Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem

Victoria J Wright1 and Ravi Kunjwal2

1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain
2Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium

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The Kochen–Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen–Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of the KS theorem for multiqubit systems. In particular, unentangled but nonlocal measurements—whose eigenstates exhibit ''nonlocality without entanglement"—are insufficient for such proofs. This also implies that proving Gleason's theorem on a multiqubit system necessarily requires entangled projections, as shown by Wallach [Contemp Math, 305: 291-298 (2002)]. Secondly, we show that a multiqubit state admits a statistical (state-dependent) proof of the KS theorem if and only if it can violate a Bell inequality with projective measurements. We also establish the relationship between entanglement and the theorems of Kochen–Specker and Gleason more generally in multiqudit systems by constructing new examples of KS sets. Finally, we discuss how our results shed new light on the role of multiqubit contextuality as a resource within the paradigm of quantum computation with state injection.

Very small physical systems, such as photons of light, behave in ways that contradict the theories of physics scientists used before the advent of quantum theory. Quantum theory was developed to describe these very small systems and does so very successfully. Broadly, the theories predating quantum theory, often called classical theories, are all noncontextual. A theory is noncontextual if every observable property of a system, such as its position, can be assumed to have a definite value at all times such that whenever and however this property is measured one will find this value. The Kochen-Specker theorem demonstrates how the predictions of quantum theory cannot be explained in a noncontextual way.

Quantum theory also has other major differences from classical theories, with two prominent examples being Bell nonlocality and entanglement. Unlike Kochen-Specker contextuality described above which involves a single quantum system, Bell nonlocality and entanglement are properties only present when we study multiple quantum systems together. In this work, however, we show that for systems of multiple qubits (as in a quantum computer) both Bell nonlocality and entanglement are essential for the presence of Kochen–Specker contextuality.

As well as relevance to the foundations of physics, we discuss how our findings may lead to a better understanding of quantum advantage in quantum computing. Quantum advantage must stem from the differences between the quantum and classical physics that describes quantum and classical computers, respectively. Therefore, understanding the nonclassicality of the multiqubit systems we study presents a path a harnessing the power of quantum advantage.

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[4] Tim Chan and Andrei Constantin, "The Contextual Fraction as a Measure of Entanglement", arXiv:2403.06896, (2024).

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