The allocation of Entropy of a Bipartite system has been proved to form a Convex cone. This can be found in the IEEE paper authored by Nicholas Pippenger. Here I am taking two bipartite system as:

1. Mixture of |0> and |1>

2. Mixture of |0> and |+>

In each case of the bipartite system case, the percentage of the two states are varied and entropy calculated. The entropies were then plotted in Matlab to obtain the following 3D view of the allocation of Entropy.

fig: Allocation of Entropy for Bipartite Quantum Systems

fig: allocation of Entropies of a two Bipartite System, the blue color plot is for the |0> and |+> mixed system, the green color plot is for |0> and |1> mixed system(click on the picture to enlarge)

The following plot is for the |0> and |+> mixed bipartite system:

fig: Allocation of Entropy for |0> and |+> bipartite system in 3D

Another view:

In this figure, the red part at the top is the area of maximum entropy for the bipartite system, while the blue at the bottom represents the minimum entropy of the composite system. Any two points inside the area satisfies the convex function criteria as well as the convex cone criteria. Hence the allocation of entropy is a Convex Cone.

Fig: 3D graph of allocation of Entropy for Bipartite System