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Wigner Measures and Coherent Quantum Control

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Abstract

We introduce Wigner measures for infinite-dimensional open quantum systems; important examples of such systems are encountered in quantum control theory. In addition, we propose an axiomatic definition of coherent quantum feedback.

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Notes

  1. The definition of a pseudodifferential operator \( \widehat{F}{} \) in \( {\mathcal L} _2(Q,\mu)\) with symbol \(F\) can be found in [5].

  2. The Kolmogorov integral is the trace in the tensor product of the space of functions on \(Q\) and the space of measures on \(Q\); \(\rho_T^2\) is an element of this space (for the original definition, which involves neither the tensor product nor the trace, see [8]).

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Funding

J. E. Gough was supported by the French National Research Agency (ANR) grant Q-COAST ANR 19-CE48-0003, and through a European Research Council (ERC) project under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 884762). T. S. Ratiu was partially supported by the National Natural Science Foundation of China grant no. 11871334 and by the NCCR Swiss MAP grant of the Swiss National Science Foundation. O. G. Smolyanov was supported by the Ministry of Science and Higher Education of the Russian Federation within the Russian Academic Excellence Project “5-100.”

Author information

Authors and Affiliations

  1. Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Penglais, Aberystwyth, Ceredigion, SY23 3BZ, UK

    J. E. Gough

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai, 200240, China

    T. S. Ratiu

  3. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    T. S. Ratiu

  4. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia

    O. G. Smolyanov

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

    O. G. Smolyanov

Authors
  1. J. E. Gough
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  2. T. S. Ratiu
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  3. O. G. Smolyanov
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Corresponding author

Correspondence to J. E. Gough.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 313, pp. 59–66 https://doi.org/10.4213/tm4181.

Translated by I. Nikitin

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Gough, J.E., Ratiu, T.S. & Smolyanov, O.G. Wigner Measures and Coherent Quantum Control. Proc. Steklov Inst. Math. 313, 52–59 (2021). https://doi.org/10.1134/S0081543821020061

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  • DOI: https://doi.org/10.1134/S0081543821020061

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