Geometry of entanglement witnesses and local detection of entanglement

Abstract

Let H [N] = H [d 1 ] ⊗ · · · ⊗ H [dn] be a tensor product of Hilbert spaces and let τ0 be the closest separable state in the Hilbert-Schmidt norm to an entangled state ρ0. Letτ0 denote the closest separable state to ρ0 along the line segment from I/N to ρ0 where I is the identity matrix. Following [1] a witness W0 detecting the entanglement of ρ0 can be constructed in terms of I, τ0 andτ0. If representations of τ0 andτ0 as convex combinations of separable projections are known, then the entanglement of ρ0 can be detected by local measurements. Gühne et. al. in [2] 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, τ0 =τ0. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which τ0 =τ0 and we show our approach works for n qubits.In [3] we elaborated on the role of a "far face" of the separable states relative to a bound entangled state ρ0 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 µ0 on the far face from ρ0. Up to a normalization this coincides with the witness obtained in [2] for the particular example analyzed there. I. MOTIVATION AND NOTATIONAn important question for quantum information theory is how to determine if a given state is entangled. Physically, one would like to do this using local measurements and classical communications. Testing for entanglement is closely related to Bell inequalities [4] and subsequent elaborations of Bell's inequalities [5]. Recently other tests have been suggested, such as that in [6] which relies on the theory of positive operators and on eigenvalue estimation.An alternate approach, which is experimentally realizable, is to define local correlated measurements motivated by some knowledge of the structure of ρ itself, and this approach has been elaborated in [2]. To describe the problem, we first define the mathematical context. Specifically, we assume we are working with n distinct systems so that ρ is represented as an N × N density operating on the tensor product Hilbert space H [N ] = H [d1] ⊗ · · · ⊗ H [dn] . The set D of such N × N densities operating on H [N ] is a compact convex subset of the real Hilbert space M of N × N Hermitian matrices where the inner product is defined by A, B = T r A † B . (Since the matrices are assumed to be Hermitian, the notation " † " denoting the Hermitian conjugate appears to be redundant. However, we will have occasion to use the inner product for more general matrices.) The set of separable densities S is defined as the convex hull of the separable projections π 1 ⊗ · · · ⊗ π n , where π k is a projection on H [d k ] . Since S is a compact convex subset of D one can test for entanglement by show...

Entanglement detection

“…Also, if k is even (odd), an explicit decomposition into 2k (2k − 1) measurements has been found. For the case that |ψ is a projector onto a maximally entangled state in a d × d-system and d is prime, it has been shown that W can indeed be decomposed into d + 1 measurements [384], hence for this special case the problem of the optimal decomposition is solved. A different interesting problem, which is somehow reverse to the original problem is the following: If we assume that some correlation measurements like σ x ⊗ σ x , σ x ⊗ σ z , σ z ⊗ σ x and σ x ⊗ σ x have been done, which states can then be detected with that data?…”

On Duality between Quantum Maps and Quantum States

“…In certain cases this quantity was estimated analytically by Terhal [100] and numerically by Gühne et al [31,32] in the context of characterizing the entanglement witnesses. In fact a non-positive dynamical matrix D, which describes a non completely positive map, may be just considered as an entanglement witnesses -an operator D such TrDρ is not negative for all separable states and negative for a given entangled state [14,65,66,83,101].…”

Optimal entanglement witnesses for qubits and qutrits

“…Similar methods for constructing an entanglement witness can be found in Ref. [28]; for other approaches see, e.g., Refs. [29,30,31].…”