A quantum version of Wielandt's inequality

2009/09/30 by M. Sanz, D. Perez-Garcia, M. M. Wolf +1 · 1 citation
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP

paper · pdf · doi:10.1109/tit.2010.2054552

published as IEEE Trans. Inf. Theory 56 (2010) 4668-4673 · 6 pages, no figures

arxiv created 2010/05/28 · arxiv updated 2011/06/02

Abstract

In this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.

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