# Entanglement of assistance

Definition: The formula defining the entanglement of assistance is dual to that for entanglement of formation:

$$\; E_{ass}(\varrho) = sup_{\{p_i,\psi_i \}}\sum_i p_i S(\varrho_i^A) ,$$
where $\; \varrho_i^A$ is the reduced density matrix of ψi and the supremum is taken over all possible decompositions of $\; \varrho$.

Entanglement of assistance was initially defined for tripartite pure states as the maximal entropy of entanglement that can be created between Alice and Bob if Charlie helps them by measuring his own subsystem and telling them the outcomes. For tripartite pure states entanglement of assistance is simply a function of Alice and Bob’s bipartite state $\; \varrho^{AB}$, but it reflects entanglement properties of the joint state $\; \varrho^{ABC}$. Since it could be increased by cooperation among Alice, Bob and Charlie, entanglement of assistance is not a LOCC monotone for tripartite states. Moreover, it is also nonadditive. However, in the limit of many copies, entanglement of assistance becomes a monotone, in the sense that it reduces to the minimum of the entropies of Alice’s and Bob’s subsystems.

More precisely, suppose that Alice, Bob and Charlie possess many copies of a tripartite quantum system each described by the pure state  ∣ΨABC and that Alice and Bob want to use their shared (generally mixed) quantum system $\; \varrho^{AB}=Tr_C[|\Psi^{ABC}\rangle\langle\Psi^{ABC}|]$ for some task such as teleportation. Even if Charlie is not immediately available to pass over his subsystem for their use, he can help Alice and Bob anyway by performing measurements on his subsystems so that Alice and Bob will have pure states conditioned on his outcomes. So in general, for a measurement that creates states  ∣ΨiAB with probabilities pi, i.e. an ensemble  E = {pi, ∣ΨiAB⟩⟨ΨiAB∣}, the average rate of conversion to singlets will be given by the entropy of entanglement S:

$$\; E(\mathcal{E})=\sum_i p_i S(\varrho_i^A)=-\sum_i p_iTr[\varrho_i^A log_2(\varrho_i^A)],$$
with $\; \varrho_i^A=Tr_B[|\Psi_i^{AB}\rangle\langle\Psi_i^{AB}|]$.

The entanglement of assistance Eass for $\; \varrho^{AB}$ is then defined as the maximum of this average rate, maximized over all possible ensembles  E consistent with $\; \varrho^{AB}=\sum_i p_i |\Psi_i^{AB}\rangle\langle\Psi_i^{AB}|$:

$$\; E_{ass}(\varrho^{AB})=max_{\mathcal{E}}E(\mathcal{E}).$$

In the context of spin chains, entanglement of assistance can be generalized to another entanglement measure in the multipartite setting: localizable entanglement, which is defined as the maximal amount of entanglement that can be localized, on average, between two spins by doing local measurements on the rest of the spins in the system.

### Properties and bounds

Since the entanglement of an ensemble is invariant under local unitary transformations and non-increasing under general local operations with classical communication (LOCC), entanglement of assistance also has these properties. Moreover, since the definition of the entanglement of assistance involves a maximization over all possible pure-state realizations of the density matrix, Eass is concave, i.e.

$$\; E_{ass}(\sum_i p_i \varrho_i) \geq \sum_i p_i E_{ass}(\varrho_i).$$
Note that the entanglement of assistance is the least of all concave functions coinciding on pure states with their partial entropy (i.e. entropy as seen from one side).

The entanglement of assistance is superadditive, i.e.

$$\; E_{ass}(\varrho_1^{AB}\otimes \varrho_2^{AB}) &gt; E_{ass}(\varrho_1^{AB}) + E_{ass}(\varrho_2^{AB}) .$$

#### Upper bounds

Two upper bounds for the entanglement of assistance were found:

1. the entropic bound which holds in general;
2. the fidelity bound which holds in the simple case of one qubit on Alice's side and one qubit on Bob's side.

Entropic bound:

$$\; E_{ass}(\varrho) \leq min\{S(Tr_A[\varrho]), S(Tr_B[\varrho])\},$$
where $\; S(Tr_A[\varrho]), S(Tr_B[\varrho])$ is the partial entropy as seen by Alice and Bob respectively.

Fidelity bound:

$$\; E_{ass}(\varrho) \leq F(\varrho, \tilde{\varrho}),$$
where $\; F(\varrho, \tilde{\varrho})$ is the fidelity of the density matrix relative to its complex conjugate in the magic basis $\; \tilde{\varrho}$.

#### Lower bounds

In general we can get lower bounds by using the fact that the average entanglement of any pure-state realization of the density matrix gives a lower bound to the entanglement of assistance and by making systematic pure-state decompositions of the density matrix. For example, applying the above idea to density matrices diagonal in a product basis we get the diagonal lower bound:

$$\; E_{ass}(\varrho) \geq (r_1 + r_4)H_2(r_1/(r_1 + r_4)) + (r_2 + r_3)H_2(r_2/(r_2 + r_3)),$$
where r1, …, r4 are the diagonal entries of $\; \varrho$ and H2(x) =  − xlog2(x) − (1 − x)log2(1 − x) is the Shannon entropy.

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