entanglement monotone

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

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

Some authors also add the condition that $\backslash mu(\backslash varrho)=1$ over the maximally entangled state $\backslash varrho$. 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.

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Category:Quantum information theory

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