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Title: Optimal entanglement witnesses for qubits and qutrits

Abstract

We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states--and their generalizations to arbitrary dimensions--where we calculate the optimal entanglement witnesses explicitly.

Authors:
; ; ;  [1]
  1. Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna (Austria)
Publication Date:
OSTI Identifier:
20786483
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.72.052331; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; BELL THEOREM; DISTANCE; ENERGY LEVELS; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; QUBITS

Citation Formats

Bertlmann, Reinhold A., Durstberger, Katharina, Hiesmayr, Beatrix C., and Krammer, Philipp. Optimal entanglement witnesses for qubits and qutrits. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Bertlmann, Reinhold A., Durstberger, Katharina, Hiesmayr, Beatrix C., & Krammer, Philipp. Optimal entanglement witnesses for qubits and qutrits. United States. doi:10.1103/PHYSREVA.72.0.
Bertlmann, Reinhold A., Durstberger, Katharina, Hiesmayr, Beatrix C., and Krammer, Philipp. Tue . "Optimal entanglement witnesses for qubits and qutrits". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786483,
title = {Optimal entanglement witnesses for qubits and qutrits},
author = {Bertlmann, Reinhold A. and Durstberger, Katharina and Hiesmayr, Beatrix C. and Krammer, Philipp},
abstractNote = {We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states--and their generalizations to arbitrary dimensions--where we calculate the optimal entanglement witnesses explicitly.},
doi = {10.1103/PHYSREVA.72.0},
journal = {Physical Review. A},
number = 5,
volume = 72,
place = {United States},
year = {Tue Nov 15 00:00:00 EST 2005},
month = {Tue Nov 15 00:00:00 EST 2005}
}
  • We consider cloning transformations of equatorial qubits vertical bar {psi}{sub {phi}}>=1/{radical}(2)(vertical bar 0>+e{sup i{phi}} vertical bar 1>) and qutrits vertical bar {psi}{sub {phi},{theta}}>=1/{radical}(3)(vertical bar 0>+e{sup i{phi}} vertical bar 1>+e{sup i{theta}} vertical bar 2>), with the transformation covariant for rotation of the phases {phi} and {theta}. The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.
  • We provide a class of optimal nondecomposable entanglement witnesses for 4Nx4N composite quantum systems or, equivalently, another construction of nondecomposable positive maps in the algebra of 4Nx4N complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.
  • We show that the entanglement witnesses based on local orthogonal observables, which were introduced by Yu and Liu [Phys. Rev. Lett. 95, 150504 (2005)] and Guehne et al. [Phys. Rev. A 74, 010301(R) (2006)] in linear and nonlinear forms, respectively, can be optimized. As applications, we use our method to calculate the optimal nonlinear witnesses of pure bipartite states and to show a lower bound on the I concurrence of bipartite higher-dimensional systems.
  • We provide a class of indecomposable entanglement witnesses. In 4x4 case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the 'weakest' quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of 2kx2k complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].
  • In a recent paper [D. Chruscinski and F. A. Wudarski, Open Sys. Information Dyn. (unpublished)], it was conjectured that the entanglement witnesses arising from some generalized Choi maps are optimal. We show that this conjecture is true. Furthermore, we show that they provide a one-parameter family of indecomposable optimal entanglement witnesses.