Suppose Bob performs a $k$-outcome measurement, characterised by the POVM $E = \{E_j\}_{j=1}^k$, on $l$ qudits $\{\rho_i\}_{i=1}^l$ sent by Alice. This results in an $k \times l$ column-stochatstic matrix $A$ with each element $a_{ij}= \mathrm{Tr}(E_i\rho_j)$. Let us call this matrix a $d$-dimensional quantum communication matrix, $Q_d$.
Similarly, we can think of a $d$-dimensional classical communication matrix, $C_d$. More precisely, in the classical case each state is defined by a probability distribution: $\{p^{(i)}\}_{i=1}^l$, correspondingly we have a diagonal density matrix $\rho_i=\sum_{\alpha=1}^d p^{(i)}(\alpha) |\alpha\rangle\langle\alpha|$, and the decoder will be a classical $k$-outcome measurement defined by a stochastic map. For example, a 3-outcome measurement on a bit would be $0\to 0$ and $1\to 1$ with probability 0.8, and $1\to 2$ with probability 0.2. Effectively, Bob will apply the (classical) POVM $E=\{E_0=|0\rangle\langle 0|, E_1=0.8 |1\rangle\langle 1|, E_2=0.2 |1\rangle\langle 1|\}$. Additionally, Alice and Bob can share some large (in principle unbounded) amount of randomness: $\sigma_{AB}=\frac{1}{N}\sum_{\beta=1}^{N\to \infty}|\beta\rangle\langle\beta|_A\otimes |\beta\rangle\langle\beta|_B$, so that they can synchronize and possibly realise some convex sum of different $C_d$.
Now, in this paper, Frenkel and Weiner showed (See Theorem 3) that the convex hull of $Q_d$, exactly matches the convex hull of $C_d$. In other words, if unlimited shared randomness is available, any $Q_d$ can be simulated using a convex mixture of classical $d$-dimensional $k \times l$ matrices.
I am trying to understand the proof of Theorem 3. One direction is trivial, for the other direction, $\mathrm{Conv}(Q_d)\subseteq \mathrm{Conv}(C_d)$, the author uses a concept called the ‘supply-demand theorem of bipartite graphs’. What is this theorem, and how is it connected to the above result? Also, I am trying to have an intuition as to why the proof requires notions like mixed discriminant and tools from graph theory. Can't we find any alternative tools, e.g. tools from convex analysis, or matrix majorisation, to attack this problem?
PS, the primary motivation behind this communication setting (no entanglement but shared classical randomness) is, can a set of quantum states and a multi-outcome measurement generate something non-trivial compared to a strategy where everything is classical? In some sense, Theorem 3 is a generalisation of Holevo's theorem (also see Theorem 5 and the remark after that), which is a particular communication task where instead of generating the entire communication matrix, Bob has to optimise a particular function (Holevo quantity) mapping the matrix to a real number.
