Entanglement monotone

In quantum information and quantum computation, an entanglement monotone is a function that quantifies the amount of entanglement present in a quantum state. Any entanglement monotone is a nonnegative function whose value does not increase under local operations and classical communication.

Definition

Let [math]\displaystyle{ \mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B) }[/math]be the space of all states, i.e., Hermitian positive semi-definite operators with trace one, over the bipartite Hilbert space [math]\displaystyle{ \mathcal{H}_A\otimes\mathcal{H}_B }[/math]. An entanglement measure is a function [math]\displaystyle{ \mu:{\displaystyle {\mathcal {S}}({\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B})}\to \mathbb{R}_{\geq 0} }[/math]such that:

1. [math]\displaystyle{ \mu(\rho)=0 }[/math] if [math]\displaystyle{ \rho }[/math] is separable;
2. Monotonically decreasing under LOCC, viz., for the Kraus operator [math]\displaystyle{ E_i\otimes F_i }[/math] corresponding to the LOCC [math]\displaystyle{ \mathcal{E}_{LOCC} }[/math], let [math]\displaystyle{ p_i=\mathrm{Tr}[(E_i\otimes F_i)\rho (E_i\otimes F_i)^{\dagger}] }[/math] and [math]\displaystyle{ \rho_i=(E_i\otimes F_i)\rho (E_i\otimes F_i)^{\dagger}/\mathrm{Tr}[(E_i\otimes F_i)\rho (E_i\otimes F_i)^{\dagger}] }[/math]for a given state [math]\displaystyle{ \rho }[/math], then (i) [math]\displaystyle{ \mu }[/math] does not increase under the average over all outcomes, [math]\displaystyle{ \mu(\rho)\geq \sum_i p_i\mu(\rho_i) }[/math] and (ii) [math]\displaystyle{ \mu }[/math] does not increase if the outcomes are all discarded, [math]\displaystyle{ \mu(\rho)\geq \sum_i \mu(p_i\rho_i) }[/math].

Some authors also add the condition that [math]\displaystyle{ \mu(\varrho)=1 }[/math] over the maximally entangled state [math]\displaystyle{ \varrho }[/math]. If the nonnegative function only satisfies condition 2 of the above, then it is called an entanglement monotone.