# Entanglement measure

Entanglement measure quantifies how much entanglement is contained in a quantum state. Formally it is any nonnegative real function of a state which can not increase under local operations and classical communication (LOCC) (so called monotonicity), and is zero for separable states.

There are operational entanglement measures such as distillable entanglement, distillable key and entanglement cost, as well as abstractly defined measures such as ones based on convex roof construction (e.g. concurrence and entanglement of formation) or based on distance from set of separable states such as relative entropy of entanglement.

Often the abstract measures are bounds for operational measures. For example relative entropy of entanglement is an upper bound for distillable entanglement and distillable key.

One of typical applications of abstract EM's is to show that certain task can not be achieved by means of LOCC. One does it by showing that if the task could be done, then some EM would increase.

EM's are not linearly ordered that is there exists entanglement measures E1 and E2 two states ρ and σ such that

EM1(ρ) < EM2(ρ) andEM1(σ) > EM2(σ)

Different entanglement measures determine different types of entanglement. All EMs for pure states are classified.

Entanglement measures are also studied and classified according to their properties, e.g. additivity, convexity and continuity. This approach to entanglement measures is known as axiomatic approach.