有限非确定性结果分配的广义科钦-斯派克定理

IF 6.6 1区 物理与天体物理 Q1 PHYSICS, APPLIED npj Quantum Information Pub Date : 2024-10-10 DOI:10.1038/s41534-024-00895-w
Ravishankar Ramanathan
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引用次数: 0

摘要

科钦-斯派克(KS)定理是量子基础的奠基性成果,它确定了维数 d ≥ 3 的希尔伯特空间中的量子相关性无法用(一致的)隐变量理论来解释,该理论为每次测量分配了一个单一的确定性结果。具体地说,在这些维度中存在着有限的向量集,以至于没有任何非上下文确定性({0, 1})结果赋值能够遵守排他性和完备性规则--即对每一组相互正交向量的赋值之和≤1,以及对任意 d 个相互正交向量的值赋值之和等于 1。量子基础的另一个核心结果是格里森定理,它证明了玻恩法则是量子形式主义的数学结果。KS 定理可视为格里森定理和逻辑紧凑性定理的结果。与此类似,格里森定理也指出了 KS 型有限矢量构造的存在,以排除 {0, 1} 情况之外的其他有限字母结果赋值。在这里,我们提出了KS定理的一般化,即在p∈ [0, 1/d) ∪ (1/d, 1/2] 的情况下,排除结果赋值在集合{0, p, 1 - p, 1}中的隐变量理论。p = 1/2 的情况尤其具有物理意义。我们证明,在这种情况下,结果排除了从根本上说是二元的(一致的)隐变量理论,即每次测量从根本上说最多有两种结果的理论(与 KS 排除的每次测量只有一个确定性结果相反)。我们提出了这个广义 KS 定理的一个独立于设备的应用,构建了一个双人非局域博弈,其中存在完美的量子获胜策略(伪心灵感应博弈),而即使为博弈者提供额外的 PR-box 类型的无信号资源(边际为 {0, 1/2, 1}),也不存在完美的经典策略。
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Generalised Kochen–Specker theorem for finite non-deterministic outcome assignments

The Kochen–Specker (KS) theorem is a cornerstone result in quantum foundations, establishing that quantum correlations in Hilbert spaces of dimension d ≥ 3 cannot be explained by (consistent) hidden variable theories that assign a single deterministic outcome to each measurement. Specifically, there exist finite sets of vectors in these dimensions such that no non-contextual deterministic ({0, 1}) outcome assignment is possible obeying the rules of exclusivity and completeness—that the sum of assignments to every set of mutually orthogonal vectors be ≤1 and the sum of value assignments to any d mutually orthogonal vectors be equal to 1. Another central result in quantum foundations is Gleason’s theorem that justifies the Born rule as a mathematical consequence of the quantum formalism. The KS theorem can be seen as a consequence of Gleason’s theorem and the logical compactness theorem. In a similar vein, Gleason’s theorem also indicates the existence of KS-type finite vector constructions to rule out other finite-alphabet outcome assignments beyond the {0, 1} case. Here, we propose a generalisation of the KS theorem that rules out hidden variable theories with outcome assignments in the set {0, p, 1 − p, 1} for p [0, 1/d) (1/d, 1/2]. The case p = 1/2 is especially physically significant. We show that in this case the result rules out (consistent) hidden variable theories that are fundamentally binary, i.e., theories where each measurement has fundamentally at most two outcomes (in contrast to the single deterministic outcome per measurement ruled out by KS). We present a device-independent application of this generalised KS theorem by constructing a two-player non-local game for which a perfect quantum winning strategy exists (a Pseudo-telepathy game) while no perfect classical strategy exists even if the players are provided with additional no-signaling resources of PR-box type (with marginals in {0, 1/2, 1}).

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来源期刊
npj Quantum Information
npj Quantum Information Computer Science-Computer Science (miscellaneous)
CiteScore
13.70
自引率
3.90%
发文量
130
审稿时长
29 weeks
期刊介绍: The scope of npj Quantum Information spans across all relevant disciplines, fields, approaches and levels and so considers outstanding work ranging from fundamental research to applications and technologies.
期刊最新文献
Trainability barriers and opportunities in quantum generative modeling Interferometry of quantum correlation functions to access quasiprobability distribution of work Lightcone bounds for quantum circuit mapping via uncomplexity Unified linear response theory of quantum electronic circuits Cavity-assisted resonance fluorescence from a nitrogen-vacancy center in diamond
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