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arXiv:2502.13669

Induced Quantum Divergence: A New Lens on Communication and Source Coding

Gilad Gour

Published Feb 19, 2025• 0 citations
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Abstract

This paper introduces the induced divergence, a new quantum divergence measure that replaces the hypothesis testing divergence in position-based decoding, simplifying the analysis of quantum communication and state redistribution while yielding tighter achievability bounds. Derived from a parent quantum relative entropy, it retains key properties such as data processing inequality and Löwner monotonicity. Like the hypothesis testing divergence, it depends on a smoothing parameter and interpolates between the parent relative entropy (as the smoothing parameter approaches one) and the min-relative entropy (as it approaches zero), the latter holding when applied to the sandwiched Rényi relative entropy of order $α\in[0,2]$. This framework refines the position-based decoding lemma, extending its applicability to a broader class of states and improving decoding success probabilities. Two key applications are considered: classical communication over quantum channels, where the induced divergence improves lower bounds on the distillable communication rate, and quantum state redistribution, where it leads to sharper bounds on communication costs. These results provide new insights into fundamental single-shot quantum information protocols and enhance existing analytical techniques.

DOI
10.48550/arxiv.2502.13669

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Quantum state redistribution is a fundamental quantum information theoretic primitive that captures a generic quantum communication scenario. In this work, we study the problem of entanglement assisted quantum state redistribution in one-shot setting and provide a new achievability result on the quantum communication required. Our bounds are in terms of max relative entropy and Rényi relative entropy of order 2. We show that our result is upper bounded by the result obtained in Berta, Christandl, Touchette (2016) (which is in terms of smooth conditional max and min entropies). We use the techniques of convex split and position based decoding (through pretty good measurement) to arrive at our result. Furthermore, in order to clarify the connection between our result and other recent results that use convex split and position based decoding, we prove a new relation between the hypothesis testing relative entropy and Rényi relative entropy of order 2.
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