[论文解读] Catalytic and asymptotic equivalence for quantum entanglement

Ray Ganardi, Tulja Varun Kondra+1 more|arXiv (Cornell University)|May 5, 2023

Quantum Computing Algorithms and Architecture被引用 4

一句话总结

该论文建立了可 distill 的量子态下催化与渐近纠缠转换之间的基本等价性，证明使用纠缠催化剂无法提高渐近 singlet 提纯率。作者证明，在允许多个副本间存在关联的条件下，两种框架完全等价，解决了量子资源理论中长期存在的一个问题，并表明在渐近纠缠提纯中催化不提供超越标准渐近协议的额外优势。

ABSTRACT

Entanglement is a fundamental resource in quantum information processing, yet understanding its manipulation and transformation remains a challenge. Many tasks rely on highly entangled pure states, but obtaining such states is often challenging due to the presence of noise. Typically, entanglement manipulation procedures involving asymptotically many copies of a state are considered to overcome this problem. These procedures allow for distilling highly entangled pure states from noisy states, which enables a wide range of applications, such as quantum teleportation and quantum cryptography. When it comes to manipulating entangled quantum systems on a single copy level, using entangled states as catalysts can significantly broaden the range of achievable transformations. Similar to the concept of catalysis in chemistry, the entangled catalyst is returned unchanged at the end of the state manipulation procedure. Our results demonstrate that despite the apparent conceptual differences between the asymptotic and catalytic settings, they are actually strongly connected and fully equivalent for all distillable states. Our methods rely on the analysis of many-copy entanglement manipulation procedures which may establish correlations between different copies. As an important consequence, we demonstrate that using an entangled catalyst cannot enhance the asymptotic singlet distillation rate of a distillable quantum state. Our findings provide a comprehensive understanding of the capabilities and limitations of both catalytic and asymptotic state transformations of entangled states, and highlight the importance of correlations in these processes.

研究动机与目标

• 解决量子信息理论中催化与渐近纠缠转换之间的概念与操作关系。
• 确定纠缠催化是否能增强量子态的渐近提纯速率。
• 研究多副本纠缠操控协议中关联的作用。
• 在量子资源理论中建立边缘渐近速率与标准渐近速率的一般等价性。
• 阐明催化在纠缠提纯与资源理论背景下的局限性。

提出的方法

• 作者分析了允许多个量子态副本之间存在关联的多副本纠缠操控协议。
• 他们采用平方纠缠度量来约束转换速率，并证明涉及纠缠熵和自由操作的不等式。
• 关键技巧在于利用如平方纠缠等资源度量的超可加性与下半连续性，将边缘渐近速率与标准渐近速率关联起来。
• 证明依赖于构造 LOCC 协议，以在 n 趋近于无穷大时实现误差趋于零的近似转换。
• 作者通过假设资源度量的超可加性与下半连续性，将结果推广至任意量子资源理论。
• 他们利用纠缠度量的连续性与单调性性质，推导出可达转换速率的边界。

Figure 1: Equivalence between catalysis and reducibility. The left part of the figure shows a state $\rho$ being converted into another state $\sigma$ through a catalytic transformation by using a catalyst in the state $\tau$ . The right part of the figure shows that $\rho$ is asymptotically reducib

实验结果

研究问题

• RQ1 催化与渐近纠缠转换在可 distill 状态下是否存在根本等价性？
• RQ2 使用纠缠催化剂是否能提高量子态的渐近 singlet 提纯速率？
• RQ3 多个副本之间的关联在纠缠操控协议中起什么作用？
• RQ4 在何种条件下，量子资源理论中的边缘渐近速率与标准渐近速率一致？
• RQ5 如平方纠缠等纠缠度量的性质如何约束可达转换速率？

主要发现

• 尽管概念上存在差异，催化与渐近纠缠转换对所有可 distill 量子态完全等价。
• 添加纠缠催化剂无法提高可 distill 态的渐近 singlet 提纯速率，证明了催化在此情境下的根本局限性。
• 从态 ρ 到目标态 σ 的转换速率受其平方纠缠度量之比的约束：R_sq(ρ→σ) ≤ E_sq(ρ)/E_sq(σ)。
• 在一般量子资源理论中，若资源度量具有超可加性且在目标态处下半连续，则边缘渐近速率等于标准渐近速率。
• 证明表明渐近速率 r 满足 r < E_sq(ρ)/E_sq(σ) − ε + δ，其中 ε, δ > 0 任意小，从而得出紧致边界 R_sq(ρ→σ) ≤ E_sq(ρ)/E_sq(σ)。
• 结果确认催化在渐近纠缠提纯中不提供优势，凸显了标准渐近协议对可 distill 态的充分性。

Figure 2: Distillation of multipartite entanglement. We call a multipartite state distillable if it can be asymptotically converted into states comprising a singlet between each of the parties. The figure shows the desired final state for $3$ parties. The multipartite distillable entanglement is the