Irreversibility of Entanglement Concentration for Pure State

For a pure state $ψ$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, both the entanglement cost $E_C(ψ)$ and the distillable entanglement $E_D(ψ)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}ψ)$. Therefore, the entanglement concentration from the multiple state $ψ^{\otimes n}$ of a pure state $ψ$ to the multiple state $Φ^{\otimes L_n}$ of the EPR state $Φ$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}ψ)$. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $ψ^{\otimes M_n}$ after the concentration for $ψ^{\otimes n}$, we evaluate the asymptotic behavior of the loss number $n-M_n$ of $ψ$. This evaluation is thought to be closely related to the entanglement compression in distant parties.

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