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arXiv 2312.02004quant-ph
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Bloch球内部混合量子态的几何性质

Geometric aspects of mixed quantum states inside the Bloch sphere

Paul M. Alsing, Carlo Cafaro, Domenico Felice, Orlando Luongo

备注：
18 pages, 3 figures, 1 table
发表信息：
Quantum Reports 6, 90 (2024)
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AI总结：

本文探讨Bloch球内混合量子态的几何性质，比较Bures与Sjöqvist度量在测地线、有限距离上的差异，发现距离排名不保，影响复杂度与体积概念。

摘要：

在研究量子态的几何学时，公认混合态可以用无限多种度量来区分。不幸的是，这种自由度导致对物理上重要的几何量（如量子态的复杂度和体积）的度量依赖解释。在本文中，我们对Bloch球内部的Bures度量和Sjöqvist度量之间的差异进行了深入的讨论。首先，我们通过对每种度量的三种替代解释进行批判性讨论，开始了对这两种度量之间的形式化比较分析。其次，我们明确展示了在这两个度量流形上测地线路径的不同行为。第三，我们比较了使用这两种度量计算初始和最终混合态之间的有限距离。有趣的是，类似于在研究具有不同度量函数的实欧几里得空间的拓扑方面（例如，通常的欧几里得度量和出租车度量）所发生的情况，我们观察到基于混合量子态之间有限距离概念的相对排名在比较Bures度量和Sjöqvist度量确定的距离时不被保留。最后，我们简要讨论了这种基于度量的相对排名违反对混合量子态复杂度和体积概念的影响。

英文摘要：

When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we illustrate explicitly the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial and final mixed state when calculated with the two metrics. Interestingly, in analogy to what happens when studying topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of finite distance among mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of complexity and volume of mixed quantum states.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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