Tight any-shot quantum decoupling

TL;DR

This paper introduces a new one-shot quantum decoupling theorem based on quantum relative entropy, providing tight bounds and operational applications in quantum information tasks like state merging, entanglement distillation, and channel coding.

Contribution

It presents a novel one-shot decoupling theorem using quantum relative entropy and derives tight bounds for various quantum information processing tasks.

Findings

Bounded decoupling error by sandwiched Rényi conditional entropies

Single-letter error exponent for quantum state merging

Regularized error bounds for entanglement distillation and channel coding

Abstract

Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched R\'enyi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) a single-letter expression for the exact error exponent of quantum state merging in terms of Petz-R\'enyi conditional entropies, and (ii) regularized expressions for the achievable error exponent of entanglement distillation and quantum channel coding in terms of Petz-R\'enyi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.