Strong converse theorems using Rényi entropies

Abstract

We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [41], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where e and q denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

Non-Asymptotic Classical Data Compression With Quantum Side Information

“…We obtain finite blocklength bounds on this function (see Theorems 3 and 4), and determine exactly its asymptotic value (see Corollary 5), in terms of the sandwiched conditional Rényi entropy [19,20,21]. A non-asymptotic study of CQSW in the strong converse domain was also carried out by Tomamichel [22], and by Leditzky, Wilde, and Datta [23]. In these works, one-sided bounds were obtained, and hence the asymptotic value of the strong converse exponent was not determined.…”

Quantum resource theories

“…for all α ∈ (0, +∞) \ {1}. Both families of entropies have found applications in quantum information theory, particularly in quantum hypothesis testing (Hayashi and Tomamichel, 2016;Mosonyi and Hiai, 2011;Mosonyi and Ogawa, 2015), and various strong converse proofs (Cooney et al, 2016;König and Wehner, 2009;Leditzky et al, 2016;Wilde et al, 2014). Of special interest are certain limiting cases of D α (ρ σ) and D α (ρ σ).…”

Computable Rényi mutual information: Area laws and correlations

“…Motivated by this, we here explore alternative definitions of the Rényi mutual information, based on the notion of quantum Rényi divergences [16,17]. These are measures of distinguishability of quantum states, which play a pivotal role in information-theoretic tasks, such as single-shot communication protocols [18,19], channel coding [20][21][22][23][24][25] or hypothesis testing [26,27]. In principle, each of the many variants of quantum Rényi divergences [20,[28][29][30][31][32][33] allows us to define a mutual information as we explain in Appendix A.…”