A Quantum Version of Wielandt´s Inequality

Author

Sanz, Mikel ; Pérez-García, David ; Wolf, Michael M. ; Cirac, Juan I.

Author_Institution

Max-Planck-Inst. fur Quantenopt., Garching, Germany

Volume

56

Issue

9

fYear

2010

Firstpage

4668

Lastpage

4673

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.

Keywords

matrix algebra; quantum communication; telecommunication channels; Wielandt inequality; classical channels; dichotomy theorems; frustration-free Hamiltonians; matrix product state dimension; quantum channels; zero-error capacity; Channel capacity; Context; Eigenvalues and eigenfunctions; Electrons; Information rates; Linear matrix inequalities; Markov processes; Matrices; Memoryless systems; Probability distribution; Quantum mechanics; Stationary state; Stochastic processes; Upper bound; Classical channels; Wielandt´s inequality; information rates; quantum channels; spin systems; strongly correlated electrons;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.2010.2054552

Filename

5550282