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A Quantum Version of Wielandt's Inequality

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4 Author(s)
Sanz, M. ; Max-Planck-Inst. fur Quantenopt., Garching, Germany ; Pérez-García, D. ; Wolf, M.M. ; Cirac, J.I.

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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Information Theory, IEEE Transactions on  (Volume:56 ,  Issue: 9 )