用高阶统计解决关于密度算符的Zeh问题
Solving the Zeh problem about the density operator with higher-order statistics
AI总结：
针对Zeh提出的同密度算符混合态无法区分的问题，研究借助量子信息处理相关方法证明两类同ρ混合态可区分，消除了不合理限制且不影响密度算符常规应用，还补充了测量公理相关溯源与Landau-Feynman情形的讨论。
摘要：
自1932年冯·诺依曼的一项工作以来，人们普遍认为如果两种统计混合态由同一个密度算符ρ表示，它们实际上就应被视为同一种混合态。1970年的一篇论文中，Zeh认为这一结果是他所称的“测量公理”的推论，他提出了一个涉及中子自旋的思想实验，并证明在该实验中密度算符无法反映全部信息。此后学界尚未达成共识，相关争议至今仍在持续。受此前在量子信息处理领域工作的启发，本文证明Zeh设想的两种具有相同ρ的混合态实际上是可以区分的。我们证明这一结果消除了此前被不合理地施加在统计混合态上的限制，但不会影响ρ的常规应用（例如在量子统计力学中的应用），同时冯·诺依曼熵仍保有其价值，甚至有助于厘清测量公理这一令人困惑的推论。为避免歧义，我们在附录中梳理了该假设的提出过程——冯·诺依曼更倾向于将其视为一种普遍性质，附录中还证明Zeh将其称为测量公理并提出相关问题的做法是正确的。我们还在另一种物理场景下讨论了密度算符的用途与内涵，我们将该场景称为Landau-Feynman情形，它涉及的是纠缠概念而非混合态概念。
英文摘要：
Since a 1932 work from von Neumann, it is generally considered that if two statistical mixtures are represented by the same density operator \r{ho}, they should in fact be considered as the same mixture. In a 1970 paper, Zeh, considering this result to be a consequence of what he called the measurement axiom, introduced a thought experiment with neutron spins and showed that in that experiment the density operator could not tell the whole story. Since then, no consensus has emerged yet, and controversies on the subject still presently develop. In this paper, stimulated by our previous works in the field of Quantum Information Processing, we show that the two mixtures imagined by Zeh, with the same \r{ho}, should however be distinguished. We show that this result suppresses a restriction unduly installed on statistical mixtures, but does not affect the general use of \r{ho}, e.g. in quantum statistical mechanics, and the von Neumann entropy keeps its own interest and even helps clarifying this confusing consequence of the measurement axiom. In order to avoid any ambiguity, the identification of the introduction of this postulate, which von Neumann rather suggested to be a general property, is given in an appendix where it is shown that Zeh was right when he spoke of a measurement axiom and identified his problem. The use and content of a density operator is also discussed in another physical case which we are led to call the Landau-Feynman situation, and which implies the concept of entanglement rather than the one of mixed states.
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