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arXiv 1302.4801quant-ph
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Kochen-Specker定理在N比特Pauli群上的证明

Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group

Mordecai Waegell, P. K. Aravind

摘要:

我们提出了一些基于N量子位泡利群的Kochen-Specker(KS)定理的基于可观测性的证明,其中N >= 4,从而增加了之前已经提出的两个和三个量子位群的证明。这些证明具有吸引人的特点,它们可以以图表的形式呈现,通过检查它们是显而易见的。它们也是不可约的,因为它们不能通过忽略其中的量子比特和/或可观测量的某个子集而被简化为更小的证明。给出了一个简单的算法,用于将任何基于可观察性的KS证明转化为大量的基于投影的KS证明;如果基于可观察性的证明有O个可观察性,每个可观察性恰好出现在两个交换集中,并且任何两个交换集中最多有一个可观察性是公共的,则相关的基于投影的奇偶证明的数量是2^O。我们引入符号的观测量和投影仪为基础的KS证明,捕捉他们的重要功能,也传达了一种感觉,这两种类型的证明在N-量子位泡利群的巨大变化。我们讨论了一个无限族的基于可观测性的证明,其成员适用于从2开始的所有数量的量子位,并展示了如何使用它来生成基于投影的KS证明,该证明仅涉及N >= 2的形式为2^N的任何维度中的9个基(或实验背景)。我们的结果的一些影响进行了讨论。

英文摘要:

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the N-qubit Pauli group for N >= 4, thus adding to the proofs that have been presented earlier for the two- and three-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has O observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2^O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the N-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members apply to all numbers of qubits from two up, and show how it can be used to generate projectors-based KS proofs involving only nine bases (or experimental contexts) in any dimension of the form 2^N for N >= 2. Some implications of our results are discussed.

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