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arXiv:2104.02019 · IEEE Transactions on Information Theory

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

Simon Becker, Nilanjana Datta, Michael G. Jabbour

Published Feb 23, 2023• 16 citations
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Abstract

We prove a variety of improved 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. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

DOI
10.1109/tit.2023.3248228

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We prove a variety of improved 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. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.α\alpha α-Rényi andα\alpha α-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.
We present a continuity bound for quantum conditional entropies in terms of fidelity, with no dependence on the dimension of the conditioning system. Our results encompass both the von Neumann entropy and one version of sandwiched Rényi entropy, which is particularly relevant for applications in quantum key distribution (QKD). In such applications, this bound should provide improved results as compared to previous continuity bounds based on trace distance. We also show that the bounds can be sharpened as expected for classical-quantum states, and verify that they hold when the conditioning system has countably infinite dimension. Finally, we present several variants tailored for exponential functions of the Rényi entropies instead, which should be relevant for QKD applications.
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)].
Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing(ε,ν)(\varepsilon ,\nu)(ε,ν)–degradable channels. These are$\varepsilon –closeindiamondnormand–close in diamond norm and–closeindiamondnormand\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.
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.
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.
We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha −Reˊnyientropyfortherange-Rényi entropy for the range−Reˊnyientropyfortherange\alpha \in [0, 1).Thisdefinitionoftheconditional). This definition of the conditional).Thisdefinitionoftheconditional\alpha −Reˊnyientropyisthemostnaturaloneamongthemultipleformswhichexistintheliterature,sinceitsatisfiestwodesirablepropertiesofaconditionalentropy,namely,thefactthatconditioningreducesentropy,andthattheassociatedreductioninuncertaintycannotexceedtheinformationgainedbyconditioning.Furthermore,ithasfoundinterestingapplicationsinvariousinformationtheoretictaskssuchasguessingwithsideinformationandsequentialdecoding.ThisconditionalentropyreducestotheconditionalShannonentropyinthelimit-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit−Reˊnyientropyisthemostnaturaloneamongthemultipleformswhichexistintheliterature,sinceitsatisfiestwodesirablepropertiesofaconditionalentropy,namely,thefactthatconditioningreducesentropy,andthattheassociatedreductioninuncertaintycannotexceedtheinformationgainedbyconditioning.Furthermore,ithasfoundinterestingapplicationsinvariousinformationtheoretictaskssuchasguessingwithsideinformationandsequentialdecoding.ThisconditionalentropyreducestotheconditionalShannonentropyinthelimit\alpha \to 1,andthisinturnallowsustorecovertherecentlyobtainedtightuniformcontinuityboundforthelatterfromourresult.Finally,weapplyourresulttoobtainatightuniformcontinuityboundfortheconditional, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional,andthisinturnallowsustorecovertherecentlyobtainedtightuniformcontinuityboundforthelatterfromourresult.Finally,weapplyourresulttoobtainatightuniformcontinuityboundfortheconditional\alpha −Reˊnyientropyofaclassical−quantumstate,for-Rényi entropy of a classical-quantum state, for−Reˊnyientropyofaclassical−quantumstate,for\alpha inthesamerangeasabove.Thisagainyieldsthecorrespondingknownboundfortheconditionalentropyofthestateinthelimitin the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limitinthesamerangeasabove.Thisagainyieldsthecorrespondingknownboundfortheconditionalentropyofthestateinthelimit\alpha \to 1$.