[论文解读] Entanglement cost of bipartite quantum channel discrimination under positive partial transpose operations

Chengkai Zhu, Shuyu He+2 more|arXiv (Cornell University)|Mar 12, 2026

Quantum Information and Cryptography被引用 0

一句话总结

本论文通过 k-injective PPT 测试器在局部性约束下定义并分析区分双量子信道的纠缠成本，并给出对称信道的精确结果的一次性纠缠辅助性能的 SDP。

ABSTRACT

Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank $k$ of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce $k$-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of $k$-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers. For every $k$, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.

研究动机与目标

• 在 LOCC 约束下量化实现对全局最优判别所需的纠缠资源。
• 引入 k-injectable testers 与 PPT 松弛以使问题可计算。
• 推导一次性纠缠辅助成功概率及相应的纠缠成本的 SDP。
• 给出高度对称信道的闭式纠缠成本，并通过具体示例进行说明。

提出的方法

• 定义 k-injectable PPT tester 来建模在 LOCC/SEP/PPT 约束下的纠缠辅助辨别。
• 将一次性辨别表述为 tester 优化，并导出最大 k-ebit 辅助成功概率的半正定规划（Theorem 4.1）。
• 将 LOCC 松弛为 SEP，将 PPT tester 松弛以获得可计算的外部界限，利用 PPT 映射。
• 引入具有纠缠注入显式资源端口的纠缠辅助 tester（定义 3）。
• 推导纠缠辅助辨别的原始对偶 SDP，并对协变信道给出对称性简化以降低维度。
• 从单信道推广到信道集合的凸集，得到对最坏情形成功概率的 SDP（Proposition 4.7）。

Figure 1: Protocols for entanglement-assisted discrimination of bipartite quantum channels. (a) A framework in which both parties prepared states and performed joint measurements to achieve diamond norm distance. (b) A general framework in which shared entanglement ( $\Phi_{k}$ ) assists both the st

实验结果

研究问题

• RQ1 对于 LOCC/SEP/PPT tester，最小的纠缠维度 k 是否足以使辨别概率达到由 diamond 范数给出的全局最优？
• RQ2 如何在 PPT 约束下建立并求解 SDP，以计算双量子信道辨别的一次性纠缠成本？
• RQ3 对称性（协变性）如何简化优化，以及对称信道对的确切纠缠成本是多少？
• RQ4 纠缠对双量子信道集合的辨别与单信道辨别相比有何不同的纠缠需求？
• RQ5 如 Werner–Holevo、简并化、去极化 SWAP 信道等具有代表性的对称信道的闭式纠缠成本是多少？

主要发现

信道对	维度	纠缠成本
Werner-Holevo	d	log2 d ebits
点对点去极化	d	1 ebit
双量子去极化	d_A×d_B	0
去极化 SWAP	d	1 ebit

• 在固定纠缠维度 k 的情况下，PPT tester 下的最大 k-ebit 辅助平均成功概率可以通过 SDP 计算（Theorem 4.1）。
• 该框架为若干高度对称的信道给出精确的一次性纠缠成本，包括 Werner–Holevo 信道，其成本为 log2 d ebits。
• 某些信道对在 PPT 约束下允许为零纠缠成本，例如两个双量子信道去极化与去极化 SWAP 信道，而点对点去极化信道成本为 1 ebit。
• Werner–Holevo 信道的纠缠成本随 log2 d ebits 增长，与其他参数无关。
• 在 PPT tester 下对去极化信道对的辨别，其成本行为不同于点对点情况，体现了双量子结构对纠缠需求的影响。
• 作者提供基于 SDP 的方法和对称性简化，使在相关信道族中的一次性纠缠成本数值计算变得可行。

Figure 2: The general framework for bipartite channel discrimination is presented. Compared with the well-understood setting of channel discrimination in Section 2.2 , several refinements are introduced. First, the channels in the ensemble $\Omega=\big\{(p_{j},{\cal N}_{A_{0}B_{0}\to A_{1}B_{1}}^{(j