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 $\mathcal(\mathcal_A\otimes\mathcal_B)$be the space of all states, i.e., Hermitian positive semi-definite operators with trace one, over the bipartite Hilbert space $\mathcal_A\otimes\mathcal_B$. An entanglement measure is a function $\mu:\to \mathbb_$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 $\mathcal_$, let p_i=\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math> and \rho_i=(E_i\otimes F_i)\rho (E_i\otimes F_i)^/\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math>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

Quantum information theory

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