We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely, we prove analogues of the equation H(AB|C) = H(A|BC) + H(B|C), which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality but survives as an inequality of the form Hα(AB|C) ⩾ Hβ(A|BC) + Hγ(B|C), where the parameters α, β, γ obey the relation αα−1=ββ−1+γγ−1 and (α − 1)(β − 1)(γ − 1) > 1; if (α − 1)(β − 1)(γ − 1) < 1, the direction of the inequality is reversed.
We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely, we prove analogues of the equation H(AB|C) = H(A|BC) + H(B|C), which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality but survives as an inequality of the form Hα(AB|C) ⩾ Hβ(A|BC) + Hγ(B|C), where the parameters α, β, γ obey the relation αα−1=ββ−1+γγ−1 and (α − 1)(β − 1)(γ − 1) > 1; if (α − 1)(β − 1)(γ − 1) < 1, the direction of the inequality is reversed.
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here, we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies and derive a new entropic uncertainty relation.
We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.
Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).
For the sandwiched Rényi entropy the conditional entropy can be defined two ways: Hα↓(A∣B)ρ,Hα↑(A∣B)ρ. In the limiting case, α=1, both definitions consolidate into conditional entropy H(A∣B)=S(AB)−S(B). The continuity inequality for conditional entropy 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 Hα↑ 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 Hα↓(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.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory.
We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation.
We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.