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Geometry and Entanglement of Two-Qubit States in the Quantum Probabilistic Representation

1
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, Mexico City 04510, Mexico
2
Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia
3
Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia
4
Department of Physics, Tomsk State University, Lenin Avenue 36, Tomsk 634050, Russia
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(9), 630; https://doi.org/10.3390/e20090630
Received: 17 July 2018 / Revised: 22 August 2018 / Accepted: 22 August 2018 / Published: 24 August 2018
(This article belongs to the Special Issue Towards Ultimate Quantum Theory (UQT))
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres–Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich’s squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities. View Full-Text
Keywords: quantum entanglement; geometric representation of qudits; probability distributions; linear entropy; Bell states quantum entanglement; geometric representation of qudits; probability distributions; linear entropy; Bell states
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López-Saldívar, J.A.; Castaños, O.; Nahmad-Achar, E.; López-Peña, R.; Man’ko, M.A.; Man’ko, V.I. Geometry and Entanglement of Two-Qubit States in the Quantum Probabilistic Representation. Entropy 2018, 20, 630.

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