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Article: Generalised Kochen–Specker theorem for finite non-deterministic outcome assignments
Title | Generalised Kochen–Specker theorem for finite non-deterministic outcome assignments |
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Authors | |
Issue Date | 10-Oct-2024 |
Publisher | Nature Research |
Citation | npj Quantum Information, 2024, v. 10 How to Cite? |
Abstract | 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}). |
Persistent Identifier | http://hdl.handle.net/10722/348855 |
ISSN | 2023 Impact Factor: 6.6 2023 SCImago Journal Rankings: 2.824 |
DC Field | Value | Language |
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dc.contributor.author | Ramanathan, Ravishankar | - |
dc.date.accessioned | 2024-10-17T00:30:28Z | - |
dc.date.available | 2024-10-17T00:30:28Z | - |
dc.date.issued | 2024-10-10 | - |
dc.identifier.citation | npj Quantum Information, 2024, v. 10 | - |
dc.identifier.issn | 2056-6387 | - |
dc.identifier.uri | http://hdl.handle.net/10722/348855 | - |
dc.description.abstract | <p> <span>The Kochen–Specker (KS) theorem is a cornerstone result in quantum foundations, establishing that quantum correlations in Hilbert spaces of dimension </span><em>d</em><span> ≥ 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 </span><em>d</em><span> 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, </span><em>p</em><span>, 1 − </span><em>p</em><span>, 1} for </span><em>p</em><span> </span><span>∈</span><span> [0, 1/</span><em>d</em><span>) </span><span>∪</span><span> (1/</span><em>d</em><span>, 1/2]. The case </span><em>p</em><span> = 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}).</span> <br></p> | - |
dc.language | eng | - |
dc.publisher | Nature Research | - |
dc.relation.ispartof | npj Quantum Information | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.title | Generalised Kochen–Specker theorem for finite non-deterministic outcome assignments | - |
dc.type | Article | - |
dc.description.nature | published_or_final_version | - |
dc.identifier.doi | 10.1038/s41534-024-00895-w | - |
dc.identifier.volume | 10 | - |
dc.identifier.eissn | 2056-6387 | - |
dc.identifier.issnl | 2056-6387 | - |