Two entanglement measures

Abstract

In a previous work we introduced an entanglement number e(ψ) for a vector state ψ. This number relied on the Schmidt decomposition of ψ which may be difficult to compute. We now present a method for finding e(ψ) that is simple and efficient. We show that e(ψ) is continuous in the vector norm. We next extend e(ψ) to an entanglement number e(ρ) for a general mixed state ρ and show that ρ is separable (not entangled) if and only if e(ρ) = 0. We next define a related quantity es (ρ) ≥ e(ρ) which we call the spectral entanglement number. We argue that es (ρ) is easier to compute than e(ρ) and that the physical motivation for es (ρ) is superior to that of e(ρ). It is shown that es (ρ) is continuous in the operator norm. Although e(ρ) is a convex function, we show that es (ρ) need not be convex. An open problem is to characterize the states ρ such that e(ρ) = es (ρ). Many illustrative examples are presented.