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Title: Concurrence via entanglement witnesses

Abstract

We derive an experimentally observable lower bound on concurrence of mixed quantum states in terms of an entanglement witness, relating measurements on single states with those on two copies.

Authors:
 [1]
  1. Department of Physics, Harvard University, Cambridge, Massachusetts 02138 (United States)
Publication Date:
OSTI Identifier:
20982471
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052302; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; MIXED STATE; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; QUANTUM TELEPORTATION

Citation Formats

Mintert, Florian. Concurrence via entanglement witnesses. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.052302.
Mintert, Florian. Concurrence via entanglement witnesses. United States. doi:10.1103/PHYSREVA.75.052302.
Mintert, Florian. Tue . "Concurrence via entanglement witnesses". United States. doi:10.1103/PHYSREVA.75.052302.
@article{osti_20982471,
title = {Concurrence via entanglement witnesses},
author = {Mintert, Florian},
abstractNote = {We derive an experimentally observable lower bound on concurrence of mixed quantum states in terms of an entanglement witness, relating measurements on single states with those on two copies.},
doi = {10.1103/PHYSREVA.75.052302},
journal = {Physical Review. A},
number = 5,
volume = 75,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}
  • We show that each entanglement witness detecting a given bipartite entangled state provides an estimation of its concurrence. We illustrate our result with several well-known examples of entanglement witnesses and compare the corresponding estimation of concurrence with other estimations provided by the trace norm of partial transposition and realignment.
  • Let H{sup [N]}=H{sup [d{sub 1}]}x{center_dot}{center_dot}{center_dot}xH{sup [d{sub n}]} be a tensor product of Hilbert spaces and let {tau}{sub 0} be the closest separable state in the Hilbert-Schmidt norm to an entangled state {rho}{sub 0}. Let {tau}-tilde{sub 0} denote the closest separable state to {rho}{sub 0} along the line segment from I/N to {rho}{sub 0} where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W{sub 0} detecting the entanglement of {rho}{sub 0} can be constructed in terms of I, {tau}{sub 0}, and {tau}-tilde{sub 0}. If representations of {tau}{sub 0}more » and {tau}-tilde{sub 0} as convex combinations of separable projections are known, then the entanglement of {rho}{sub 0} can be detected by local measurements. Guehne et al. [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, {tau}{sub 0}={tau}-tilde{sub 0}. We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which {tau}{sub 0}{ne}{tau}-tilde{sub 0} and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a 'far face' of the separable states relative to a bound entangled state {rho}{sub 0} constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density {mu}{sub 0} on the far face from {rho}{sub 0}. Up to a normalization this coincides with the witness obtained by Guehne et al. for the particular example analyzed there.« less
  • We study entanglement witnesses that can be constructed with regard to the geometrical structure of the Hilbert-Schmidt space, i.e., we present how to use these witnesses in the context of quantifying entanglement and the detection of bound entangled states. We give examples for a particular three-parameter family of states that are part of the magic simplex of two-qutrit states.
  • We construct entanglement witnesses with regard to the geometric structure of the Hilbert-Schmidt space and investigate the geometry of entanglement. In particular, for a two-parameter family of two-qutrit states that are part of the magic simplex, we calculate the Hilbert-Schmidt measure of entanglement. We present a method to detect bound entanglement which is illustrated for a three-parameter family of states. In this way, we discover new regions of bound entangled states. Furthermore, we outline how to use our method to distinguish entangled from separable states.
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