Symmetric N-qubit anticoherent states

Entanglement is among the key features of quantum mechanics. In the last decade, a lot of efforts has been made to quantify the amount of entanglement of various multipartite states, either pure or mixed. In particular, the search for maximally entangled states (states maximizing certain measures of entanglement) has focused a great deal of attention, see e.g. Refs. [1–4].
In this work, we present a comprehensive study of maximally entangled symmetric N-qubit states with respect to the definition of Gisin [1]. According to this definition, a state is maximally entangled if all its one-qubit reduced density matrices are maximally mixed. These states maximize various entanglement measures, such as von Neumann and Meyer-Wallach entropies [5]. They are unique up to local unitaries within the class of states interconvertible under stochastic local operations and classical communication (SLOCC) [3]. Besides, they are conjectured to be maximally entangled with respect to the Negative Partial Transpose measure of entanglement [6]. As appreciated by B. Kraus, they play an important role in the determination of the local unitary equivalence of multiqubit states [7]. Moreover, they are maximally fragile (in the sense that they are the states which are the most sensitive to noise) and therefore have been proposed as ideal candidates for ultrasensitive sensors [1]. We provide general conditions for a symmetric state with an arbitrary number of qubits to be maximally entangled and identify families of SLOCC classes which do not contain any such states. We also compute various measure of entanglement associated with those states in order to characterize them further and find all maximally entangled states up to 4 qubits. We finally prove that maximally entangled states coincide with anticoherent states of order 1. According to the definition of Ref. [8], a symmetric state of N qubits is anticoherent to order t iff 〈(S·n)k〉 is independent of n for k = 1, . . . , t where n is a tridimensional unit vector and S is the collective spin operator associated to the N-qubit system.

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