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 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 (), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.
Highlights
Geometrical representations are of particular interest in various problems of quantum mechanics.For instance, the Bloch representation is widely used in the context of characterizing quantum correlations in multiqubit systems [1,2,3]
Using the decomposition properties of Dicke states, we show that the separable states are necessarily the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra
We discussed the role of a specific generalized Weyl–Heisenberg algebra in the algebraic structure of qubits and qudits. The use of this generalized Weyl–Heisenberg algebra was based on the fact that qubits are neither fermions nor bosons
Summary
Geometrical representations are of particular interest in various problems of quantum mechanics. The classification of multipartite entangled states was investigated from several perspectives using different geometrical tools [31,32,33] to provide the appropriate way to approach the quantum correlations in multiqubit states. Among these quantum states, j-spin coherent states are of special interest [34]. They are the most classical (in contrast to quantum) states and can be viewed as 2j-qubit states which are completely separable In this sense, spin coherent states can be used to characterize the entanglement in totally symmetric multiqubit systems [35]. We discuss the separability in terms of the permanent of the matrix of the overlap between spin coherent states
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