Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system

• URL: http://arxiv.org/abs/2111.14694v2
• Date: Sat, 16 Jul 2022 13:25:52 GMT
• Title: Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system
• Authors: Fattah Sakuldee,{\L}ukasz Cywi\'nski
• Abstract summary: We consider a quantum probe $P$ undergoing pure dephasing due to its interaction with a quantum system $S$. For $P$ being a qubit, the witness is particularly simple: observation of breaking of Kolmogorov consistency of sequential measurements on a qubit coupled to $S$ means that the accessible algebra of $S$ is noncommutative.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a quantum probe $P$ undergoing pure dephasing due to its interaction with a quantum system $S$. The dynamics of $P$ is then described by a well-defined sub-algebra of operators of $S,$ i.e. the "accessible" algebra on $S$ from the point of view of $P.$ We consider sequences of $n$ measurements on $P,$ and investigate the relationship between Kolmogorov consistency of probabilities of obtaining sequences of results with various $n,$ and commutativity of the accessible algebra. For a finite-dimensional $S$ we find conditions under which the Kolmogorov consistency of measurement on $P,$ given that the state of $S$ can be arbitrarily prepared, is equivalent to the commutativity of this algebra. These allow us to describe witnesses of nonclassicality (understood here as noncommutativity) of part of $S$ that affects the probe. For $P$ being a qubit, the witness is particularly simple: observation of breaking of Kolmogorov consistency of sequential measurements on a qubit coupled to $S$ means that the accessible algebra of $S$ is noncommutative.

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