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

TL;DR

This work investigates the non-standard Jordanian deformation U_h( sl(2,R) ) as a Hopf-algebra framework to construct h-deformed Dicke states for N-qubit systems. It contrasts the resulting h-Dicke states with their standard q-deformed counterparts, highlighting structural differences such as non-orthogonality of subspaces and h-dependent weightings that introduce extra excitation components weighted by powers of h. The authors provide explicit 2- and 3-qubit examples, develop Schmidt and Acín classifications, and establish a general algebraic approach to generate h-Dicke states for arbitrary N, including a detailed construction of h-W states and a scalable method via the simplified Δ_h^{(N)}(Z_+) coproduct. The deformation parameter h emerges as a tool to describe noise, decoherence, and tunable entanglement in quantum information protocols, with potential applicability to modeling experimental imperfections and designing novel entangled resources. The framework lays the groundwork for further comparative studies with q-deformations and extensions to larger systems and permutation symmetries.

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.

Figures (5)

• Figure 1: Coupling tree for two spin-$\tfrac{1}{2}$ particles, showing $J=1,0$ states, labeled as $D$ and $M$, respectively.
• Figure 2: Left: Density matrix for the undeformed Bell state $\ket{\Psi^-}= (\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow })/\sqrt{2}$ in the computational basis. Middle: Corresponding density matrix of the $q$-deformed state $\ket{M^0_2}_q$ for $q=0.5$. Right: Corresponding density matrix of the $h$-deformed state $\ket{M^0_2}_h$ for $h=0.5$.
• Figure 3: Coupling tree for three spin-$\tfrac{1}{2}$ particles, with final labels $D$, $M$ and $V$.
• Figure 4: Density matrix for the undeformed state $\ket{D^{-1}_3}_{h=0}$ (left) and for the $h$-Dicke deformed one $\ket{D^{-1}_3}_{h=1.5}$ (right). Density matrix for the $q$-Dicke deformed state $\ket{D^{-1/2}_3}_{q=1.5}$ (middle).
• Figure 5: Four spin-1/2 coupling tree with vertical lines pointing to state labels.