Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States. Part II: The Perma-Concurrence Parameter

Abstract

This paper deals with separable and entangled qudits | ψ d 〉 (quantum states in dimension d) constructed from Dicke states made of N = d − 1 qubits. Such qudits present the property to be totally symmetric under the interchange of the N qubits. We discuss the notion of perma-concurrence P d for the qudit | ψ d 〉 , introduced by the authors (Entropy 2018, 20, 292), as a parameter for characterizing the entanglement degree of | ψ d 〉 . For d = 3 , the perma-concurrence P 3 constitutes an alternative to the concurrence C for symmetric two-qubit states. We give several expressions of P d (in terms of matrix permanent and in terms of unit vectors of R 3 pointing on the Bloch sphere) and precise the range of variation of P d (going from separable to maximally entangled states). Numerous examples are presented for P d . Special attention is devoted to states of W type and to maximally entangled states of Bell and Greenberger−Horne−Zeilinger type.

Keywords

• generalized Weyl-Heisenberg algebra
• qubits and qudits
• Dicke states
• SU(2) coherent states
• entanglement
• concurrence
• perma-concurrence
• Majorana stars