# Alice sends random states in a channel, what Bob receives?

Suppose Alice prepares $\rho_x$ with probabilities $p_x$ and sends it to Bob. I would say this is the same thing as "Alice prepares $\rho = \sum_x p_x \rho_x$ and sends it to Bob", but Preskill's lecture and Nielsen & Chuang seems to disagree.

I will be more precise.

If I want to compress the n-fold tensor product of $\rho$ I can do it down to $n \ S(\rho)$ qubits ($S$ is the Von Neumann entropy), this is the Schumacher theorem as stated in Nielsen & Chuang, page 544.

If I want to compress the ensemble $\epsilon = \{p_x,\rho_x\}$ Schumacher-like, I can do it down to $n \ \chi(\epsilon)$ qubits, as stated in http://www.theory.caltech.edu/people/preskill/ph229/notes/chap5.pdf, page 31. (Maybe if Preskill gave a definition for "ensemble fidelity" the issue would be easier to tackle...)

Note that $n \ \chi(\epsilon)$ is ensemble-dependent and in general different from the Von-Neumann entropy. But this seems to suggest that different ways of looking at a density matrix allows different ways of encoding it: If I think I am sending a $\rho_x$ with probabilities $p_x$ I must use the $\chi$ and hence a certain number of qubits, if I send $\rho = \sum_x p_x \rho_x$ I must use the entropy, and hence another number of qubits...

This bother me because I was intimately convinced that the ensemble description is not physical: Classical and quantum probabilities in density matrices,

The key idea is that Alice prepares a sequence made out of $N$ states $\rho_j \in \{\rho_x\}$, and we want to know the sequence $\{\rho_1,\rho_2,\dots,\rho_N\}$. You can compute the average $\rho=\sum p_x\rho_x$, but this only gives (asymptotically) the average of the sequence i.e. $$\rho\approx\left<\rho_j\right>_j,$$ which is not the same as the information contained in the sequence itself.
• @giuliobullsaver Alice & Bob are communicating so both know the characteristics of the channel and have agreed on an encoding scheme. Therefore Bob knows the set of possible states $\{\rho_x\}$ & associated probabilities $p_x$. W/o compression Alice sends a string of states $S=\{\rho_1,\rho_2,\dots\}$ & that's what Bob gets. W/compression Alice takes $S$ compresses and sends a smaller string $S'$. Once received, Bob tries to decompress $S'$ to get $S$ & a necessary condition for success is that there is enough information available to encode the full string (i.e. at least $n\chi$ qbits). May 30, 2014 at 18:48