Positive phase space distributions and uncertainty relations

J.G. Krüger

Published Jan 1, 1993• 0 citations

Abstract

In contrast to a widespread belief, Wigner's theorem allows the construction of true joint probabilities in phase space for distributions describing the object system as well as for distributions depending on the measurement apparatus. The fundamental role of Heisenberg's uncertainty relations in Schroedinger form (including correlations) is pointed out for these two possible interpretations of joint probability distributions. Hence, in order that a multivariate normal probability distribution in phase space may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schroedinger form should be satisfied.

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Abstract

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Connections between Heisenberg uncertainty relations and the existence of joint probability distributions of corresponding observables in the frame of quantum logics are investigated. It is shown that, provided the set of physical states is sufficient, uncertainly relations imply the nonexistence of joint probability distributions in any state On etudie la relation entre les relations d'incertitude de Heisenberg et l'existence de distributions de probabilites multiples des observables correspondants dans le cadre des logiques quantiques. On montre que, si l'ensemble des etats physiques est suffisant, les relations d'incertitude impliquent la non existence des distributions de probabilite multiples dans quelque etat que ce soitConnections between Heisenberg uncertainty relations and the existence of joint probability distributions of corresponding observables in the frame of quantum logics are investigated. It is shown that, provided the set of physical states is sufficient, uncertainly relations imply the nonexistence of joint probability distributions in any state On etudie la relation entre les relations d'incertitude de Heisenberg et l'existence de distributions de probabilites multiples des observables correspondants dans le cadre des logiques quantiques. On montre que, si l'ensemble des etats physiques est suffisant, les relations d'incertitude impliquent la non existence des distributions de probabilite multiples dans quelque etat que ce soit

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{\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.

x

, $\ensuremath{\mu}$ for a particle in a pure state $\ensuremath{\psi}(x)$. Suppose that the phase part $\ensuremath{\Psi}(x,z)$ of its Fourier transform ${T}_{F}[p(x,\ensuremath{\mu})]\ensuremath{\equiv}\ensuremath{\mid}G(x,z)\ensuremath{\mid}\mathrm{exp}[i\ensuremath{\Psi}(x,z)]$ is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function $\ensuremath{\Psi}(x,z)$. The extremum solution gives an output PDF $p(x,\ensuremath{\mu})$ that is the convolution of the Wigner ${p}_{W}(x,\ensuremath{\mu})$ with an instrument function defining uncertainty in either position

x

or momentum $\ensuremath{\mu}$. The convolution arises naturally out of the approach, and is one dimensional, in comparison with the ad hoc two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes $\ensuremath{\psi}(x)$ are the same at their boundaries [examples: states $\ensuremath{\psi}(x)$ with positive parity; with periodic boundary conditions; free particle trapped in a box].