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arXiv:2102.05827 · Annales Henri Poincaré

A Universal Representation for Quantum Commuting Correlations

Roy Araiza, Travis B. Russell, Mark Tomforde

Published Jun 11, 2022• 6 citations
arXivHTMLPDF
DOI
10.1007/s00023-022-01197-7

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We present in this paper that the CHSH game admits one and only one optimal state and so remove all ambiguity of representations. More precisely, we use the well-known universal description of quantum commuting correlations as state space on the universal algebra for two player games, and so allows us to unambigiously compare quantum strategies as states on this common algebra. As such we find that the CHSH game leaves a single optimal state on this common algebra. In turn passing to any non-minimal Stinespring dilation for this unique optimal state is the only source of ambiguity (including self-testing): More precisely, any state on some operator algebra may be uniquely broken up into its minimal Stinespring dilation as an honest representation for the operator algebra followed by its vector state. Any other Stinespring dilation however arises simply as an extension of the minimal Stinespring dilation (i.e., as an embedding of the minimal Hilbert space into some random ambient one). As such this manifests the only source of ambiguity appearing in most (but not all!) traditional self-testing results such as for the CHSH game as well as in plenty of similar examples. We then further demonstrate the simplicity of our arguments on the Mermin--Peres magic square and magic pentagram game. Most importantly however, we present this article as an illustration of operator algebraic techniques on optimal states and their quotients, and we further pick up the results of the current article in another following one (currently under preparation) to derive a first robust self-testing result in the quantum commuting model.
The possible quantum state of wormholes or little baby universes should be described by a nonfactorizable density matrix given by the path integral over the class of asymptotically flat four-geometries and asymptotically vanishing matter-field configurations which suitably match the prescribed data on three-surfaces which do not divide the manifold on the inner boundary. An instanton is here obtained which can represent such a nonsimply connected wormhole manifold, and is used to evaluate the asymptotic effective interaction of the resulting correlated baby universes with ordinary quantum fields at low energies in the Fock representation. It is argued that the demand of locality on the interacting quantum field commutators is no longer valid for correlated baby universes, and it is therefore concluded that quantum coherence in the matter-field sector is lost as a consequence of the interaction with nonsimply connected wormholes. A proposal is advanced that wormholes may provide us with a complementary quantum state sector that would induce the collapse of the state vector in the quantum measurement of any observable for ordinary microscopic matter systems.