arXiv 1111.54252011-11-23quant-phcs.ITmath-phmath.ITmath.MP

量子信息理论中的问题是（不可）判定的吗？

Are problems in Quantum Information Theory (un)decidable?

Michael M. Wolf, Toby S. Cubitt, David Perez-Garcia

备注：: 11 pages

摘要：

本说明旨在促进讨论量子信息理论中出现的典型问题在多大程度上是算法可判定的（原则上而不是实践中）。纠缠理论和量子信道中的各种问题都可以通过量词消去来判定，只要它们允许一个紧凑的公式，而不需要对整数进行量化。然而，对于许多渐近定义的性质，它们必须对所有或一个整数N成立，有效的程序似乎很难找到。我们回顾了一些主要的工具（disproving）的可判定性，并将其应用到量子信息理论的问题。我们发现，像“我们能克服1/2的忠诚度吗？一个两量子比特的单态“很容易变得不可决定。仔细研究这些问题可能会排除量子信息理论中寻求的一些“单字母”公式。

英文摘要：

This note is intended to foster a discussion about the extent to which typical problems arising in quantum information theory are algorithmically decidable (in principle rather than in practice). Various problems in the context of entanglement theory and quantum channels turn out to be decidable via quantifier elimination as long as they admit a compact formulation without quantification over integers. For many asymptotically defined properties which have to hold for all or for one integer N, however, effective procedures seem to be difficult if not impossible to find. We review some of the main tools for (dis)proving decidability and apply them to problems in quantum information theory. We find that questions like "can we overcome fidelity 1/2 w.r.t. a two-qubit singlet state?" easily become undecidable. A closer look at such questions might rule out some of the "single-letter" formulas sought in quantum information theory.

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