Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements

TL;DR

This paper studies Markov chains with doubly stochastic matrices, introduces universal convex polytopes for probability vectors, and applies these concepts to analyze non-selective quantum measurements and the quantum Zeno effect.

Contribution

It introduces universal convex polytopes based on Birkhoff-von Neumann expansion and applies them to quantum measurement sequences, linking classical and quantum probabilistic systems.

Findings

Convex polytopes shrink over time, increasing minimum entropy.

Non-selective measurements can destroy off-diagonal density matrix elements.

Quantum Zeno effect emerges with rapid repeated measurements.

Abstract

A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff-von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with $d$-dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This `hermaphrodite' system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with `frozen discrete paths' (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).