Exponential Analysis for Entanglement Distillation

TL;DR

This paper introduces a comprehensive analysis of the reliability function in entanglement distillation, extending the framework to black-box scenarios and connecting it to quantum hypothesis testing, with implications for various free operation classes.

Contribution

It characterizes the reliability function of entanglement distillation using the quantum Hoeffding divergence and extends the analysis to black-box settings with multiple state hypotheses.

Findings

Exact finite blocklength result linked to composite hypothesis testing

Reliability function characterized by regularized quantum Hoeffding divergence

Constructed optimal distillation protocol under full state knowledge

Abstract

Historically, the focus in entanglement distillation has predominantly been on the distillable entanglement, and the framework assumes complete knowledge of the initial state. In this paper, we study the reliability function of entanglement distillation, which specifies the optimal exponent of the decay of the distillation error when the distillation rate is below the distillable entanglement. Furthermore, to capture greater operational significance, we extend the framework from the standard setting of known states to a black-box setting, where distillation is performed from a set of possible states. We establish an exact finite blocklength result connecting to composite correlated hypothesis testing without any redundant correction terms. Based on this, the reliability function of entanglement distillation is characterized by the regularized quantum Hoeffding divergence. In the special case of a pure initial state, our result reduces to the error exponent for entanglement concentration derived by Hayashi et al. in 2003. Given full prior knowledge of the state, we construct a concrete optimal distillation protocol. Additionally, we analyze the strong converse exponent of entanglement distillation. While all the above results assume the free operations to be non-entangling, we also investigate other free operation classes, including PPT-preserving, dually non-entangling, and dually PPT-preserving operations.