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01.03.2021 | Ausgabe 3/2021

Quantum Information Processing 3/2021

Entanglement types for two-qubit states with real amplitudes

Zeitschrift:
Quantum Information Processing > Ausgabe 3/2021
Autoren:
Oscar Perdomo, Vicente Leyton-Ortega, Alejandro Perdomo-Ortiz
Wichtige Hinweise
This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.

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Abstract

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states—those locally equivalent to the Bell states—form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled or so-called product states are \(\pi /4\) units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval \([0,\pi /4]\) with the point 0 corresponding to the maximally entangled states. The point \(\pi /4\) corresponds to the unentangled states which geometrically are described by a torus. Finally, for every \(0< d < \pi /4\) the point d corresponds to a disjoint pair of torus. Finally, we also show how this geometrical interpretation allows to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.

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