SU(N) x S_{m}-Invariant Eigenspaces of N^{m} x N^{m} Mean Density Matrices
Abstract
Description
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.
15 pages, LaTeX, single figure; we correct a small but systematic error in our N = 3, m = 3 analysis, leading to a much simpler set of results. Also, we have added a section (the penultimate one) entitled, "Parameterized Families of N^{m} x N^{m} Mean Density Matrices", having observed that the choice by Zyczkowski et al (quant-ph/9804024) of a uniform distribution over the (N-1)-dimensional simplex spanned by the eigenvalues is somewhat arbitrary in nature, and can be replaced by any member of a family of (symmetric) Dirichlet distributions
15 pages, LaTeX, single figure; we correct a small but systematic error in our N = 3, m = 3 analysis, leading to a much simpler set of results. Also, we have added a section (the penultimate one) entitled, "Parameterized Families of N^{m} x N^{m} Mean Density Matrices", having observed that the choice by Zyczkowski et al (quant-ph/9804024) of a uniform distribution over the (N-1)-dimensional simplex spanned by the eigenvalues is somewhat arbitrary in nature, and can be replaced by any member of a family of (symmetric) Dirichlet distributions