Irreversibility of Entanglement Concentration for Pure State

Abstract

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

6 pages, 1 figure