[论文解读] Entanglement manipulation and distillability beyond LOCC

Eric Chitambar, Julio I. de Vicente+2 more|arXiv (Cornell University)|Nov 10, 2017

Quantum Information and Cryptography参考文献 48被引用 31

一句话总结

本文通过引入基于非纠缠、对偶非纠缠和PPT保持通道的广义资源理论，研究了超越LOCC的纠缠操控。结果表明，所有NPT纠缠态均可通过使用对偶非纠缠通道转化为LOCC可提纯的态，且纠缠度量（如非难性和Rényi熵）在这些广义操作下可能增加——凸显了与LOCC的关键差异，并揭示了探测NPT束缚纠缠的新途径。

ABSTRACT

When a quantum system is distributed to spatially separated parties, it is natural to consider how the system evolves when the parties perform local quantum operations with classical communication (LOCC). However, the structure of LOCC channels is exceedingly complex leaving many important physical problems unsolved. In this paper we consider generalized resource theories of entanglement based on different relaxations to the class of LOCC. The behavior of various entanglement measures is studied under non-entangling channels, as well as the newly introduced classes of dually non-entangling and PPT-preserving channels. In an effort to better understand the nature of LOCC bound entanglement, we study the problem of entanglement distillation in these generalized resource theories. We first show that unlike LOCC, general non-entangling maps can be superactivated, in the sense that two copies of the same non-entangling map can nevertheless be entangling. On the single-copy level, we demonstrate that every NPT entangled state can be converted into an LOCC-distillable state using channels that are both dually non-entangling and having a PPT Choi representation and that every state can be converted into an LOCC-distillable state using operations belonging to any family of polytopes that approximate LOCC. We then turn to the stochastic convertibility of multipartite pure states and show that any two states can be interconverted by any polytope approximation to the set of separable channels. Finally, as an analog to $k$-positive maps, we introduce and analyze the set of $k$-non-entangling channels.

研究动机与目标

• 通过研究LOCC之外的广义操作类，理解LOCC在纠缠操控中的局限性与结构特征。
• 通过检验某些纠缠态在更广类操作下是否仍不可提纯，探究NPT束缚纠缠是否存在。
• 分析纠缠度量（如可塑性、Rényi熵和非难性）在非纠缠及类似通道类下的行为。
• 引入并表征新的通道类，包括对偶非纠缠和k-非纠缠映射，作为k-正映射的类比。
• 确定在放宽的操作类下（特别是保持PPT结构的类），是否所有纠缠态均可实现纠缠提纯。

提出的方法

• 基于非纠缠、对偶非纠缠和PPT保持通道，引入广义资源理论，作为LOCC的松弛。
• 分析这些通道类下纠缠度量（可塑性、Rényi熵、非难性）的单调性。
• 使用方向导数和积分表示法，证明Rényi熵在非纠缠映射下的单调性或非单调性。
• 应用柯西-施瓦茨不等式和积分不等式，对Rényi熵泛函的方向导数进行有界。
• 构建显式通道族（如对偶非纠缠通道），实现任意纠缠态向LOCC可提纯态的转换。
• 引入k-非纠缠映射作为广义完全非纠缠映射的层级结构，类比于k-正映射。

实验结果

研究问题

• RQ1 是否所有NPT纠缠态均可通过LOCC之外的操作（如对偶非纠缠或PPT保持通道）转化为LOCC可提纯态？
• RQ2 Rényi α-熵等纠缠度量是否在非纠缠映射下增加，这对可提纯性意味着什么？
• RQ3 是否存在一个比LOCC更大的操作类，仍无法提纯某些NPT态，从而可能表明NPT束缚纠缠的存在？
• RQ4 PPT通道类与PPT保持通道类在保持或增加纠缠能力方面有何差异？
• RQ5 k-非纠缠映射是否能实现LOCC或可分操作下不可能的态转换，且是否保持纠缠度量的单调性？

主要发现

• 所有NPT纠缠态均可通过使用对偶非纠缠通道转化为LOCC可提纯态。
• 对于α ∈[0, 1/2)，纠缠的Rényi α-熵可在非纠缠通道下被任意增加，表明其非单调性。
• 对于α ∈[1/2, +∞]，Rényi α-熵与相对Rényi熵度量一致，因此在非纠缠通道下保持单调性。
• 在对偶非纠缠映射下，Schmidt秩可被增加，证明了纠缠的超激活现象。
• 在PPT保持通道下，非难性可被任意大的分数倍增加，表明其在该类中不具单调性。
• 所有多体纯态在可分通道集的任意多面体近似下均可相互转换，意味着在这些近似中随机可转换性具有普遍性。