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

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

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