Exponents for classical-quantum channel simulation in purified distance

TL;DR

This work derives exact large-deviation exponents for channel simulation in classical and classical-quantum settings under purified distance, covering non-signaling, shared randomness, and entanglement-assisted scenarios. It provides single-letter formulas: the error exponent $\frac{1}{2}\sup_{\alpha \ge 0} \alpha (r-\widetilde{I}_{1+\alpha}(W))$ and the strong converse exponent $\sup_{\frac{1}{2} \le \alpha \le 1} \frac{1-\alpha}{\alpha}(\widetilde{I}_{\alpha}(W)-r)$, with appropriate substitutions for classical channels. The analysis uses meta-converse expansions, auxiliary-channel constructions, Chebyshev approximations, and fidelity-based techniques to handle non-commutativity in the quantum setting, and establishes that entanglement-assisted and non-signaling exponents align in the CQ case via a rounding argument. A key implication is the absence of a critical rate in the error exponent, unlike some classical results, and the results provide a complete large-deviation picture for channel simulation under purified distance. These findings deepen the understanding of simulation limits in quantum information processing and offer precise benchmarks for resource-assisted channel simulation performance.

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 Rényi divergences of order $α\in [1, \infty)$, 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 Rényi divergences of order $α\in [\frac{1}{2},1]$. 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.

Theorems & Definitions (24)

• Definition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Corollary 6
• Proposition 7
• Theorem 8
• Proposition 9
• Lemma 10