermal construction on the original complete Hilbert space

D5/S3/Quantum/Analysis/TraceClass/ContinuousEnergyHamiltonian constructs a Hamiltonian on any supplied complete full spectral basis of a normalized positive trace-class contraction sigma in a complete separable complex Hilbert space. For a nonempty compact positive trace-one family A, the conditional premises are finiteness of actual singular spectralEnergy(sigma,rho) and continuity of its real value on A. Convexity is not required here.

An injective natural code retains all basis vectors. Set a(i)=-log(s(i)) when s(i)>0 and a(i)=(code(i)+1)log(2) when s(i)=0. Finite singular energy forces zero mass on every reference-kernel vector. Only scalar energy terms are extended by zero to the natural numbers. Dini’s theorem for real finite prefixes supplies uniform extended tails. UniformTailAmplification gives a natural monotone diverging weight w with both an additive cost bound and uniform weighted tails. The generic real-series tail theorems in OrliczCommonEnergy supply actual summability and continuity without their unrelated stronger moment premises. The Hamiltonian coefficients are h(i)=1+w(code(i))*a(i) on the original basis.

The dyadic code estimate makes exp(-a(i)) summable. For each beta>0, the exceptional indices with beta*w(code(i))<1 form a finite set, so the same Hamiltonian has summable heat coefficients at every positive inverse temperature. DiagonalHeat and DiagonalEnergy supply its dense full-space self-adjoint operator domain, full graph and weighted-l2 characterization, positive trace-class heat and actual complex/real trace HasSum, full strong generator graph equivalence and semigroup uniqueness, and the lower form bound by the identity. The actual resolvent-cap mixed-state energy is finite and continuous in both ENNReal and real values. It equals one plus the amplified diagonal series and is at most 1+spectralEnergy(sigma,rho)+delta. Actual trace-one diagonal summation supplies the constant one.

The same module derives the original-premise endpoint. A nonempty closed real-convex family of positive trace-one states with finite continuous actual entropy has an entropy maximizer gamma0. Its center-energy inequality embeds the family in a compact full energy sublevel; closedness then gives compactness. The existing fullCompactCommonReference yields gamma in A and sigma in A with finite continuous reference energy bounded by realEntropy(gamma)+log(2). Applying the new construction to sigma’s native full basis with delta=1 yields the strictly positive finite uniform bound 2+realEntropy(gamma)+log(2). No compactness, bounded entropy or common reference is assumed in this original-premise endpoint. Separability of the ambient H is explicit, as in the source conjecture. Singular references, finite-dimensional H, infinite reference kernels and noncommuting families are included. No genuine relative-entropy D or Conjecture 2 conclusion follows merely from the energy continuity statement, and no historical priority claim is made here.

Normalized finite-entropy trace-log bridge

FiniteEntropyRelativeEntropy defines D only for actual positive trace-one operators rho and sigma with finite actual spectral entropy of rho. On any complete complex Hilbert space it uses the liminf of the real trace of rho times the difference of the bounded CFC logarithms of rho+epsilon I and sigma+epsilon I, followed by ENNReal.ofReal, at epsilon=1/(n+1). The proved extended characterization is E_sigma(rho)-S(rho). Finite energy gives the nonnegative unclamped real value and the lower bound norm(rho-sigma)^2/4; infinite energy and failed kernel inclusion give infinity. For arbitrary full spectral resolutions at finite energy, the signed double logarithmic series is absolutely summable and has the same value. Zero reference eigenvalues have zero weighted overlap before the real logarithm convention is used.

The same unit derives a fixed normalized sigma in the original nonempty closed real-convex family with finite continuous entropy. Entropy attainment and the center energy bound put the family inside a compact energy sublevel; closedness gives compactness. The existing compact common-reference theorem then makes the actual D finite, continuous and bounded by S(gamma)+log(2). The Hellinger and spectral-distribution suppliers retain the attribution in Library/QuantumBounds/androulakis2023nussbaumszkola.md; the analytic trace-class suppliers retain Tom Ole Diem’s Physlib provenance and license. This is a formal mathematical implication of the stated premises, without a claim of historical novelty or present literature openness.