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On the Extension of a Family of Projections to a Positive Operator-Valued Measure

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Abstract

In this paper, the problem of constructing a measure which accepts values within a positive cone of bounded operators in a Hilbert space is considered. The assumption is made that a projection-valued function defined on a subset X0 of the original set X is initially given. The goal of this paper is to find such a scalar measure μ on set X as well as the extension of a projection-valued function from X0 to X, and, as a result, to obtain an operator-valued measure which has a projection-valued density relative to μ. In general, the solution to this problem is found for |X| = 4 and |X0| = 2. As an example, a function on X0 is considered which accepts values within a set of projections onto coherent states. For this case, the problem which concerns the informational completeness of the measurement determined by the constructed measure is investigated. In other words, the possibility is considered of restoring a quantum state (a positive unit-trace operator) based on trace values of the matrix formed from the product of a measure and a quantum state. In the case of the constructed measure, it is demonstrated that it is possible to restore a quantum state only if it represents a projection. A restriction imposed on the probability distribution is also found; with this restriction satisfied, the probability distribution can be obtained as a result of measuring a certain quantum state.

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Funding

This work was supported by the Russian Science Foundation under grant no. 19-11-00086, https://rscf.ru/en/project/19-11-00086/.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, 119991, Moscow, Russia

    A. O. Alekseev & G. G. Amosov

Authors
  1. A. O. Alekseev
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  2. G. G. Amosov
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Correspondence to A. O. Alekseev or G. G. Amosov.

Additional information

Translated by A.V. Shishulin

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Alekseev, A.O., Amosov, G.G. On the Extension of a Family of Projections to a Positive Operator-Valued Measure. Vestnik St.Petersb. Univ.Math. 56, 1–8 (2023). https://doi.org/10.1134/S1063454123010028

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  • DOI: https://doi.org/10.1134/S1063454123010028

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