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A quantum version of Wielandt's inequality

M. Sanz, D. Perez-Garcia, M. M. Wolf, J. I. Cirac

IEEE Trans. Inf. Theory 4668-4673 **, (2010)

DOI: 10.1109/TIT.2010.2054552 Pfeil
**: 0909.5347 Pfeil

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.