A quantum version of Wielandt's inequality

Mikel Sanz; David Perez-Garcia; Michael Marc Wolf; Juan I. Cirac

doi:10.1109/TIT.2010.2054552

A quantum version of Wielandt's inequality

Mikel Sanz, David Perez-Garcia, Michael Marc Wolf, Juan I. Cirac

52 Citations (Scopus)

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.

Original language	English
Journal	I E E E Transactions on Information Theory
Volume	56
Issue number	9
Pages (from-to)	4668-4673
Number of pages	5
ISSN	0018-9448
DOIs	https://doi.org/10.1109/TIT.2010.2054552
Publication status	Published - Sept 2010

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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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AU - Sanz, Mikel

AU - Perez-Garcia, David

AU - Wolf, Michael Marc

AU - Cirac, Juan I.

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N2 - 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.

AB - 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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DO - 10.1109/TIT.2010.2054552

M3 - Journal article

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VL - 56

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EP - 4673

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 9

ER -

A quantum version of Wielandt's inequality

Abstract

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