Generalization of the Spectral Theorem to the Case of Families of Noncommuting Operators and a~Linear Programming Problem

R. A. Roshchin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 241-250

Résumé

The aim of the present work is to describe, for a given quantum-mechanical system and a noncommutative) family of observables $A_\nu$, density matrices $\rho$ that possess the following property: In a certain probability space, there exists a family of random variables $\xi _\nu$ such that, for any set of pairwise commuting operators $A_{\nu _1}, A_{\nu _2}, \dots,A_{\nu _n}$, the quantum-mechanical correlation coefficient of observables is equal to the classical correlation coefficient of random variables: $\mathrm {Sp}(\rho A_{\nu _1}A_{\nu _2}\dots A_{\nu _n})=\mathbb E(\xi _{\nu _1}\xi _{\nu _2} \dots \xi _{\nu _n})$. It turns out that the existence of such random variables can be expressed in terms of a solution to a special optimization problem, a linear programming problem. The technique developed allows one to construct an earlier unknown solution to an important specific problem of the classical representation of a correlation function of the form $g\cos (\alpha -\beta )$ as the classical correlation of random processes $\xi _\alpha$ and $\eta _\beta$ such that $|\xi _\alpha| \le 1$ and $|\eta _\beta| \le 1$, in the parameter range ${2}/{\pi }$.