Figure 4: The sign distribution of ∂gSα in Ising model and XY model for different bipartitions.
The sign distribution of ∂gSα on the α−g plane for different bipartitions on the systems. Panels (a), (b) and (c) correspond to Ising model of size N=12, where Alice possesses 6,5 and 4 of the spins, respectively. ∂gSα is negative in lighter regions and positive in the red regions. Clearly, regardless of choice of bipartition, ∂gSα is always negative for g>1 and takes on both negative and positive values otherwise. Note that for very small g, ∂gSα only becomes negative for very large α and thus appears completely positive in the graph above. The existence of negative ∂gSα can be verified by analysis of ∂gSα in the α→∞ limit. The choice of bipartition affects only the shape of the ∂gSα=0 boundary, which is physically unimportant. Panels (d), (e) and (f) correspond to XY model with fixed 
N=14 and L=7, 6 and 5, respectively. Panels (g), (h) and (i) correspond to XY model with fixed 
N=18 and L=9, 8 and 7, respectively. Here, the value of L represents a bipartition in which L qubits are placed in one bipartition and N−L qubits in the other. The only region in which ∂Sα/∂g takes on both negative and positive values is in phase 1A. Note that the transition between phase 1A and 1B occurs at g=0.5 for 
(Point E in Fig. 3) and g=0.75 for 
(Point D in Fig. 3).
