We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light. The concept of spin is ubiquitous in quantum theory and all related fields of research, such as solidstate physics, molecular, atomic, nuclear or high-energy physics [1][2][3][4][5]. It has profound implications for the structure of matter as a consequence of the celebrated spinstatistics theorem [6]. The spin of a quantum system, be it an electron, a nucleus or an atom, has also been proven to be a key resource for many applications such as in spintronics [7], quantum information theory [8] or nuclear magnetic resonance [9]. Simple geometrical representations of spin states [10] allow one to develop physical insight regarding their general properties and evolution. Particularly well studied is the case of a single twolevel system, formally equivalent to a spin-1/2. In this case, the geometric representation is particularly simple. Indeed, the density matrix can be expressed in a basis formed of Pauli matrices and the identity matrix, leading to a parametrization in terms of a vector in R 3 . Pure states correspond to points on a unit sphere, the so-called Bloch sphere, and mixed states fill the inside of the sphere, the "Bloch ball". The simplicity of this representation help visualize the action and geometry of all possible spin-1/2 quantum channels [11]. For arbitrary pure spin states, another nice geometrical representation has been developed by Majorana in which a spin-j state is visualized as 2j points on the Bloch sphere [12]. This so-called Majorana or stellar representation has been exploited in various contexts (see, e. g., [11,[13][14][15][16][17]), but cannot be generalized to mixed spin states.Given the importance of geometrical representations, there have been numerous attempts to extend the previous representations to arbitrary mixed states. The former rely on a variety of sophisticated mathematical concepts such as su(N )-algebra generators [10,18,19], polarization operator basis [20][21][22] [26]. In the present Letter we propose an elegant generalisation to arbitrary spin-j of the spin-1/2 Bloch sphere representation based on matrices introduced by Weinberg in the context of relativistic quantum field theory [27]. The main result of the paper is theorem 2, which allows us to express any spin-j density matrix as a linear combination of matrices with convenient properties. The remarkable features of our representation are especially reflected in the simple coo...

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

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 (SLOCC). 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.

The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order t: A spin state is said to be anticoherent to order t if the moment of order k of the spin components along any directions are equal for k = 1, 2, . . . , t. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-j states and symmetric states of 2j spin-1/2 (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.