Two entanglement measures

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 e s ( ρ ) ≥ e ( ρ ) which we call the spectral entanglement number. We argue that e s ( ρ ) is easier to compute than e ( ρ ) and that the physical motivation for e s ( ρ ) is superior to that of e ( ρ ). It is shown that e s ( ρ ) is continuous in the operator norm. Although e ( ρ ) is a convex function, we show that e s ( ρ ) need not be convex. An open problem is to characterize the states ρ such that e ( ρ ) = e s ( ρ ). Many illustrative examples are presented.