### Logarithmic Negativity of Bipartitions as KPI  
To quantify the bipartite entanglement, we use the standard logarithmic negativity measure as the key performance indicator (KPI), which is an entanglement monotone and normalized to yield 0 for separable (or PPT) states, and 1 for maximally entangled Bell states. The negativity of a bipartite state \(\rho\) is calculated via the absolute sum of its negative eigenvalues \(\mu_{i}\) of its partial transpose \(\rho^{\Gamma_{A}}\) with respect to subsystem \(A\), or equivalently as [35; 36]  
\[N(\rho)\equiv\frac{||\rho^{\Gamma_{A}}||_{1}-1}{2} \tag{10}\]  
where \(||X||_{1}\) is the trace norm of operator \(X\)[37]. Logarithmic negativity is calculated based on negativity as [38]  
\[LN(\rho)=\log[1+2N(\rho)]. \tag{11}\]  
In our second scenario, which is the major contribution of the present work, we show that by implementing the QZD with a sufficient number of iterations, all the eight qubits in five nodes of the network are entangled without requiring quantum state teleportation, Bell measurements, or even the \(z\)-basis measurements as in the final step of the first scenario.  
Witnessing all-particle genuine entanglement requires inseparability (or non-zero entanglement) among all bipartite cuts. Inseparability in all bipartite cuts is also stated in [13] as a necessary condition for entanglement distillation between \(E\) and \(D\). For the eight-qubit system, we take the following seven bipartite cuts \(\rho_{iS-i}\) with \(i=1,2,\ldots,7\). Due to the existence of PPT in systems beyond two-qutrits or four-qubits, inseparability does not imply distillable entanglement or a non-zero entanglement measure value. However, a non-zero entanglement value such as \(LN(\rho)>0\) implies inseparability.  
Bipartite cuts assume that qubits in each cut are in one site so that global operations such as Bell measurements are applicable in each cut. Hence, considering the \(\rho_{1:7}\) cut, for example, the result of a Bell measurement on \(C_{2}\) and \(E\) qubits can be communicated to [PERSON] and Bob so that entanglement between \(A_{2}\) and \(B_{2}\) qubits can be made distillable, yielding \(LN(\rho_{1:7})=1\). Therefore, while the logarithmic negativity of bipartite cuts of \(\rho^{ini}\) are \(\{1,0,1,0,1,0,1\}\), after rewriting the state, the logarithmic negativity of bipartite cuts of \(\rho\), as illustrated in Figure 2, yield \(\{1,2,1,0,1,2,1\}\). Because the entanglement between \(A_{1}\) and \(C_{1}\), also between \(B_{1}\) and \(D\), illustrated with orange lines in Figure 2d, is unknown to the parties, the bipartite system of \(A_{1}A_{2}B_{1}B_{2}\) and \(C_{1}C_{2}DE\) is separable, yielding \(LN(\rho_{4:4})=0\). Therefore, due to this separability, considering the five nodes spatially separated, no entanglement can be distilled from \(\rho\), implying bound entanglement.  
After presenting the results of Scenario 1 in Figures 3 and 4, our aim will be to show that it is possible to make all the bipartite cuts of the eight qubits in five nodes inseparable through a simple QZD procedure, illustrated with the transition of a network's state from Figure 5a,b. Note that the blue lines connecting qubits in Figure 5b do not imply a cyclic entanglement but rather show that all the qubits in the system are entangled.  
Figure 3: Bar plots of density matrices \(\rho_{DE}^{1}\), \(\rho_{DE}^{40}\), and \(\rho_{DE}^{345}\) for visualizing how distillable entanglement is obtained between \(D\) and \(E\) throughout QZD.  
Figure 2: Bipartite cuts \(\rho_{i:8-i}\) with \(i=1,2,\ldots,7\) of the eight-qubit system \(\rho\) are presented from (**a–g**). Cuts are denoted with dashed red lines. The \(\rho_{44}\) cut shown in (**d**) is separable. Please see text for details.