Non-standard quantum algebra $\mathcal{U}_h (\mathfrak{sl}(2, \mathbb{R}))$ and $h$-Dicke states

TL;DR

This paper explores the use of a non-standard quantum algebra to generate and analyze $h$-deformed Dicke states, revealing unique features and potential applications in quantum information and noise modeling.

Contribution

It introduces a novel $h$-deformation of Dicke states using Jordanian quantum algebra, differing from standard $q$-deformations, and discusses their properties and potential experimental relevance.

Findings

$h$-deformed Dicke states exhibit distinct features from $q$-Dicke states.

Comparison shows similarities between $h$-Dicke states and experimental states, suggesting noise modeling.

Explicit constructions for small $N$ and a method for arbitrary $N$ are provided.

Abstract

We discuss the application of the Jordanian quantum algebra $\mathcal{U}_h (\mathfrak{sl}(2, \mathbb{R}))$, a Hopf algebra deformation of the Lie algebra $\mathfrak{sl}(2, \mathbb{R})$, in order to generate sets of $N$ qubit quantum states. We construct the associated $h$-deformed Dicke states using the Clebsch-Gordan coefficients for $\mathcal{U}_h (\mathfrak{sl}(2, \mathbb{R}))$, showing that the former exhibit completely different features than the $q$-Dicke states obtained from the standard quantum deformation $\mathcal U_q (\mathfrak{sl}(2, \mathbb{R}))$. Moreover, the density matrices of these $h$-deformed Dicke states are compared to the experimental realizations of those of Dicke states, and several similarities are observed, indicating that the $h$-deformation could be used to describe noise and decoherence effects in experimental settings, as well as to control the degree of entanglement of the state in quantum computing protocols. In particular, $h$-Dicke states for $N=2,3,4$ are presented, a method to construct the $h$-deformed analogs of $W$-states for arbitrary $N$ is given, and some algebraic considerations for the explicit derivation of generic $h$-Dicke states are provided.