dorsal/arxiv
View SchemaSU(N) x S_{m}-Invariant Eigenspaces of N^{m} x N^{m} Mean Density Matrices
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9806089 |
| URL | https://arxiv.org/abs/quant-ph/9806089 |
Abstract
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.
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"abstract": "We extend to additional probability measures and scenarios, certain of the\nrecent results of Krattenthaler and Slater (quant-ph/9612043), whose original\nmotivation was to obtain quantum analogs of seminal work on universal data\ncompression of Clarke and Barron. KS obtained explicit formulas for the\neigenvalues and eigenvectors of the 2^m x 2^m density matrices derived by\naveraging the m-fold tensor products with themselves of the 2 x 2 density\nmatrices. The weighting was done with respect to a one-parameter (u) family of\nprobability distributions, all the members of which are spherically-symmetric\n(SU(2)-invariant) over the ``Bloch sphere\u0027\u0027 of two-level quantum systems. For u\n= 1/2, one obtains the normalized volume element of the minimal monotone\n(Bures) metric. In this paper, analyses parallel to those of KS are conducted,\nbased on an alternative \"natural\" measure on the density matrices recently\nproposed by Zyczkowski, Horodecki, Sanpera, and Lewenstein (quant-ph/9804024).\nThe approaches of KS and that based on ZHSL are found to yield [1 + m/2]\nidentical SU(2) x S_{m}-invariant eigenspaces (but not coincident eigenvalues\nfor m \u003e 3). Companion results, based on the SU(3) form of the ZHSL measure, are\nobtained for the twofold and threefold tensor products of the 3 x 3 density\nmatrices. We find a rather remarkable limiting procedure (selection rule) for\nrecovering from these analyses, the (permutationally-symmetrized) multiplets of\nSU(3) constructed from two or three particles. We also analyze the scenarios\n(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\naddition, generalize the ZHSL measure, so that it incorporates a family of\n(symmetric) Dirichlet distributions (rather than simply the uniform\ndistribution), defined on the (N-1)-dimensional simplex of eigenvalues.",
"arxiv_id": "quant-ph/9806089",
"authors": [
"Paul B. Slater"
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"title": "SU(N) x S_{m}-Invariant Eigenspaces of N^{m} x N^{m} Mean Density Matrices",
"url": "https://arxiv.org/abs/quant-ph/9806089"
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