Let be the boson Fock space over a finite dimensional Hilbert space . It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup of bounded operators in . This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization , where Z1 is an element of and has the form on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in , ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in , , and [αrs] is a symmetric matrix in ; (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its -parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in by the estimation of its parameters using O(n2) measurements with a finite number of outcomes.
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February 2021
Research Article|
February 22 2021
A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications
Tiju Cherian John
;
Tiju Cherian John
a)
Indian Statistical Institute, Delhi Centre
, 7, SJSS Marg, New Delhi 110059, India
a)Author to whom correspondence should be addressed: tijucherian@gmail.com
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K. R. Parthasarathy
K. R. Parthasarathy
b)
Indian Statistical Institute, Delhi Centre
, 7, SJSS Marg, New Delhi 110059, India
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a)Author to whom correspondence should be addressed: tijucherian@gmail.com
b)
Electronic mail: krp@isid.ac.in
J. Math. Phys. 62, 022102 (2021)
Article history
Received:
June 23 2020
Accepted:
January 07 2021
Citation
Tiju Cherian John, K. R. Parthasarathy; A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications. J. Math. Phys. 1 February 2021; 62 (2): 022102. https://doi.org/10.1063/5.0019413
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