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Quantum Gaussian maximizers and log-Sobolev inequalities

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Abstract

In the present paper, we give a proof of optimality of Gaussian encodings for the ultimate communication rate of generalized noisy homodyne and heterodyne receivers under the oscillator energy constraint, without any additional “threshold condition” onto the signal power. Rather remarkably, the proof is based on various generalizations of the celebrated log-Sobolev inequality motivated by a solution of the problem of Gaussian maximizers for the capacity of quantum measurement channels.

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Notes

  1. The fact that the log-Sobolev inequality implies Lieb’s bound for the Wehrl entropy was observed by Carlen [1], see also [25]. Another related result [3] concerns minimization of the Wehrl entropy with a fixed entropy. We are indebted to the referee who pointed out these references.

  2. See, e.g., [10, 13], for the mathematical treatment of expressions with the unbounded operators related to the bosonic position–momentum observables.

  3. We denote by Sp the trace of \(2s\times 2s\)-matrices as distinct from the trace of operators on \({\mathcal {H}}\).

  4. We denote by \(I_{s}\) the unit \(s\times s-\)matrix.

  5. Notably, the expression (36) is of the same type as the one obtained in [7] by optimizing the information from applying sharp position measurement to noisy optimally squeezed states (A.H. is indebted to M. J. W. Hall for this observation).

  6. The Wehrl inequality proved by Lieb [23] corresponds to \(\beta _{q}=\beta _{p}=1/2\) and gives the minimal entropy \(\ln 2\pi e\) attained on the coherent states.

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Acknowledgements

The work was supported by the grant of Russian Science Foundation 19-11-00086, https://rscf.ru/project/19-11-00086/.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow, Russia, 119991

    Alexander S. Holevo & Sergey N. Filippov

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  1. Alexander S. Holevo
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Holevo, A.S., Filippov, S.N. Quantum Gaussian maximizers and log-Sobolev inequalities. Lett Math Phys 113, 10 (2023). https://doi.org/10.1007/s11005-023-01634-6

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