Geometry of entanglement witnesses and local detection of entanglement
- Department of Mathematics and Statistics and Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250 (United States)
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} 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.
- OSTI ID:
- 20634105
- Journal Information:
- Physical Review. A, Vol. 67, Issue 1; Other Information: DOI: 10.1103/PhysRevA.67.012327; (c) 2003 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
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