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    Grothendieck bound in a single quantum system

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    Publication date
    2022-11
    End of Embargo
    2023-11-07
    Author
    Vourdas, Apostolos
    Keyword
    Grothendieck
    Single quantum system
    Rights
    © 2022 IOP Publishing. This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1751-8121/ac9dcf.
    Peer-Reviewed
    Yes
    Open Access status
    embargoedAccess
    
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    Abstract
    Grothendieck's bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a 'classical' quadratic form ${\cal C}$ that uses complex numbers in the unit disc, and takes values less than 1. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the 'quantum' quadratic form ${\cal Q}$ might take values greater than 1, up to the complex Grothendieck constant $k_\mathrm G$. The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the 'Grothendieck region' $(1,k_\mathrm G)$, which is a classically forbidden region in the sense that ${\cal C}$ cannot take values in it. Necessary (but not sufficient) conditions for ${\cal Q}$ taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with six and 12-dimensional Hilbert space, are shown to lead to ${\cal Q}$ in the Grothendieck region $(1,k_\mathrm G)$. They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.
    URI
    http://hdl.handle.net/10454/19241
    Version
    Accepted manuscript
    Citation
    Vourdas A (2022) Grothendieck bound in a single quantum system. Journal of Physics A: Mathematical and Theoretical. 55: 435206.
    Link to publisher’s version
    https://doi.org/10.1088/1751-8121/ac9dcf
    Type
    Article
    Notes
    The full-text of this article will be released for public view at the end of the publisher embargo on 07 Nov 2023.
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