Exponents for classical-quantum channel simulation in purified distance

TL;DR

This paper derives exact error and strong converse exponents for entanglement-assisted classical-quantum channel simulation using purified distance, providing single-letter formulas optimized over sandwiched Rènyi divergences, advancing understanding of quantum channel simulation limits.

Contribution

It presents the first exact formulas for error and strong converse exponents in entanglement-assisted quantum channel simulation, utilizing novel techniques to handle non-commutativity and optimize over Rènyi divergences.

Findings

Error exponent expressed as a single-letter formula over Rènyi divergences for $\alpha \in [1, \infty)$

Strong converse exponent formulated over Rènyi divergences for $\alpha \in [rac{1}{2}, 1]$

Methodology involves quantum fidelity properties, auxiliary channels, Chebyshev inequalities, and entropic bounds.

Abstract

We determine the exact error and strong converse exponent for entanglement-assisted classical-quantum channel simulation in worst case input purified distance. The error exponent is expressed as a single-letter formula optimized over sandwiched R\'enyi divergences of order $\alpha \in [1, \infty)$, notably without the need for a critical rate--a sharp contrast to the error exponent for classical-quantum channel coding. The strong converse exponent is expressed as a single-letter formula optimized over sandwiched R\'enyi divergences of order $\alpha\in [\frac{1}{2},1]$. As in the classical work [Oufkir et al., arXiv:2410.07051], we start with the goal of asymptotically expanding the meta-converse for channel simulation in the relevant regimes. However, to deal with non-commutativity issues arising from classical-quantum channels and entanglement-assistance, we critically use various properties of the quantum fidelity, additional auxiliary channel techniques, approximations via Chebyshev inequalities, and entropic continuity bounds.