Mixed discrete-variable Gaussian states

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states.

[1]  M. Everitt,et al.  Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies , 2021, Advanced Quantum Technologies.

[2]  B. Sanders,et al.  Qudits and High-Dimensional Quantum Computing , 2020, Frontiers in Physics.

[3]  Paulo E. M. F. Mendonca,et al.  Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators , 2013, 1304.0253.

[4]  D. Dragoman,et al.  Finite oscillator obtained through finite frame quantization , 2013, 1301.0704.

[5]  M. A. Marchiolli,et al.  Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles , 2012 .

[6]  D. Dragoman,et al.  Properties of finite Gaussians and the discrete-continuous transition , 2012, 1205.6302.

[7]  M. A. Marchiolli,et al.  Quasidistributions and coherent states for finite-dimensional quantum systems , 2011 .

[8]  Jean-Pierre Gazeau,et al.  Finite-dimensional Hilbert space and frame quantization , 2011 .

[9]  Nicolae Cotfas,et al.  Finite tight frames and some applications , 2008, 0803.0077.

[10]  K. Wolf,et al.  A discrete quantum model of the harmonic oscillator , 2007, 0711.3089.

[11]  M. Hayashi,et al.  Quantum information with Gaussian states , 2007, 0801.4604.

[12]  M. Ruzzi.  Jacobi ϑ-functions and discrete Fourier transforms , 2006, math-ph/0604018.

[13]  M. A. Marchiolli,et al.  Quasiprobability distribution functions for periodic phase spaces: I. Theoretical aspects , 2005, quant-ph/0602216.

[14]  A. Vourdas,et al.  Analytic representation of finite quantum systems , 2005, quant-ph/0504014.

[15]  K. Wolf,et al.  The canonical Kravchuk basis for discrete quantum mechanics , 2000 .

[16]  M. A. Kutay,et al.  The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform , 2000 .

[17]  T. Hakioǧlu.  Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase , 1998, quant-ph/9809074.

[18]  J. Peřina,et al.  Discrete Wigner function for finite-dimensional systems , 1998 .

[19]  J. Tolar,et al.  Quantization on and coherent states over , 1997 .

[20]  M. A. Marchiolli,et al.  Discrete Coherent States and Probability Distributions in Finite-Dimensional Spaces , 1996 .

[21]  V. B. Uvarov,et al.  Classical Orthogonal Polynomials of a Discrete Variable , 1991 .

[22]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[23]  P. Combe,et al.  A stochastic treatment of the dynamics of an integer spin , 1988 .

[24]  D. Galetti,et al.  An extended Weyl-Wigner transformation for special finite spaces , 1988 .

[25]  M. L. Mehta,et al.  Eigenvalues and eigenvectors of the finite Fourier transform , 1987 .

[26]  J. Schwinger.  UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[27]  V I Man 'ko,et al.  Introduction to Quantum Optics , 1995 .