[论文解读] Norms and Cones in the Theory of Quantum Entanglement

Nathaniel Johnston|arXiv (Cornell University)|Jul 5, 2012

Quantum Mechanics and Applications参考文献 247被引用 12

一句话总结

本文引入了一类范数——具体为 $ s(k) $-向量范数和 $ S(k) $-算子范数——作为量子纠缠理论中纠缠判据的推广。通过将这些范数与最小算子空间及对偶性联系起来，该研究确立了其在表征量子通道保真度、束缚纠缠以及纠缠几何度量方面的作用，为量子信息问题提供了新的分析工具。

ABSTRACT

There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.

研究动机与目标

• 开发一族范数，使其在量子纠缠理论中起到与纠缠判据相对偶的角色。
• 通过将其与最小算子空间及对偶性联系起来，为这些范数建立严格的数学基础。
• 将这些范数与量子信息理论中的关键问题相联系，包括非正部分转置束缚纠缠态的存在性问题。
• 利用新范数表征最小门保真度、最大输出纯度及几何纠缠度量。
• 提供一个统一框架，将正映射锥、范数与量子信息理论中的线性保持问题联系起来。

提出的方法

• 将 $ s(k) $-向量范数与 $ S(k) $-算子范数定义为 $ k $-正映射锥的对偶，推广了迹范数与钻石范数。
• 利用 Choi–Jamiołkowski 同构将超算子上的范数与算子锥联系起来，从而实现对偶性分析。
• 证明这些范数源自最小算子空间，类似于纠缠判据源自最小算子系统。
• 应用对偶性理论，表明 $ S(k) $-范数是 $ k $-超正映射锥的对偶，并推导其等距群与谱性质。
• 利用半定规划通过凸优化计算这些范数，特别利用对称扩展与 $ k $-正映射。
• 证明非 PPT 束缚纠缠态的存在性等价于涉及 $ S(1) $-范数的特定范数不等式。

实验结果

研究问题

• RQ1 能否构造一族范数，使其推广纠缠判据在检测纠缠中的作用？
• RQ2 这些新范数与量子信息理论中现有范数（如迹范数与钻石范数）有何关系？
• RQ3 这些新范数与非正部分转置束缚纠缠态的存在性之间存在何种联系？
• RQ4 能否利用 $ S(1) $-范数表征量子通道的最小门保真度？
• RQ5 这些范数与多体系统中几何纠缠度量之间有何关联？

主要发现

• $ s(k) $-向量范数与 $ S(k) $-算子范数被引入为纠缠判据的推广，其来源于最小算子空间。
• 证明 $ S(k) $-范数是 $ k $-超正映射锥的对偶，其等距群由酉与反酉变换刻画。
• 非 PPT 束缚纠缠态的存在性等价于涉及 $ S(1) $-范数的特定不等式，提供了新的操作性判据。
• 量子通道的最小门保真度被表征为其中 Choi 矩阵的 $ S(1) $-范数的倒数。
• 三体与四体态的几何纠缠度量可用 $ S(k) $-范数表示，扩展了已知的两体结果。
• 本文证明 $ ext{CP}(V(M_n)) $ 构成半群当且仅当 $ V(M_n) $ 是超齐次算子系统，将范数理论与量子通道中的半群结构联系起来。