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Journal of Mathematical Physics

Orbits of the rotation group on the density matrices of spin-1 particles

Michel Daumens, Marcel Perroud

Published Jul 1, 1977• 0 citations
Publisher

Abstract

The SO(3) orbits of the spin-one mixed states, contained in each SU(3) orbit are shown to be characterized by the squares (defined by means of a symmetric bilinear form) of the eigenvectors of the density matrices. The orbits of matrices diagonalizable by a rotation in a spherical basis are deduced, and quadrupole matrices are considered as a special case.

DOI
10.1063/1.523433

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arXiv:quant-ph/9806089•Jun 26, 1998•2 citations
The SO(3) orbits of the spin-one mixed states, contained in each SU(3) orbit are shown to be characterized by the squares (defined by means of a symmetric bilinear form) of the eigenvectors of the density matrices. The orbits of matrices diagonalizable by a rotation in a spherical basis are deduced, and quadrupole matrices are considered as a special case.
We consider one-particle spin states. We show that all the information contained in the state can be written in an equivalent form using three completely decohered mixed states. The density matrix of such a system has the form of tensor product of three diagonal matrices. The unitary operators defined in the space of one-particle spin states are represented by some transformation of tensor products of diagonal matrices. We determine the explicit form of the unitary transformations acting on the representing three-qubit system and corresponding basic operators acting in the space of the initial qubit states.
The density matrix of an arbitrary pure state of a system consisting of two spin‐1/2 particles is derived from the Pauli spin angular momentum operators. Mixed singlet and triplet states are then formed from linear combinations of pure states and their corresponding density matrices constructed. Singlet and triplet states are exemplified by the spin isomers parahydrogen and orthohydrogen, respectively. Partial mixing is illustrated with the example of bilinear spin–spin coupling. Various properties of the density matrices of pure and mixed states are discussed, including idempotence, factoring, and spin correlation.
The k-particle reduced density matrices γ k for a singlet state or a spin-orientation averaged ensemble are invariant under the spin rotation group SU(2), i.e. they are independent of spin, even if they are expressed in terms of spin orbitals. The γ k contain the same information as their spinfree counterparts Γ k expressed in terms of spinfree orbitals. However, the relation between the two quantities is, by no means, elementary. The summation over spin is particularly demanding and surprisingly complicated for the k-particle density cumulants λ k and for the formulation of a generalised normal ordering with respect to an arbitrary reference state, including the generalised Wick theorem. Plausible extrapolations from small k turned out to be unjustified and some published expressions for k = 4 need to be revised. A compact general result, valid for all k has not yet been found.
Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.
This paper considers homomorphism of the Lie group SU(2) to the Lie group SO(3) of all rotations of 3- dimensional Euclidean space from Observers Mathematics point of view. In our work, we proved that in Observers Mathematics the probability of spin-j transformation is a homomorphism (representation) of SU(2 ) to the set of matrix transformations of a linear space of polynomial functions is less than 1, and got corresponding results for elementary fermions and bosons. As a continuation of these results we proved here the following theorems: Theorem 1. In Observers Mathematics the probability of two-to-one transformation of SU(2) to SO(3) is Lie groups homomorphism (representation) is less than 1. Theorem 2. The probability of two-to-one transformation and spin-j transformation (j = 1) are equivalent in Observers Mathematics is less than 1.
We extend to additional probability measures and scenarios, certain of the recent results of Krattenthaler and Slater (quant-ph/9612043), whose original motivation was to obtain quantum analogs of seminal work on universal data compression of Clarke and Barron. KS obtained explicit formulas for the eigenvalues and eigenvectors of the 2^m x 2^m density matrices derived by averaging the m-fold tensor products with themselves of the 2 x 2 density matrices. The weighting was done with respect to a one-parameter (u) family of probability distributions, all the members of which are spherically-symmetric (SU(2)-invariant) over the ``Bloch sphere'' of two-level quantum systems. For u = 1/2, one obtains the normalized volume element of the minimal monotone (Bures) metric. In this paper, analyses parallel to those of KS are conducted, based on an alternative "natural" measure on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera, and Lewenstein (quant-ph/9804024). The approaches of KS and that based on ZHSL are found to yield [1 + m/2] identical SU(2) x S_{m}-invariant eigenspaces (but not coincident eigenvalues for m > 3). Companion results, based on the SU(3) form of the ZHSL measure, are obtained for the twofold and threefold tensor products of the 3 x 3 density matrices. We find a rather remarkable limiting procedure (selection rule) for recovering from these analyses, the (permutationally-symmetrized) multiplets of SU(3) constructed from two or three particles. We also analyze the scenarios (all for m = 2) N = 2 x 3, N= 2 x 3 x 2, N= 3 x 2 x 2 and N = 4 and, in addition, generalize the ZHSL measure, so that it incorporates a family of (symmetric) Dirichlet distributions (rather than simply the uniform distribution), defined on the (N-1)-dimensional simplex of eigenvalues.