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From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Simon Becker, Nilanjana Datta, Michael G. Jabbour

TL;DR

A variety of improved uniform continuity bounds are proved for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems and an open problem raised by Shirokov regarding the characterisation of states with finite entropy is settled.

Abstract

We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinite-dimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon- and von Neumann entropies. Hence, to deal with more general entropies, e.g. $α$-Rényi and $α$-Tsallis entropies, with $α\in (0,1)$, for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions and the equivalence of all Schatten norms in special spectral subspaces of the Hamiltonian. This approach is, as we show, motivated by continuity bounds for $α$-Rényi and $α$-Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. Bounds for $α>1$ are provided, too. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov on the so-called Finite-dimensional Approximation (FA) property.

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

TL;DR

A variety of improved uniform continuity bounds are proved for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems and an open problem raised by Shirokov regarding the characterisation of states with finite entropy is settled.

Abstract

We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energy-constrained continuity bound for the von Neumann entropy of states of infinite-dimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinite-dimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon- and von Neumann entropies. Hence, to deal with more general entropies, e.g. -Rényi and -Tsallis entropies, with , for which continuity bounds are known only for finite-dimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions and the equivalence of all Schatten norms in special spectral subspaces of the Hamiltonian. This approach is, as we show, motivated by continuity bounds for -Rényi and -Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. Bounds for are provided, too. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov on the so-called Finite-dimensional Approximation (FA) property.

Paper Structure

This paper contains 19 sections, 17 theorems, 129 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Entropies
  4. Quantitative continuity measures for functions
  5. Quantitative continuity measures for functions of operators
  6. Analytical background
  7. Main results
  8. Shannon entropies
  9. A mean-constrained Fano's inequality
  10. A mean-constrained continuity bound for the entropy of random variables with a countably infinite alphabet
  11. von Neumann entropies
  12. Classical $\alpha$-Rényi and $\alpha$-Tsallis entropies
  13. Discrete random variables
  14. Continuous random variables
  15. Quantum Rényi and Tsallis entropies
  16. ...and 4 more sections

Key Result

Proposition 2.3

Let $\hat{H}$ be a self-adjoint operator that is bounded from below with purely discrete spectrum. Let $E_1\le E_2\le E_3\le\cdots$ be the eigenvalues of $\hat{H}$; then where $\operatorname{lin}$ denotes the linear hull. In particular, this implies that for $\pi_1,..,\pi_n$ being the first $n$ eigen-projections of $\hat{H}$, counting multiplicity, and $\tilde{\pi}_1,..,\tilde{\pi}_n$ any other m

Figures (3)

  • Figure 1: We compare our tight bound \ref{['eq:theo:ContinuityQuantum']} to the bound \ref{['eq:BoundWinter3']} by Winter for general Hamitonians specialized to the single-mode number operator and also to \ref{['eq:BoundWinter2']} for different values of $\alpha.$ Due to the piecewise definition of \ref{['eq:BoundWinter2']} the latter curves also show a jump discontinuity. We see that for a wide range of $\alpha$ and low energies, \ref{['eq:BoundWinter3']} outperforms \ref{['eq:BoundWinter2']}, but for high energies, there exist values of $\alpha$ for which \ref{['eq:BoundWinter2']} outperforms \ref{['eq:BoundWinter3']}.
  • Figure 2: The upper left figure illustrates the right-hand side of our tight bound \ref{['eq:theo:ContinuityQuantum']}. The upper right figure illustrates the difference of the bound \ref{['eq:BoundWinter']} obtained by Winter W15 to our bound on the von Neumann entropy. The lower two figures compare the bound \ref{['eq:BoundWinter2']} found by Winter, for fixed $\alpha$, to our bound. The improvement is in all cases particularly significant for high energies and large trace distances.
  • Figure 3: These figures illustrate our findings of Corollary \ref{['corr:contdisc']}. We compute for 5000 random distributions $p$ on $\{1,2,...,1000\}$ the $\alpha$-Tsallis and $\alpha$-Rényi entropy by perturbing the distribution by $\varepsilon q$ where $q$ is another random vector. In the fourth plot we compute the bounds for different values of $\beta$ and observe that low values of $\beta$ yield better bounds. We choose the weights $w_i=i.$ In this histogram, the $x$-axis depicts the absolute value of the difference of the $\alpha$-Tsallis and $\alpha$-Rényi entropy for the realizations of our 5000 sample distributions and also the value of our bound for these realizations. The $y$-axis shows the number of times this value on the $x$-axis was achieved among the 5000 realizations.

Theorems & Definitions (37)

  • Remark 1
  • Definition 2.1: Entropies: Discrete random variables
  • Definition 2.2: Quantum entropies
  • Proposition 2.3: Courant-Fisher
  • Lemma 2.4
  • proof
  • Theorem 1: Mean-constrained Fano's inequality on $\mathbb N_0$
  • proof
  • Theorem 2: Fano's inequality for countably infinite alphabet with a general constraint
  • Definition 3.1: Maximal coupling
  • ...and 27 more