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arXiv:1706.02212 · Journal of Mathematical Physics

Maximum and minimum entropy states yielding local continuity bounds

Eric P. Hanson, Nilanjana Datta

Published Apr 1, 2018• 23 citations
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Abstract

Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρε*(σ) [respectively, ρ*,ε(σ)] which has the maximum (respectively, minimum) entropy among all states which lie in a specified neighborhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρε*(σ) and ρ*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the minimum- and maximum-entropies, providing explicit formulas for certain smoothed quantities. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely, a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s rule, and majorization theory.

DOI
10.1063/1.5000120

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Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρε*(σ) [respectively, ρ*,ε(σ)] which has the maximum (respectively, minimum) entropy among all states which lie in a specified neighborhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρε*(σ) and ρ*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the minimum- and maximum-entropies, providing explicit formulas for certain smoothed quantities. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely, a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s rule, and majorization theory.
We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the ααα-Rényi entropy, SαS_αSα​, for α∈(0,1)α\in (0,1)α∈(0,1). We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood (ε\varepsilonε-ball) of a given state, and thus was the minimal state in majorization order in the ε\varepsilonε-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over εεε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance εεε, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the ααα-Rényi entropy, for which we obtain novel bounds in the case α>1α> 1α>1. We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.
We establish a tight upper bound for the difference in von Neumann entropies between two quantum states,ρ1\rho _{1}ρ1​andρ2\rho _{2}ρ2​. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator(ρ1−ρ2)(\rho _{1} - \rho _{2})(ρ1​−ρ2​). This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.
Entropies play a fundamental role in classical and quantum information theory. There are a plethora of quantum entropies and they satisfy a host of interesting properties. The von Neumann entropy, the first quantum entropy to be defined and arguably the most important of them, satisfies a powerful property called strong subadditivity. It was proved by Lieb and Ruskai in 1973 and has turned out to be an essential tool in establishing various important results in quantum information theory. We briefly review this property and its applications. Our main focus, however, is on another desirable property of entropies, namely, continuity. We introduce a notion of majorization flow and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for Schur concave functions with a particular emphasis on entropic functions relevant in information theory. In particular, for the case of the α\alphaα-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case α>1\alpha>1α>1, thus resolving an open problem and providing a substantial improvement over previously known bounds.
Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].
We construct for a Schur concave function fff on the set of quantum states a tight upper bound on the difference f(ρ)−f(σ)f(ρ)-f(σ)f(ρ)−f(σ) for a quantum state ρρρ with finite f(ρ)f(ρ)f(ρ) and any quantum state σσσ mmm-partially majorized by the state ρρρ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition 12∥ρ−σ∥1≤ε\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon21​∥ρ−σ∥1​≤ε and find simple sufficient conditions for vanishing this bound with  min⁡{ε,1/m}→0 \,\min\{\varepsilon,1/m\}\to0\,min{ε,1/m}→0. The obtained results are applied to the von Neumann entropy. The concept of ε\varepsilonε-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.