Perron-Frobenius Theory for Positive Maps on Trace Ideals

Details Perron-Frobenius Theory for Positive Maps on Trace Ideals Perron-Frobenius Theory for Positive Maps on Trace Ideals

2000-07-14

arxiv.org/abs/math-ph/0007020v1

Physics

Mathematical Physics

15 pages AMS-latex, submitted for Publication to the Fields Institute Communication Series in a volume dedicated to the 60th B

Scientific paper

This article provides sufficient conditions for positive maps on the Schatten classes $\mathcal J_{p}, 1\le p<\infty$ of bounded operators on a separable Hilbert space such that a corresponding Perron-Frobenius theorem holds. With applications in quantum information theory in mind sufficient conditions are given for a trace preserving, positive map on $\mathcal J_{1}$, the space of trace class operators, to have a unique, strictly positive density matrix which is left invariant under the map. Conversely to any given strictly positive density matrix there are trace preserving, positive maps for which the density matrix is the unique Perron-Frobenius vector.

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