A quantum version of Wielandt's inequality

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

doi:10.1109/TIT.2010.2054552

A quantum version of Wielandt's inequality

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

Research output: Contribution to journal › Article › peer-review

78 Scopus citations

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
Article number	5550282
Pages (from-to)	4668-4673
Number of pages	6
Journal	IEEE Transactions on Information Theory
Volume	56
Issue number	9
DOIs	https://doi.org/10.1109/TIT.2010.2054552
State	Published - Sep 2010
Externally published	Yes

Keywords

Classical channels
Wielandt's inequality
information rates
quantum channels
spin systems
strongly correlated electrons

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10.1109/TIT.2010.2054552

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title = "A quantum version of Wielandt's inequality",

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 = "Classical channels, Wielandt's inequality, information rates, quantum channels, spin systems, strongly correlated electrons",

author = "Mikel Sanz and David P{\'e}rez-Garc{\'i}a and Wolf, {Michael M.} and Cirac, {Juan I.}",

note = "Funding Information: Manuscript received September 30, 2009; revised May 29, 2010. Date of current version August 18, 2010. This work was supported in part by the QCCC Program of the Elitenetzwerk Bayern and in part by the DFG (project FOR635). D. P{\'e}rez-Garc{\'i}a was supported by the Spanish grants I-MATH, MTM2008-01366, S2009/ESP-1594, and the EU project QUEVADIS. M. M. Wolf was supported by QUANTOP, the Danish Natural Science Research Council (FNU), and the EU projects QUEVADIS and COQUIT.",

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language = "English",

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AU - Pérez-García, David

AU - Wolf, Michael M.

AU - Cirac, Juan I.

N1 - Funding Information: Manuscript received September 30, 2009; revised May 29, 2010. Date of current version August 18, 2010. This work was supported in part by the QCCC Program of the Elitenetzwerk Bayern and in part by the DFG (project FOR635). D. Pérez-García was supported by the Spanish grants I-MATH, MTM2008-01366, S2009/ESP-1594, and the EU project QUEVADIS. M. M. Wolf was supported by QUANTOP, the Danish Natural Science Research Council (FNU), and the EU projects QUEVADIS and COQUIT.

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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.

KW - Classical channels

KW - Wielandt's inequality

KW - information rates

KW - quantum channels

KW - spin systems

KW - strongly correlated electrons

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EP - 4673

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 9

M1 - 5550282

ER -

A quantum version of Wielandt's inequality

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