$q$-analog qudit Dicke states

TL;DR

This paper introduces $q$-deformed qudit Dicke states using quantum algebra, providing a compact permutation-based expression, analyzing their entanglement entropy, and discussing their preparation on quantum computers without increasing circuit complexity.

Contribution

It defines $q$-analog qudit Dicke states with a permutation-based formulation and explores their entanglement and preparation on quantum computers.

Findings

States can be expressed as weighted sums over permutations with $q$-factors.

The bipartite entanglement entropy can be computed explicitly.

Preparation circuit complexity remains unchanged with $q$-dependence.

Abstract

Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their $d$-level generalization. We define here $q$-deformed qudit Dicke states using the quantum algebra $su_q(d)$. We show that these states can be compactly expressed as a weighted sum over permutations with $q$-factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a $q$-dependence does not change the circuit gate count.