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Entropy 2015, 17(7), 5063-5084; doi:10.3390/e17075063

Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States

1
Departamento de Física, Cinvestav, AP 14-740, Mexico DF 07000, Mexico
2
SEPI-UPIITA, Instituto Politécnico Nacional, Av. IPN No. 2580, Col. La Laguna Ticomán, C.P. Mexico DF 07340, Mexico
3
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, Gliwice 44-100, Poland
4
Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4, Kraków PL-30-059, Poland
5
Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, Warsaw PL-02-668, Poland
*
Author to whom correspondence should be addressed.
Academic Editors: Demosthenes Ellinas, Giorgio Kaniadakis, Jiannis Pachos and Antonio M. Scarfone
Received: 15 June 2015 / Accepted: 10 July 2015 / Published: 20 July 2015
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
View Full-Text   |   Download PDF [1675 KB, uploaded 20 July 2015]   |  

Abstract

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit. View Full-Text
Keywords: quantum entanglement; Rényi–Ingarden–Urbanik entropy; three-tangle; hyperdeterminants; random states quantum entanglement; Rényi–Ingarden–Urbanik entropy; three-tangle; hyperdeterminants; random states
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Enríquez, M.; Puchała, Z.; Życzkowski, K. Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States. Entropy 2015, 17, 5063-5084.

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