Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions

There are different inequivalent ways to define the R\'enyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csisz\'ar has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of R\'enyi capacity, defined in terms of the sandwiched quantum R\'enyi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that $$sc(W,R,P)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^*(W,P)\right],$$ where $\chi_{\alpha}^*(W,P)$ is the $P$-weighted sandwiched R\'enyi divergence radius of the image of the channel.Comment: 46 pages. V7: Added the strong converse exponent with cost constrain

The structure of Renyi entropic inequalities

We investigate the universal inequalities relating the alpha-Renyi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropy (alpha=1) which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is non-negativity: In other words, any collection of non-negative numbers assigned to the nonempty subsets of n parties can be arbitrarily well approximated by the alpha-entropies of the 2^n-1 marginals of a quantum state. For alpha>1 we show analogously that there are no non-trivial homogeneous (in particular no linear) inequalities. On the other hand, it is known that there are further, non-linear and indeed non-homogeneous, inequalities delimiting the alpha-entropies of a general quantum state. Finally, we also treat the case of Renyi entropies restricted to classical states (i.e. probability distributions), which in addition to non-negativity are also subject to monotonicity. For alpha different from 0 and 1 we show that this is the only other homogeneous relation.Comment: 15 pages. v2: minor technical changes in Theorems 10 and 1

Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication

This paper studies the difficulty of discriminating between an arbitrary quantum channel and a "replacer" channel that discards its input and replaces it with a fixed state. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein's lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Renyi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel N as the optimal type II error exponent when discriminating between a large number of independent instances of N and an arbitrary "worst-case" replacer channel chosen from the set of all replacer channels.Comment: v3: 35 pages, 4 figures, accepted for publication in Communications in Mathematical Physic

Some continuity properties of quantum R\'enyi divergences

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel R\'enyi $\alpha$-divergences over all $\alpha>1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched R\'enyi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) R\'enyi divergences, which may be of independent interest.Comment: v4: Continuity is studied on more general sets of the form $\rho\le f(\sigma)$ for a large class of functions $f$. 44 page

Geometric relative entropies and barycentric R\'enyi divergences

We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum R\'enyi divergences starting from a set of monotone quantum relative entropies. Despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies. Here we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum R\'enyi quantities corresponding to any finite set of quantum relative entropies $(D^{q_x})_{x\in X}$ and signed probability measure $P$, as $$Q_P^{\mathrm{b},\mathbf{q}}((\rho_x)_{x\in X}):=\sup_{\tau\ge 0}\left\{\text{Tr}\,\tau-\sum_xP(x)D^{q_x}(\tau\|\rho_x)\right\}.$$ We show that monotone quantum relative entropies define monotone R\'enyi quantities whenever $P$ is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical R\'enyi $\alpha$-divergence in the 2-variable case ($X=\{0,1\}$, $P(0)=\alpha$). We show that if both $D^{q_0}$ and $D^{q_1}$ are monotone and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric R\'enyi divergences are strictly between the log-Euclidean and the maximal R\'enyi divergences, and hence they are different from any previously studied quantum R\'enyi divergence.Comment: v4: Extended "Conclusion and Outlook". 68 page

On the error exponents of binary quantum state discrimination with composite hypotheses

We consider the asymptotic error exponents in the problem of discriminating two sets of quantum states. It is known that in many relevant setups in the classical case (commuting states), the Stein, the Chernoff, and the direct exponents coincide with the worst pairwise exponents of discriminating arbitrary pairs of states from the two sets. On the other hand, counterexamples to this behaviour in finite-dimensional quantum systems have been demonstrated recently for the Chernoff and the Stein exponents of composite quantum state discrimination with a simple null-hypothesis and an alternative hypothesis consisting of continuum many states. In this paper we provide further insight into this problem by showing that the worst pairwise exponents may not be achievable for any of the Stein, the Chernoff, or the direct exponents, already when the null-hypothesis is simple, and the alternative hypothesis consists of only two non-commuting states. This finiteness of the hypotheses in our construction is especially significant, because, as we show, with the alternative hypothesis being allowed to be even just countably infinite, counterexamples exits already in classical (although infinite-dimensional) systems. On the other hand, we prove the achievability of the worst pairwise exponents in two paradigmatic settings: when both the null and the alternative hypotheses consist of finitely many states such that all states in the null-hypothesis commute with all states in the alternative hypothesis (semi-classical case), and when both hypotheses consist of finite sets of pure states.Comment: 26 page

Quantum Rényi Divergences and the Strong Converse Exponent of State Discrimination in Operator Algebras

The sandwiched R\'enyi $\alpha$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of R\'enyi divergences as the tight quantifiers of the trade-off between the two error probabilities in the strong converse domain of state discrimination. In this paper we show the same for the sandwiched R\'enyi divergences of two normal states on an injective von Neumann algebra, thereby establishing the operational significance of these quantities. Moreover, we show that in this setting, again similarly to the finite-dimensional case, the sandwiched R\'enyi divergences coincide with the regularized measured R\'enyi divergences, another distinctive feature of the former quantities. Our main tool is an approximation theorem (martingale convergence) for the sandwiched R\'enyi divergences, which may be used for the extension of various further results from the finite-dimensional to the von Neumann algebra setting. We also initiate the study of the sandwiched R\'enyi divergences of pairs of states on a $C^*$-algebra, and show that the above operational interpretation, as well as the equality to the regularized measured R\'enyi divergence, holds more generally for pairs of states on a nuclear $C^*$-algebra.Comment: 35 page