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Some Algebraic and Geometric Aspects of Quantum Measurements

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Abstract

We study positive operator-valued measures by algebraic and geometric methods. We prove that positive operator-valued measures are parametrized by a Poisson manifold. Also, we show how to obtain symplectic leaves of this Poisson manifold in terms of parameters of the measures. In addition, we study the interaction of two projection-valued measures by the methods of algebraic geometry.

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Acknowledgments

We are grateful to Grigory Amosov, Alexei Bondal, and Ilya Karzhemanov for very fruitful discussions and support.

Funding

The second author was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00908. The work was also supported by the HSE Basic Research Program and the Russian Academic Excellence Project “5-100.”

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701, Russia

    A. S. Kocherova & I. Yu. Zhdanovskiy

  2. Laboratory of Algebraic Geometry and Its Applications, HSE University, ul. Usacheva 6, Moscow, 119048, Russia

    I. Yu. Zhdanovskiy

Authors
  1. A. S. Kocherova
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  2. I. Yu. Zhdanovskiy
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Corresponding author

Correspondence to A. S. Kocherova.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 313, pp. 109–123 https://doi.org/10.4213/tm4167.

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Kocherova, A.S., Zhdanovskiy, I.Y. Some Algebraic and Geometric Aspects of Quantum Measurements. Proc. Steklov Inst. Math. 313, 99–112 (2021). https://doi.org/10.1134/S0081543821020103

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