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arXiv:1707.04249

Tight uniform continuity bound for a family of entropies

Eric P. Hanson, Nilanjana Datta

Published Jul 13, 2017• 8 citations
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Abstract

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. 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 ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.

DOI
10.48550/arxiv.1707.04249

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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.
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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$.
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.
For the sandwiched Rényi entropy the conditional entropy can be defined two ways: H~α↓(A∣B)ρ,H~α↑(A∣B)ρ\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{H}^\uparrow_α(A|B)_ρHα↓​(A∣B)ρ​,Hα↑​(A∣B)ρ​. In the limiting case, α=1α=1α=1, both definitions consolidate into conditional entropy H(A∣B)=S(AB)−S(B)H(A|B)=S(AB)-S(B)H(A∣B)=S(AB)−S(B). The continuity inequality for conditional entropy H(A∣B)H(A|B)H(A∣B), called the Alicki-Fannes-Winter (AWF) inequality, shows that if the states are close in trace-distance, then the conditional entropies are also close. Having the AWF inequality for conditional entropy, we show that the channel entropy defined through the relative entropy is continuous with respect to the diamond-distance between channels. Inspired by this, similar continuity inequalities for the Rényi conditional entropy H~α↑\widetilde{H}^\uparrow_αHα↑​ were obtained in the work [A. Marwah and F. Dupuis, J. Math. Phys. 63, 052201 (2022)]. We provide continuity bounds for the sandwiched Rényi and Tsallis conditional entropies H~α↓(A∣B)ρ,T~α↓(A∣B)ρ\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{T}^\downarrow_α(A|B)_ρHα↓​(A∣B)ρ​,Tα↓​(A∣B)ρ​ for states with the same marginal on the conditioning system. Similar to the previous bounds, our bound depends only on the dimension of the conditioning system. We apply this result to prove continuity of the channel entropy for Rényi and Tsallis channel entropies defined through the sandwiched Rényi and Tsallis relative entropies.
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We consider families of tight upper bounds on the difference S(ρ)−S(σ)S(\rho)-S(\sigma)S(ρ)−S(σ) with a rank/energy constraint imposed on the state ρ\rhoρ, which are valid provided that the state ρ\rhoρ partially majorizes the state σ\sigmaσ and is close to the state σ\sigmaσ with respect to the trace norm. The upper bounds within these families depend on the parameter mmm of partial majorization. The upper bounds corresponding to m=0m=0m=0 coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of ε\varepsilonε-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.