Entanglement monotone: Difference between revisions

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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 ${\mathcal {S}}({\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B})$ 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:

$\mu (\rho )=0$ if $\rho$ is separable;
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.

References

^ 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.
^ 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.

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[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.

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@@ Line 5: / Line 5: @@
 == Definition ==
-Let <math>\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>\mathcal{H}_A\otimes\mathcal{H}_B</math>. An entanglement measure is a function <math>\mu:{\displaystyle {\mathcal {S}}({\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B})}\to \mathbb{R}_{\geq 0}</math>such that:
+Let <math>\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)</math>be the space of all [[Quantum state|states]], i.e., [[Hermitian matrix|Hermitian]] [[Positive semi-definite matrix|positive semi-definite]] operators with trace one, over the bipartite [[Hilbert space]] <math>\mathcal{H}_A\otimes\mathcal{H}_B</math>. An entanglement measure is a function <math>\mu:{\displaystyle {\mathcal {S}}({\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B})}\to \mathbb{R}_{\geq 0}</math>such that:
-# <math>\mu(\rho)=0</math> if <math>\rho</math> is separable;
+# <math>\mu(\rho)=0</math> if <math>\rho</math> is [[Separable state|separable]];
-# Monotonically decreasing under LOCC, viz., for the [[Kraus operator]] <math>E_i\otimes F_i</math> corresponding to the LOCC <math>\mathcal{E}_{LOCC}</math>, let <math>p_i=\mathrm{Tr}[(E_i\otimes F_i)\rho (E_i\otimes F_i)^{\dagger}]</math> and <math>\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>\rho</math>, then (i) <math>\mu</math> does not increase under the average over all outcomes, <math>\mu(\rho)\geq \sum_i p_i\mu(\rho_i)</math> and (ii) <math>\mu</math> does not increase if the outcomes are all discarded, <math>\mu(\rho)\geq \sum_i \mu(p_i\rho_i)</math>.
+# Monotonically decreasing under [[LOCC]], viz., for the [[Kraus operator]] <math>E_i\otimes F_i</math> corresponding to the LOCC <math>\mathcal{E}_{LOCC}</math>, let <math>p_i=\mathrm{Tr}[(E_i\otimes F_i)\rho (E_i\otimes F_i)^{\dagger}]</math> and <math>\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>\rho</math>, then (i) <math>\mu</math> does not increase under the average over all outcomes, <math>\mu(\rho)\geq \sum_i p_i\mu(\rho_i)</math> and (ii) <math>\mu</math> does not increase if the outcomes are all discarded, <math>\mu(\rho)\geq \sum_i \mu(p_i\rho_i)</math>.
 Some authors also add the condition that <math>\mu(\varrho)=1</math> over the maximally entangled state <math>\varrho</math>. If the nonnegative function only satisfies condition 2 of the above, then it is called an entanglement monotone.