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

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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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Notes

  1. The Hamiltonian in this example is time dependent; however, thanks to the commutative nature of the system, the analysis is the same as the one from Eq. (2) with mutually commuting conditioned Hamiltonians.

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Acknowledgements

We would like to thank Piotr Szańkowski for critical reading and valuable suggestions. Fruitful discussions with Simon Milz, Philip Taranto, and Dariusz Chruściński are appreciated. We acknowledge support by the Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). Initial work on this topic was supported by funds from Polish National Science Center, Grant No. 2015/19/B/ST3/03152.

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  1. The International Centre for Theory of Quantum Technologies, University of Gdańsk, Jana Bażyńskiego 1A, Gdańsk, 80-309, Poland

    Fattah Sakuldee

  2. Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw, 02-668, Poland

    Fattah Sakuldee & Łukasz Cywiński

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Sakuldee, F., Cywiński, Ł. Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system. Quantum Inf Process 21, 244 (2022). https://doi.org/10.1007/s11128-022-03576-9

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