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arXiv:2007.05049 · IEEE Transactions on Information Theory

A Tight Uniform Continuity Bound for the Arimoto-Rényi Conditional Entropy and its Extension to Classical-Quantum States

Michael G. Jabbour, Nilanjana Datta

Published Jan 13, 2022• 11 citations
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Abstract

We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha $-Rényi entropy for the range$\alpha \in [0, 1$). This definition of the conditional$\alpha $-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit$\alpha \to 1$, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional$\alpha $-Rényi entropy of a classical-quantum state, for$\alpha $in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit$\alpha \to 1$.

DOI
10.1109/tit.2022.3142812

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We prove a tight uniform continuity bound for Arimoto’s version of the conditionalα\alpha α-Rényi entropy for the rangeα∈[0,1\alpha \in [0, 1α∈[0,1). This definition of the conditionalα\alpha α-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 limitα→1\alpha \to 1α→1, 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 conditionalα\alpha α-Rényi entropy of a classical-quantum state, forα\alpha αin the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limitα→1\alpha \to 1α→1.
The Rényi entropy of general order unifies the well-known Shannon entropy with several other entropy notions, like the min-entropy or collision entropy. In contrast to the Shannon entropy, there seems to be no commonly accepted definition for the conditional Rényi entropy: several versions have been proposed and used in the literature. In this paper, we reconsider the definition for the conditional Rényi entropy of general order as proposed by Arimoto in the seventies. We show that this particular notion satisfies several natural properties. In particular, we show that it satisfies monotonicity under conditioning, meaning that conditioning can only reduce the entropy, and (a weak form of) chain rule, which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on. None of the other suggestions for the conditional Rényi entropy satisfies both these properties. Finally, we show a natural interpretation of the conditional Rényi entropy in terms of (unconditional) Rényi divergence, and we show consistency with a recently proposed notion of conditional Rényi entropy in the quantum setting.
In this short note, I show how a recent result of Alhejji and Smith (A tight uniform continuity bound for equivocation, 2019. arXiv:1909.00787v1) regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical–quantum states. The bound is optimal in the sense that there always exists a pair of classical–quantum states saturating the bound, and so, no further improvements are possible. An immediate application is a uniform continuity bound for the entanglement of formation that improves upon the one previously given by Winter (Commun Math Phys 347(1):291–313, 2016. arXiv:1507.07775). Two intriguing open questions are raised regarding other possible uniform continuity bounds for conditional entropy: one about quantum–classical states and another about fully quantum bipartite states.
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 −Reˊnyiand-Rényi and−Reˊnyiand\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.
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.
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 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.