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.
Highlights
Entanglement in multi-qubit systems is a valuable resource in implementing several quantum protocols offering significant advantages in the communication and processing of information [1].Quantum teleportation [2,3] and quantum cryptography or quantum key distribution [4] constitute promising applications in quantum information science.The characterisation, classification and identification of genuine entanglement in multipartite quantum systems remain a challenging issue and continue presently to attract attention in the field of quantum information theory
This parameter was introduced by Ollivier and Zurek [10] and by Henderson and Vedral [11]; it goes beyond entanglement and provide a proper tool to investigate the quantum correlations in an arbitrary bipartite state even for a separable state
When N → ∞, a convenient limiting procedure shows that the space G∞ corresponds to the space of the states of the harmonic oscillator
Summary
Entanglement in multi-qubit systems is a valuable resource in implementing several quantum protocols offering significant advantages in the communication and processing of information [1]. The Wootters concurrence and the entanglement of formation are two examples of quantitative measures of entanglement [9]; the concurrence is a simple tool to characterize the degree of entanglement of two-qubit systems in pure states Another indicator of quantum correlations in bipartite quantum systems is the quantum discord based on the von Neumann entropy. Developing the tools to distinguish between different classes of multipartite entangled states is essential in quantum information theory In this respect, the classification of mutli-entangled states was approached via local unitary operations, local operations and classical communication, and stochastic local operations assisted by classical communication (see the list of references quoted in [14]). We use the following notations: A† , [ A, B], I, δi,j , and CN of the operator A, the commutator of the operators A and B, the identity operator, the Kronecker δ, and the binomial coefficient, respectively
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