On pseudo-stochastic matrices, pseudo-positive maps, and hidden correlation phenomena

We introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic and bistochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with "diamond" sets of stochastic matrices and pseudo-positive maps are dealt with. We introduce the notion of hidden quantum correlations. We present the mean values of observables depending on one classical random variable described by the probability distribution in the form of correlation functions of two (three, etc.) random variables described by the corresponding joint probability distributions. We develop analogous constructions for the density matrices of quantum states and quantum observables. We consider examples of four-dimensional Hilbert space corresponding to the "quantum roulette" and "quantum compass."