Tensor Representation of Spin States

Abstract

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...

Tensor eigenvalues and entanglement of symmetric states

“…(see [22] for detail). The A µ1µ2...µ N are invariant under permutation of the indices and enjoy the property that for any…”

“…In order to conveniently deal with expansions of quantum states over spin coherent states, we use a representation suited to this purpose, that has recently been introduced in [22]. We express a spin-j density matrix ρ in the following way.…”

“…This feature is particularity important in our case, because a rotated spin-j quantum state ρ =R † ρR, witĥ R = exp(−iθ J · n) the spin-j representation of a rotation, has a tensor representation given by A µ1...µ N = R µ1,ν1 · · · R µ N ,ν N A ν1...ν N with R the 4 × 4 matrix whose 3 × 3 lower-right block is the orthogonal matrix associated with the rotation of axis n and angle θ, and R µ,0 = R 0,µ = δ 0,µ [22].…”

“…The tensor associated with a spin coherent state |α pointing in direction n is simply given by (9) with n 0 = 1 and n = (n 1 , n 2 , n 3 ) [22]. The definition of classicality, Eq.…”

“…Other well-known entanglement criteria are based on bounds of spin-correlations [20,21], which in turn exploit the positive-partial transpose (PPT) criterion. In [22] we introduced a tensorial representation for spin states. In this representation, a spin-j density matrix is expanded as a sum over matrices of dimensions (2j + 1) × (2j + 1), and the expansion coefficients take the form of a tensor A µ1µ2...µ N with N = 2j indices.…”