Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States.

A relation is established in the present paper between Dicke states in a -dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension . 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 (-level) states are represented by points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a -dimensional space) is describable by a -qubit vector (in a -dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the qubits makes it possible to measure the entanglement between the qubits forming the qudit. This is confirmed by a Fubini-Study metric analysis. A new parameter, proportional to the permanent and called , is introduced for characterizing the entanglement of a symmetric qudit arising from qubits. For d = 3 ( ⇔ 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.