Smooth Entropy Bounds on One-Shot Quantum State Redistribution

In quantum state redistribution as introduced in [Luo and Devetak (2009)] and [Devetak and Yard (2008)], there are four systems of interest: the $A$ system held by Alice, the $B$ system held by Bob, the $C$ system that is to be transmitted from Alice to Bob, and the $R$ system that holds a purification of the state in the $ABC$ registers. We give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one-shot setting. Our bounds are in terms of the smooth conditional min- and max-entropy, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work [Oppenheim (2008)]: it decomposes the quantum state redistribution task into two simpler quantum state merging tasks by introducing a coherent relay. In the independent and identical (iid) asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information $I(C:R|B)$, and our bounds for the total cost converge to the conditional entropy $H(C|B)$. This yields an alternative proof of optimality of these rates for quantum state redistribution in the iid asymptotic limit. In particular, we obtain a strong converse for quantum state redistribution, which even holds when allowing for feedback.Comment: v3: 29 pages, 1 figure, extended strong converse discussio

The apex of the family tree of protocols: Optimal rates and resource inequalities

We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the Fully Quantum Slepian-Wolf protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in Quantum Information Theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic i.i.d. scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results, yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previously bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.Comment: 31 pages, 2 figures. Published versio

Strong converse theorems using R\'enyi entropies

We use a R\'enyi 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 [arXiv:1404.5940], 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 R\'enyi 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 R\'enyi divergence. In particular, we obtain novel bounds relating these quantities, as well as the R\'enyi conditional mutual information, to the fidelity of two quantum states.Comment: 40 pages, 5 figures; v4: Accepted for publication in Journal of Mathematical Physic

The Fidelity of Recovery is Multiplicative

Fawzi and Renner [Commun. Math. Phys. 340(2):575, 2015] recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e. the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brandao et al. [Phys. Rev. Lett. 115(5):050501, 2015] of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.Comment: v2: 9 pages, published versio

Convex-split and hypothesis testing approach to one-shot quantum measurement compression and randomness extraction

We consider the problem of quantum measurement compression with side information in the one-shot setting with shared randomness. In this problem, Alice shares a pure state with Reference and Bob and she performs a measurement on her registers. She wishes to communicate the outcome of this measurement to Bob using shared randomness and classical communication, in such a way that the outcome that Bob receives is correctly correlated with Reference and Bob's own registers. Our goal is to simultaneously minimize the classical communication and randomness cost. We provide a protocol based on convex-split and position based decoding with its communication upper bounded in terms of smooth max and hypothesis testing relative entropies. We also study the randomness cost of our protocol in both one-shot and asymptotic and i.i.d. setting. By generalizing the convex-split technique to incorporate pair-wise independent random variables, we show that our one shot protocol requires small number of bits of shared randomness. This allows us to construct a new protocol in the asymptotic and i.i.d. setting, which is optimal in both the number of bits of communication and the number of bits of shared randomness required. We construct a new protocol for the task of strong randomness extraction in the presence of quantum side information. Our protocol achieves error guarantee in terms of relative entropy (as opposed to trace distance) and extracts close to optimal number of uniform bits. As an application, we provide new achievability result for the task of quantum measurement compression without feedback, in which Alice does not need to know the outcome of the measurement. This leads to the optimal number of bits communicated and number of bits of shared randomness required, for this task in the asymptotic and i.i.d. setting.Comment: version 5: 29 pages, 1 figure. Added applications to randomness extraction (against quantum side information) and measurement compression without feedbac