Multiparticle entanglement purification for two-colorable graph states

We investigate multiparticle entanglement purification schemes which allow one to purify all two colorable graph states, a class of states which includes, e.g., cluster states, Greenberger-Horne-Zeilinger states, and code words of various error correction codes. The schemes include both recurrence protocols and hashing protocols. We analyze these schemes under realistic conditions and observe for a generic error model that the threshold value for imperfect local operations depends on the structure of the corresponding interaction graph, but is otherwise independent of the number of parties. The qualitative behavior can be understood from an analytically solvable model which deals only with a restricted class of errors. We compare direct multiparticle entanglement purification protocols with schemes based on bipartite entanglement purification and show that the direct multiparticle entanglement purification is more efficient and the achievable fidelity of the purified states is larger. We also show that the purification protocol allows one to produce private entanglement, an important aspect when using the produced entangled states for secure applications. Finally we discuss an experimental realization of a multiparty purification protocol in optical lattices which is issued to improve the fidelity of cluster states created in such systems.

Figure 1

Graphs with $N=7$ corresponding to (a) GHZ state, (b) linear cluster state, and (c) seven-qubit Steane code.

Figure 2

(a) Minimal value of fidelity $F_{\min }+[\circ ]$ and parameter $q_{\min }\times [\ast ]$ for linear cluster states [GHZ states] for different number of particles $N$ and perfect local operations.

Figure 3

The required initial fidelity for linear cluster states as a function of the number $N$ of parties. The dotted curve is an exponential fit to the exact values (circles).

Figure 4

Maximal reachable fidelity $F_{\max }$ and minimal required fidelity $F_{\min }$ plotted against the error parameter $p$ (local operations) for density operators arising from single-qubit white noise. Curves from top to bottom correspond to linear cluster states with $N=2,4,6,8,10$ particles.

Figure 5

Threshold value for errors in local operations $p_{\min }$ for GHZ states (∗) and linear cluster states (×) with a different number of particles, $N$.

Figure 6

Threshold value $q_{\mathrm{crit}}$ for imperfect local operations as a function of number of the particles, $M=N∕2$, for $M$ odd.

Figure 7

Achievable fidelity of a linear cluster state with $N=4$ using direct multiparty entanglement purification (solid line) and conservative upper bound for methods based on bipartite entanglement purification (dashed line) for different errors in local operations $p$.

Figure 8

The fidelity and the conditional fidelity as a function of the number of steps in the purification protocol.