Showing papers in "Theoretical and Mathematical Physics in 1976"

Generalized uncertainty relations and efficient measurements in quantum systems

Abstract: We consider two variants of a quantum-statistical generalization of the Cramer-Rao inequality that establishes an invariant lower bound on the mean square error of a generalized quantum measurement. The proposed complex variant of this inequality leads to a precise formulation of a generalized uncertainty principle for arbitrary states, in contrast to Helstrom’s variant [1] in which these relations are obtained only for pure states. A notion of canonical states is introduced and the lower mean square error bound is found for estimating of the parameters of canonical states, in particular, the canonical parameters of a Lie group. It is shown that these bounds are globally attainable only for canonical states for which there exist e¢ cient measurements or quasimeasurements.

Construction of an exact solution of the Dyson equation for the mean value of the Green's function

Abstract: Many problems of theoretical physics lead to the solution of stochastic equations, i.e., differential equations whose coefficients are random functions of the time or coordinates. Such problems arise in particular in the investigation of the propagation of waves in stochastic media, in statistical physics, in problems of detection and filtration of weak signals, and for optimal control. A number of methods are known for obtaining closed equations for the moments of the solution of stochastic systems with large fluctuations of the parameters and also the kinetic equations for the corresponding probability densities (see, for example, [1-4] and the bibliography in [31). These methods use some assumption about the rapidity (large or small scale) of the fluctuations compared with the characteristic dimensions of the system (field). Under the assumption that the fluctuations of the parameters have a delta-functional correlation, one can solve a considerable number of problems [2, 3]; but if such an idealization is inadequate, serious difficulties arise in the derivation of closed equations for the moments (even for the average field) and their solution. It should be noted that in the case when the fluctuations of the parameters of the system can be approximated by Markov processes with a finite number of states (for example, by a telegraph signal with Poisson statistics of the discontinuities), one can sometimes find an exact solution for the moments [5, 61. In the case of Gaussian fluctuations of the parameters, the equation for the first moment is usually solved only in Bourret's approximation [1, 7], which is effective only for sufficiently rapid fluctuations. The wellknown approximation of Kraichnan [8] is somewhat stronger, although its practical use and the establishment of the conditions of applicability give rise to difficulties. In the present paper, we prove the possibility of constructing an exact solution of the Dyson equation for the mean value of the Green's function of a stochastic linear system of general form with Gaussian fluctuations of the parameters without assuming that these fluctuations have a small scale. We show that for exponentially correlated fluctuations the exact solution has the form of an infinite continued fraction; we establish the correspondence between the first stages of this continued fraction and the results of wellknown approximations (diffusion [2, 3], Bourret, "excess" [9]). As an example, we find the dynamical parameters of an harmonic oscillator that has fluctuations of the eigenfrequency and the losses. In particular, we show that the higher approximations of the Dyson equation in this case take into account the influence of the higher bands of parametric resonance, and we estimate the conditions of applicability of the Bourret approximation.