Geometric properties of qudit systems

TL;DR

This paper explores the geometric visualization of qudit systems, especially thermal states, using generalized Bloch vectors and invariants, revealing quantum phase transitions and adiabatic evolutions in specific Hamiltonian models.

Contribution

It introduces a geometric framework for analyzing qudit thermal states and characterizes phase transitions using Bloch vectors and group invariants.

Findings

Paths in simplex representations show quantum phase transitions.

Explicit quantum phase diagrams at different temperatures.

Geometric visualization aids understanding of adiabatic evolutions.

Abstract

We discuss in general how to geometrically visualize a qudit system, with a particular interest in thermal states. The principle of maximum entropy is used to study the geometric properties of an ensemble of finite dimensional Hamiltonian systems with known average energy. These geometric characterizations are given in terms of the generalized diagonal Bloch vectors and the invariants of the special unitary group in $n$ dimensions. As examples, Hamiltonians written in terms of linear and quadratic generators of the angular momentum algebra are considered with $J= 1$ and $J=3/2$. For these cases, paths as functions of the temperature are established in the corresponding simplex representations, which show first- and second-order quantum phase transitions, as well as the adiabatic evolution of the interaction strengths (control parameters) of the Hamiltonian models. For the Lipkin-Meshkov-Glick Hamiltonian the quantum phase diagram is explicitly shown for different temperature values in parameter space.