Entanglement monotone

In quantum information and quantum computation, an entanglement monotone or entanglement measure 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.

Various entanglement monotones exist for bipartite systems as well as for multipartite systems. Common entanglement monotones are the entropy of entanglement, concurrence, negativity, squashed entanglement, entanglement of formation and tangle.

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. S2CID 59577352.
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. S2CID 119194947.