Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems

TL;DR

This paper introduces a symmetry-adapted tensor basis for representing and visualizing multi-spin quantum systems, enabling clearer insights into their structure through spherical plots, with applications to systems of up to six qubits and mixed spin systems.

Contribution

It develops a novel symmetry-adapted tensor basis for finite-dimensional quantum systems and demonstrates its application to complex multi-spin systems including qubits and qudits.

Findings

Effective decomposition of multi-spin operators into spherical harmonics

Visualization of quantum states using multiple spherical plots

Extension of methods to coupled spins with arbitrary spin numbers

Abstract

We study the representation and visualization of finite-dimensional quantum systems. In a generalized Wigner representation, multi-spin operators can be decomposed into a symmetry-adapted tensor basis and they are mapped to multiple spherical plots that are each assembled from linear combinations of spherical harmonics. We apply two different approaches based on explicit projection operators and coefficients of fractional parentage in order to obtain this basis for up to six spins 1/2 (qubits), for which various examples are presented. An extension to two coupled spins with arbitrary spin numbers (qudits) is provided, also highlighting a quantum system of a spin 1/2 coupled to a spin 1 (qutrit).