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Acta Physica Sinica

Bivariate normal distribution of coherent state in parameterized phase space

Fan Hong-Yi

Published Jan 1, 2014• 4 citations

Abstract

Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.

DOI: 10.7498/aps.63.020302

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Combining quantum mechanics and the normal distribution in statistics we study the coherent state from the point of view of statistics and by using the integration method within ordered product of operators. We find that the pure coherent state |z >z| exhibits a bivariate normal distribution of randon variables in (q,p) phase space, z=(q+ip)/√2, with a real k-parameter which is related to the quantization scheme, and the correlation coefficient is ik. For k=±1, |z >z| respectively is arranged as P-ordering (all P stand on the left of all Q) and Q-ordering (all Q stand on the left of all P), while in the case of k=0, |z >z| is arranged as the Weyl-ordering. In the cases of P-ordering and Q-ordering, in the classical correspondence function of |z >z||z=(q+ip)/√2 the bivariates (q,p) are correlated, only in the case of Weyl correspondece, (q,p) are independent. In other words, the Weyl ordering of operators is liable to decouple the correlation in bivariates.

Based on the technique of integration within an ordered product of operators we investigate a completeness relation of pure states (such as the coordinate eigenstate, the momentum eigenstate and the coherent state) into normally ordered Gaussian forms. The Weyl ordering invariance under similarity transformations is employed to reveal physical meaning of a kind of normally ordered Gaussian operators, which have the similar forms to the bivariate normal distributions in statistics, i.e., the thermo mixed state density matrix.

In order that a bivariate normal probability distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$

{\mathit{p}}^{2}

\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.

Based on the technique of integration within an ordered product of operators we have demonstrated that single-mode mixed states' density matrices can be recast into the normally ordered Gaussian forms [Chin. Phys. Lett. 24 (2007) 3322]. Here we employ the Weyl ordering invariance under similar transformations to show that some two-mode mixed states with entanglement involved can be put into normally ordered form in the bivariate normal distribution too and its marginal distributions can be analysed. In this way, density operators of quantum statistics can be analogous to mathematical statistics, and calculation of variances can be simplified.