Quantum Whigmaleerie I

We know that a classical information can be described by a 2D PDF, while a quantum information or quantum channel can be described by a 3D PDF.

If we transmit a classical information through a quantum channel, that means we need to convert the 2D information to a 3D information by applying a state \rho_x and then do a measurement (i.e. POVM) to project the 3D information back to 2D information. This process is almost certainly increases the randomness. The Holevo bound states that: H(X:Y) \leq S(\rho)-\sum_x p(x)S(\rho_x) . And we can compare with classical information: H(X:Y) \leq H(X), in general, the quantum channel capacity is less than classical one iff \rho_x is orthogonal and pure, which is actually degenerate the quantum channel to classical one (classical information is a special case of quantum information).

Then it become no sense to transmit a classical information  via a quantum channel. Of course, the most important reason of why we doing that is it increases the security, i.e. quantum key distribution. In my view, it looks like a temporary  solution based on the current technology: if we transmit a quantum information via a quantum channel, it has the same property I think.

Let’s move back to the idea that we want to transmit classical information through quantum channel. Assume the \rho_x is not a pure state, how can we get the maximum mutual information? The first thing come to my mind is to apply PCA to the 3D information. I’m not sure how to do it nor someone has already done it, but I think it probably is a good idea to start with.