Exponential Analysis for Entanglement Distillation
Zhiwen Lin, Ke Li, Kun Fang
TL;DR
This work develops a black-box framework for entanglement distillation, deriving an exact finite-blocklength link between distillation error and composite correlated hypothesis testing, and showing the reliability function is captured by the regularized quantum Hoeffding divergence $E^{\operatorname{rf}}(\mathscr{R},\mathcal{F}_{\operatorname{NE}},r)=H_r^\infty(\mathscr{R}\|\mathscr{F})$ for $0<r<D^{\infty}(\mathscr{R}\|\mathscr{F})$. It recovers Hayashi et al.'s pure-state results in the appropriate limit and provides a strong-converse bound via the regularized Hoeffding anti-divergence, with extensions to PPT-preserving, dually non-entangling, and dually PPT-preserving operations. When the initial state is known, the authors construct an explicit non-entangling distillation protocol that achieves the reliability function, and they connect the rate-versus-error trade-off to composite hypothesis testing without extraneous correction terms. The framework covers both general state ensembles and the pure-state case, offering precise, operation-class dependent exponents and paving the way for robust distillation under state uncertainty, including potential applications to quantum communication and cryptography. Overall, the results provide a rigorous, finite-blocklength foundation for entanglement distillation under uncertainty and suggest several avenues for future work, such as continuity properties of regularized Petz Rényi divergences and multipartite resource theories.
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.