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 Trudy Mat. Inst. Steklova, 2021, Volume 313, Pages 78–86 (Mi tm4161)

Structure of a General Quantum Gaussian Observable

A. S. Holevo

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

Abstract: A structure theorem is established which shows that an arbitrary multimode bosonic Gaussian observable can be represented as a combination of four basic cases whose physical prototypes are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes a connection between the descriptions of a Gaussian observable in terms of the characteristic function and in terms of the density of a probability operator-valued measure (POVM) and has remarkable parallels with the treatment of bosonic Gaussian channels in terms of their Choi–Jamiołkowski form. Along the way we give the “most economical” (in the sense of minimal dimensions of the quantum ancilla) construction of the Naimark extension of a general Gaussian observable. We also show that the Gaussian POVM has bounded operator-valued density with respect to the Lebesgue measure if and only if its noise covariance matrix is nondegenerate.

DOI: https://doi.org/10.4213/tm4161

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English version:
Proceedings of the Steklov Institute of Mathematics, 2021, 313, 70–77

Bibliographic databases:

UDC: 519.248.3
Revised: September 2, 2020
Accepted: October 29, 2020

Citation: A. S. Holevo, “Structure of a General Quantum Gaussian Observable”, Mathematics of Quantum Technologies, Collected papers, Trudy Mat. Inst. Steklova, 313, Steklov Math. Inst., Moscow, 2021, 78–86; Proc. Steklov Inst. Math., 313 (2021), 70–77

Citation in format AMSBIB
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