Eight types of qutrit dynamics generated by unitary evolution combined with 2+1 projective measurement

TL;DR

This paper classifies the types of Markov chain dynamics that can arise in a qutrit system under unitary evolution combined with specific projective measurements, providing a clear framework for understanding possible state evolutions.

Contribution

It introduces a classification scheme for qutrit dynamics based on eigenvalues and numerical range analysis, linking unitary operators to possible chain types.

Findings

Eight possible chain types identified

Classification based on eigenvalues of 2x2 dynamics matrix

Partition of numerical range determines chain generation

Abstract

We classify the Markov chains that can be generated on the set of quantum states by a unitarily evolving 3-dim quantum system (qutrit) that is repeatedly measured with a projective measurement (PVM) consisting of one rank-2 projection and one rank-1 projection. The dynamics of such a system can be visualized as taking place on the union of a Bloch ball and a single point, which correspond to the respective projections. The classification is given in terms of the eigenvalues of the 2x2 matrix that describes the dynamics arising on the Bloch ball, i.e., on the 2-dim subsystem. We also express this classification as the partition of the numerical range of the unitary operator that governs the evolution of the system. As a result, one can easily determine which of the eight possible chain types can be generated with the help of any given unitary.