Multiqubit symmetric states with maximally mixed one-qubit reductions

We present a comprehensive study of maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. A general criterion is provided to easily identify whether given symmetric states are maximally entangled in that respect or not. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin of the system vanishes, as well as in corollary the dipole moment of the Husimi function. We establish the link between this kind of maximal entanglement, the anticoherence properties of spin states, and the degree of polarization of light fields. We analyze the relationship between the MES states and the classes of states equivalent through stochastic local operations with classical communication (SLOCC). We provide a nonexistence criterion of MES states within SLOCC classes of qubit states and show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES states of this system are identified and characterized. Finally the entanglement content of MES states is analyzed with respect to the geometric and barycentric measures of entanglement, as well as to the generalized N-tangle. We show that the geometric entanglement of MES states is ensured to be larger than or equal to 1/2, but also that MES states are not in general the symmetric states that maximize the investigated entanglement measures.Comment: 12 pages, 4 figure

Anticoherence measures for spin states

Among all possible spin states, spin-coherent states are the most classical because the spin expectation value in these states yields a vector of maximal norm pointing in a well defined direction. In contrast, anticoherent spin state to order t are such that is independent of the unit vector n for k = 1, ..., t [1]. By construction, coherent and anticoherent spin states are at both ends of the spectrum of classicality. The aim of this work is to position all possible spin states on such a spectrum, that is to provide measures of anticoherence. To this aim, we introduce an axiomatic definition of anticoherence measures to any order t. In particular, we show that the total variance of a pure spin state, first introduced in [2] can be used to define a measure of anticoherence to order 1. We describe a systematic way of constructing anticoherence measures to any order that relies on the mapping between spin-j states and symmetric states of N = 2j spin-1/2. In particular, we exploit the fact that anticoherent spin states to order t have maximally mixed t-spin-1/2 reduced density matrices in the symmetric subspace [3]. [1] J. Zimba, Electron. J. Theor. Phys. 3, 143 (2006). [2] A. A. Klyachko, B. Öztop, and A. S. Shumovsky, Phys. Rev. A 75, 032315 (2007). [3] D. Baguette, T. Bastin, and J. Martin, Phys. Rev A 90, 032314 (2014)

Highly non-classical symmetric states of an N-qubit system

In this work, we consider two measures of non-classicality for pure symmetric N-qubit states : Wehrl entropy (S) and Wehrl participation ratio (R). Measures of non-classicality help to the understanding of the mechanisms responsible for the transition from quantum to classical physics and are usefull in the context of information processing and quantum-enhanced measurements

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. [1] N. Gisin, H. Bechmann-Pasquinucci, Phys. Lett. A 246 (1998). [2] A. Higuchi, A. Sudbery, Phys. Lett. A, 272, 213 (2000). [3] F. Verstraete, J. Dehaene, B. De Moor, Phys. Rev. A 68, 012103 (2003). [4] J. Martin, O. Giraud, P. A. Braun, D. Braun and T. Bastin, Phys. Rev. A 81, 062347 (2010). [5] D. A. Meyer, N. R. Wallach, J. Math. Phys. 43, 4273 (2002). [6] I. D. K. Brown, S. Stepney, A. Sudbery, and S. L. Braunstein, J. Phys. A 38, 1119 (2005). [7] B. Kraus, Phys. Rev. Lett. 104, 020504 (2010). [8] J. Zimba, EJTP 3, 10 (2006)

On the Identication of Symmetric N-qubit Maximally Entangled States

Maximally entangled states can serve as a useful resource in many different contexts. It is therefore important to identify those states. Here we are interested in the identification of maximally entangled states in the symmetric subspace of an N-qubit system. By maximally entangled states, we refer to symmetric states characterized by a one qubit reduced density matrix proportional to the identity. These states maximise various entanglement measures [1] such as von Neumann and Meyer-Wallach entropy and are unique up to LU in their SLOCC class [2]. We identify and characterize all maximally entangled symmetric states up to 4 qubits. 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 maximally entangled states. [1] F. Verstraete, J. Dehaene, B. De Moor, Phys. Rev. A 68, 012103 (2003). [2] G. Gour, N. Wallach, N. J. Phys. 13, 073013 (2011

Anticoherence of spin states with point-group symmetries

We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then investigate in detail the case of small numbers of qubits and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order

Anticoherence and entanglement of spin states

We investigate multiqubit permutation-symmetric states with maxi- mally mixed reduced density matrices in the symmetric subspace [1]. Such states can be viewed as particular spin states, namely anticoher- ent spin states [2]. Using the Majorana representation of spin states in terms of points on the unit sphere [3], we analyze the consequences of degeneracies of the Majorana points and of a point-group symmetry in their arrangement on the existence of anticoherent spin states. We provide different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and SLOCC classes [4]. We consider in detail the case of small numbers of qubits and solve the 4-qubit case completely by identifying and characterizing all 4-qubit anticoherent states. [1] D. Baguette, T. Bastin, and J. Martin, Phys. Rev. A 90, 032314 (2014); O. Giraud et al., Phys. Rev. Lett. 114, 080401 (2015); D. Baguette et al., Phys. Rev. A 92, 052333 (2015). [2] J. Zimba, Electron. J. Theor. Phys. 3, 143 (2006). [3] E. Majorana, Nuovo Cimento 9, 43 (1932). [4] SLOCC classes : Classes of states equivalent through stochastic local operations with classical communication