Entanglement measures and the Hilbert-Schmidt distance

As classical information arises from probability correlation between two random variables, quantum information arises from entanglement [1, 2]. Motivated by the finding of an entangled state which does not violate Bell’s inequality, the problem of quantifying entanglement has received an increasing interest recently. Vedral et. al. [3] proposed three necessary conditions that any measure of entanglement has to satisfy and showed that if a “distance” between two states has the property that it is nonincreasing under every completely positive trace preserving map (to be referred to as the CP nonexpansive property), the “distance” of a state to the set of disentangled states satisfies their conditions. It has been shown that the quantum relative entropy and the Bures metric have the CP nonexpansive property [3], and it has been conjectured that so does the Hilbert-Schmidt distance [4]. In the interesting Letter [5], Witte and Trucks claimed that the Hilbert-Schmidt distance really has the CP nonexpansive property and conjectured that the distance generates a measure of entanglement satisfying even the stronger condition posed later by Vedral and Plenio [4]. However, it can be readily seen that their suggested proof includes a serious gap. In this Letter, it will be shown that, contrary to their claim, the Hilbert-Schmidt distance does not have the CP nonexpansive property by presenting a counterexample. Let H = H1⊗H2 be the Hilbert space of a quantum system consisting of two subsystems with Hilbert spaces H1 and H2. We assume that H1 and H2 have the same finite dimension. We shall consider the notion of entanglement with respect to the above two subsystems. Let T be the set of density operators on H. The set D of disentangled states is the set of all convex combinations of pure tensor product states. There are several requirements that every measure of entanglement, E, should satisfy [3, 4]: (E1) E(σ) = 0 for all σ ∈ D.