Figure 1

Illustration of the protocol for a three-colorable LME state where color $A$ is purified (see Sec. 3a). The target state is given by $|{\Psi }_{\mathbf{0}}\rangle ={\prod }_{i=2}^{n-1}{U}_{i-1,i,i+1}{|+\rangle }^{\otimes n}$. The parties in color A entangle copy 1 with copy 2 by applying cnot gates. All other parties make a GHZ measurement on one qubit of each copy and one qubit of a maximally entangled state, $|{\Phi }^{+}\rangle$. This measurement is depicted as a triangle. Then a measurement in the $X$ basis is performed on the qubits of color $A$ of the second copy. This measurement is displayed by an arrow pointing in the upper right direction. The qubits of the remaining state are colored in a darker fashion (bright red).Reuse & Permissions

Figure 2

$\parallel {\left(E_{123}^{{\Phi }^i}\right)}^{\Gamma }{\parallel }_{tr}$ for randomly generated maps ${\Phi }_{\mathrm{dep}}^i$ $(1\le i\le 1000)$. In all cases it holds that $\parallel {\left(E_{123}^{{\Phi }^i}\right)}^{\Gamma }{\parallel }_{tr}>1$, indicating that the CJ state is entangled and, therefore, that the corresponding map cannot be implemented by LOCC.Reuse & Permissions

Figure 3

Illustration of two methods to realize the CPM ${\mathcal{E}}_i=\frac{1}{2}(\rho +S_i\rho S_i^{†})$ of the depolarization process given by Eq. (14), where $|\Psi \rangle ={\prod }_{i=2,4,6,...}{U}_{i-1,i,i+1}{|+\rangle }^{\otimes n}$. Panel (a) depicts how $S_i$ is implemented using the auxillary state $|\varphi \rangle$ (see main text). The measurement in the GHZ basis is pictured as the triangle which acts on $\rho$, the auxillary state, and one qubit of a maximally entangled state. The circle marks that an $X$ operator is applied. In order to realize the CPM ${\mathcal{E}}_i$ the parties choose randomly to apply this transformation or not. Panel (b) shows how ${\mathcal{E}}_i$ is implemented by employing the LME state $|{\psi }^{\prime }\rangle$ and tracing out the qubit which is marked with a cross.Reuse & Permissions

Figure 4

The noise threshold $p$ of the depolarizing and dephasing channel given in Eqs. (25) and (26) and the minimal required fidelity for the purification process for a state subject to global white noise of Eq. (27) is shown. We consider three-colorable LME states of three, four, five, and six qubits which are generated by three-qubit phase gates. The interaction pattern were given by $U_{123}{|+\rangle }^{\otimes 3}$, $U_{123}{U}_{234}{|+\rangle }^{\otimes 4}$ for three and four qubits, respectively $(\times )$. For the five-qubit state we have two possible patterns, a linear one $U_{123}{U}_{234}{U}_{345}{|+\rangle }^{\otimes 5}$ $(\times )$ and a GHZ-like one $U_{123}{U}_{124}{U}_{125}{|+\rangle }^{\otimes 5}$ $(\circ )$. In case of the six-qubit state one has three different patterns, a linear one $U_{123}{U}_{234}{U}_{345}{U}_{456}{|+\rangle }^{\otimes 6}$ $(\times )$, an intermediate one $U_{134}{U}_{235}{U}_{234}{U}_{346}{|+\rangle }^{\otimes 6}$ $(\diamond )$, and a GHZ-like one $U_{123}{U}_{124}{U}_{125}{U}_{126}{|+\rangle }^{\otimes 6}$ $(\circ )$. For the dephasing channel $(\square )$ the pattern does not strongly influence the threshold. For the global white-noise channel $(*)$ we show the fidelity for a target LME state with a linear pattern.Reuse & Permissions

Figure 5

Illustration of how one can obtain graph (bipartite) states from LME states by local $Z$ measurements. The input state is given by $\mathcal{E}\left(\right|\Psi \rangle \langle \Psi \left|\right)$ from Eq. (27) with ${|\Psi \rangle }_6=U_{123}{U}_{234}{U}_{345}{U}_{456}{|+\rangle }^{\otimes 6}$. The local measurements in the $Z$ basis are pictured by arrows pointing in the upper right direction and the numbers $+1,-1$ denote the outcomes. Panel (a) shows how one can obtain a three-qubit graph state $\mathcal{E}\left(\right|\Psi \rangle \langle \Psi \left|\right)$ with ${|\Psi \rangle }_3=U_{13}{U}_{34}{|+\rangle }^{\otimes 3}$. Panel (b) indicates how a bipartite state $\mathcal{E}\left(\right|\Psi \rangle \langle \Psi \left|\right)$ with ${|\Psi \rangle }_2=U_{12}{|+\rangle }^{\otimes 2}$ is obtained. The coloring of the qubits refer to the three colors of the LMES.Reuse & Permissions

Figure 6

Illustration of how one could connect two graph states (bipartite) states to obtain LME states (a), (b) and how one can recombine LMES (c), (d). In panel (a) the LMES $U_{123}{U}_{234}{|+\rangle }^{\otimes 4}$ is obtained by applying the projector $Q_i$ (see main text) between graph states of the form $U_{12}{U}_{24}{|+\rangle }^{\otimes 3}$ and $U_{1^{\prime }{3}^{\prime }}{U}_{3^{\prime }{4}^{\prime }}{|+\rangle }^{\otimes 3}$. In panel (b) $Q_i{H}^{\otimes 2}$ is applied to maximally entangled states $|{\Phi }^{+}\rangle$, resulting in $U_{123}{|+\rangle }^{\otimes 3}$. Panels (c) and (d) show how LMES can be recombined by employing the projector $P_i$ (see main text). The coloring of the qubits refers in all parts to the colors of the quantum state.Reuse & Permissions