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Entropy 2013, 15(6), 1963-1984; doi:10.3390/e15061963

Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control

1 Department of Mathematical Physics, National University of Ireland Maynooth, Science Building, North Campus, Maynooth, Co. Kildare, Ireland 2 School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road,Dublin 4, Ireland
* Author to whom correspondence should be addressed.
Received: 7 April 2013 / Revised: 17 April 2013 / Accepted: 17 May 2013 / Published: 23 May 2013
(This article belongs to the Special Issue Quantum Information 2012)
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We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.
Keywords: two-qubit systems; metric spaces; Haar measure two-qubit systems; metric spaces; Haar measure
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Watts, P.; O'Connor, M.; Vala, J. Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control. Entropy 2013, 15, 1963-1984.

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