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Quantum accessible information and classical entropy inequalities

A. S. Holevo, A. V. Utkin

TL;DR

This work develops a general optimality criterion for accessible information in quantum ensembles, linking measurement optimization to discrete entropy inequalities that are discrete analogues of the log-Sobolev inequality. By applying the criterion to symmetric ensembles, notably quantum pyramids, the authors derive tight, parameterized entropy bounds and propose information-optimal observables, with explicit results for acute, obtuse, and flat pyramids. The approach yields a unified framework that recovers known bounds and provides new ones, including a 1-parameter family of bounds for multidimensional distributions and a conjectured globally information-optimal measurement for quantum pyramids, supported by analytical reductions and numerical evidence. These results offer both analytical tools and numerical benchmarks for proving or approximating optimal measurements in quantum communication scenarios, and they illuminate the interplay between measurement design, entropy inequalities, and channel capacity in structured quantum systems. $A(\\mathcal{E})$ denotes accessible information and, at optimum, $A(\\mathcal{E}) = H(\\pi) - \\mathrm{Tr}\, \\overline{\\rho} \\Lambda_0$, while entropy bounds take forms such as $-\\sum_j t_j \\log t_j \\\ge \\\mu_0(p) (\\sum_j \\sqrt{t_j})^2 - \\mu_1(p)$ with suitable parameters, and in the two-state case reduce to $h(t) \\\ge (\\\lambda_0(p) + \\\lambda_1(p)) \\\sqrt{t(1-t)} - \\frac{\\lambda_1(p) - \\lambda_0(p)}{2}$.

Abstract

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the optimality criterion recently obtained in [7], when applied to specific ensembles of states, leads to nontrivial tight lower bounds for the Shannon entropy that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) put forward and numerically substantiated in [2] is reconsidered and the corresponding tight entropy inequalities are proposed. Via the optimality criterion, this suggests also a proof of the conjecture concerning globally information-optimal observables for quantum pyramids put forward in [2].

Quantum accessible information and classical entropy inequalities

TL;DR

This work develops a general optimality criterion for accessible information in quantum ensembles, linking measurement optimization to discrete entropy inequalities that are discrete analogues of the log-Sobolev inequality. By applying the criterion to symmetric ensembles, notably quantum pyramids, the authors derive tight, parameterized entropy bounds and propose information-optimal observables, with explicit results for acute, obtuse, and flat pyramids. The approach yields a unified framework that recovers known bounds and provides new ones, including a 1-parameter family of bounds for multidimensional distributions and a conjectured globally information-optimal measurement for quantum pyramids, supported by analytical reductions and numerical evidence. These results offer both analytical tools and numerical benchmarks for proving or approximating optimal measurements in quantum communication scenarios, and they illuminate the interplay between measurement design, entropy inequalities, and channel capacity in structured quantum systems. denotes accessible information and, at optimum, , while entropy bounds take forms such as with suitable parameters, and in the two-state case reduce to .

Abstract

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the optimality criterion recently obtained in [7], when applied to specific ensembles of states, leads to nontrivial tight lower bounds for the Shannon entropy that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) put forward and numerically substantiated in [2] is reconsidered and the corresponding tight entropy inequalities are proposed. Via the optimality criterion, this suggests also a proof of the conjecture concerning globally information-optimal observables for quantum pyramids put forward in [2].

Paper Structure

This paper contains 12 sections, 13 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Accessible information: a criterion for optimality
  3. Example: two pure states
  4. Multidimensional case
  5. The lower bound for the Shannon entropy: formulation and discussion
  6. The conjecture: “ quantum pyramids”
  7. Checking the optimality conditions: the case of acute pyramids
  8. The case of obtuse pyramids
  9. The case of flat pyramid
  10. Suggestions for proofs
  11. The case of theorem \ref{['th2']}
  12. The case of theorem \ref{['th3']}

Key Result

Theorem 1

The minimization problem has the dual problem The following statements are equivalent: The solution of the dual problem (dprob) is unique on the support of the operator $\overline{\rho }$ i.e. $\Lambda _{0}\overline{\rho }=\Lambda _{0}^{\prime}\overline{\rho }$ for any two solutions $\Lambda _{0},\Lambda _{0}^{\prime}.$ In particular, if $\overline{\rho }$ is non-degenerate, then the

Theorems & Definitions (26)

  • Remark 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • Remark 2
  • Remark 3
  • ...and 16 more