Entanglement purification with double selection

We investigate an entanglement purification protocol with double-selection process, which works under imperfect local operations. Compared with the usual protocol with single selection, this double-selection method has higher noise thresholds for the local operations and quantum communication channels and achieves higher fidelity of purified states. It also provides a yield comparable to that of the usual protocol with single selection. We discuss on general grounds how some of the errors which are introduced by local operations are left as intrinsically undetectable. The undetectable errors place a general upper bound on the purification fidelity. The double selection is a simple method to remove all the detectable errors in the first order, so that the upper bound on the fidelity is achieved in the low-noise regime. The double selection is further applied to purification of multipartite entanglement such as two-colorable graph states.

Figure 1

Bipartite entanglement purification with single selection.

Figure 2

Bipartite entanglement purification with double selection.

Figure 3

Maximum achievable fidelity $F_{\text{max}}$ (upper curves) and minimum required channel fidelity $F_{\text{min}}$ (lower curves) are plotted as functions of the error probability $p=p_g=p_m$ for the double and single selections.

Figure 4

Working range $\left(p_g,p_m\right)$ of local operations. Each purification protocol $\mathcal{A}$ ($\mathcal{S}$ or $\mathcal{D}$) achieves $F_{\text{max}}$ for $F_{\text{ch}}>F_{\text{min}}$ when the error probabilities $p_g$ and $p_m$ are below the threshold curve (solid line for $\mathcal{D}$ and dotted line for $\mathcal{S}$).

Figure 5

The yield $Y_{\mathcal{A}}\left(F,F_{\text{ch}}=0.8\right)$ is plotted as a function of the target fidelity $F$ for each protocol with $p_g=p_m=0.02$ (upper curves) and $p_g=p_m=0.04$ (lower curves).

Figure 6

Setup of a protocol of bipartite entanglement purification. An upper bound on the fidelity is determined in the first order by the undetectable errors (indicated by black stars) introduced by the final two CNOT gates. Similar bounds are obtained with other configurations of two-qubit gates.

Figure 7

Purification of two-colorable graph states with double selection.

Figure 8

Maximum achievable fidelity $F_{\text{max}}$ (upper curves) and minimum required initial fidelity $F_{\text{min}}$ (lower curves) for the Steane seven-qubit code state $|0_L⟩$ are plotted as functions of the error probability $p=p_g=p_m$ for the single and double selections.

Figure 9

The yield $Y_{\mathcal{A}}\left(F,F_{\text{ch}}=0.9\right)$ is plotted as a function of the target fidelity $F$ for each protocol with $p_g=p_m=0.02$ (upper curves) and $p_g=p_m=0.04$ (lower curves).