Geometric measure of entanglement and applications to bipartite and multipartite quantum states

Abstract

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings (see Shimony [1] and Barnum and Linden [2]), is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary twoqubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of threequbit GHZ, W and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.

A comparative study of relative entropy of entanglement, concurrence and negativity

“…The problem was raised by Eisert and Plenio [2] on the numerical example of the concurrence and negativity and then studied by others [3,5,6,7,8,9,10,11]. The ordering problem is closely related to existence of the upper and lower bounds of one entanglement measure versus the other [5,7,11,31].…”

“…However, such pure states can be transformed into each other by local operations. To show this, first we note that any pure state, given by (7), can be transformed by local rotations into the superposition |ψ P (p) = √ p|01 + √ 1 − p|10 (0 ≤ p ≤ 1), for which the concurrence and negativity are equal to 2 p(1 − p), as a special case of (8). The same value of these entanglement measures occurs also for |ψ P (1 − p) , but this state can be transformed into |ψ P (p) by applying NOT gate to each of the qubits.…”

“…The problem was then analyzed by others [3,4,5,6,7,8,9,10,11]. In particular, Virmani and Plenio [4] proved that all good asymptotic entanglement measures are either identical or fail to impose consistent orderings on the set of all quantum states.…”

Geometric analysis of entangled qubit pairs

“…However, even for perfectly entangled states with C = 1 the sum of variances (14) is non-zero, because | is never an eigenstate of cosine and sine operators simultaneously 3 .…”

“…Nevertheless, numerous approaches have been developed so far in order to intuitively analyse this problem in terms of some geometrical quantities. For example, the entanglement of two spin-1/2 particles can be studied in terms of the distances between states in high-dimensional manifolds [1,2]; one can define entanglement geometrically as the distance between a given state and the nearest separable state [3,4], relate entanglement to the geometrical structure of the quaternionic Hopf fibration [5]; or analyse it by the approach of operator trigonometry [6]. A review of separability criteria and entanglement measures is discussed geometrically in [7].…”