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.[1][2]

Definition

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

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

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

References

1. Horodecki, Ryszard; Horodecki, Paweł; Horodecki, Michał; Horodecki, Karol (2009-06-17). "Quantum entanglement". Reviews of Modern Physics. 81 (2): 865–942. arXiv:quant-ph/0702225. Bibcode:2009RvMP...81..865H. doi:10.1103/RevModPhys.81.865.
2. Chitambar, Eric; Gour, Gilad (2019-04-04). "Quantum resource theories". Reviews of Modern Physics. 91 (2): 025001. arXiv:1806.06107. Bibcode:2019RvMP...91b5001C. doi:10.1103/RevModPhys.91.025001.