Generalized Weyl-Heisenberg algebra, qudit systems and entanglement measure of symmetric states via spin coherent states

TL;DR

This paper establishes a link between Dicke states, generalized Weyl-Heisenberg algebra, and symmetric qudit entanglement, introducing a new entanglement measure called perma-concurrence and connecting Majorana representations with entanglement quantification.

Contribution

It introduces a novel entanglement measure for symmetric qudits based on the permanent of overlap matrices and connects Majorana representations with entanglement analysis.

Findings

Perma-concurrence effectively characterizes qudit entanglement.

Separable states correspond to Perelomov coherent states in this framework.

The approach generalizes to higher-dimensional qudits with explicit examples for d=3,4,5.

Abstract

A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N = d-1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl-Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini-Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For d=3 (i.e., N = 2), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for d=4 and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.