Manipulating multiqudit entanglement witnesses by using linear programming
Abstract
A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension d{sub 1}(multiply-in-circle sign)d{sub 2}(multiply-in-circle sign){center_dot}{center_dot}{center_dot}(multiply-in-circle sign)d{sub n}. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamiolkowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall [Phys. Rev. A 72, 022311 (2005)] are obtained.
- Authors:
- Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664 (Iran, Islamic Republic of)
- (Iran, Islamic Republic of)
- Publication Date:
- OSTI Identifier:
- 20982495
- Resource Type:
- Journal Article
- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052326; (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; EIGENFUNCTIONS; EIGENVALUES; HILBERT SPACE; INFORMATION THEORY; QUANTUM ENTANGLEMENT; QUANTUM INFORMATION; QUBITS
Citation Formats
Jafarizadeh, M. A., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Research Institute for Fundamental Sciences, Tabriz 51664, Najarbashi, G., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, and Habibian, H.. Manipulating multiqudit entanglement witnesses by using linear programming. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.052326.
Jafarizadeh, M. A., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Research Institute for Fundamental Sciences, Tabriz 51664, Najarbashi, G., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, & Habibian, H.. Manipulating multiqudit entanglement witnesses by using linear programming. United States. doi:10.1103/PHYSREVA.75.052326.
Jafarizadeh, M. A., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Research Institute for Fundamental Sciences, Tabriz 51664, Najarbashi, G., Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, and Habibian, H.. Tue .
"Manipulating multiqudit entanglement witnesses by using linear programming". United States.
doi:10.1103/PHYSREVA.75.052326.
@article{osti_20982495,
title = {Manipulating multiqudit entanglement witnesses by using linear programming},
author = {Jafarizadeh, M. A. and Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795 and Research Institute for Fundamental Sciences, Tabriz 51664 and Najarbashi, G. and Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795 and Habibian, H.},
abstractNote = {A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension d{sub 1}(multiply-in-circle sign)d{sub 2}(multiply-in-circle sign){center_dot}{center_dot}{center_dot}(multiply-in-circle sign)d{sub n}. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamiolkowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall [Phys. Rev. A 72, 022311 (2005)] are obtained.},
doi = {10.1103/PHYSREVA.75.052326},
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 use robust semidefinite programs and entanglement witnesses to study the distillability of Werner states. We perform exact numerical calculations that show two-undistillability in a region of the state space, which was previously conjectured to be undistillable. We also introduce bases that yield interesting expressions for the distillability witnesses and for a tensor product of Werner states with an arbitrary number of copies.
-
Geometry of entanglement witnesses and local detection of entanglement
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 » -
Geometric entanglement witnesses and bound entanglement
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. -
Entanglement witnesses and geometry of entanglement 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. -
Spectral conditions for entanglement witnesses versus bound entanglement
It is shown that entanglement witnesses constructed via the family of spectral conditions are decomposable, i.e., cannot be used to detect bound entanglement. It supports several observations that bound entanglement reveals highly nonspectral features.