Tight Analysis of Convex Splitting with Applications in Quantum Information Theory

Convex splitting is a powerful technique in quantum information theory that applies to proving the achievability of numerous information-processing protocols. In this paper, we establish a one-shot error exponent and a one-shot strong converse for convex splitting with trace distance as the error criterion. Our result exhibits the following features. Firstly, the derived error exponent is given in terms of a generalized sandwiched Rényi mutual information which is additive for product states. Hence, our bound directly applies to the n-fold product scenario for any blocklength. Secondly, the one-shot error bound is meaningful in the sense that the error exponent is all positive on the achievable rate region. This leads to new one-shot exponent results in various tasks in quantum information theory. Conversely, we prove a one-shot strong converse to demonstrate that the convex splitting error converges to one exponentially fast at rates outside the achievable rate region. We also establish an optimal one-shot characterization of the sample complexity for convex splitting, which yields matched second-order asymptotics, showing the tightness of our result. This then leads to a stronger one-shot analysis in quantum information theory.Our technique is to introduce a unified functional analytic approach using Kosaki’s noncommutative weighted Lp norm. Our results then follow from tight analysis based on such a norm framework.The full version of the manuscript can be found at [arXiv:2304.12055] [1].