# Exercise 6

## from Example Sheet 4

### Exercise 6.

Alice encodes classical information into $$n$$ photons which she sends to Bob through a quantum channel. What is the maximum number of bits of information that Bob can infer from the output of the channel by doing measurements on it?

Hint: Use the Holevo bound.

Let $$X$$ denote Alice’s classical information, and $$Y$$ the random variable obtained by Bob from measurements on what he obtains from the quantum channel. The amount of bits of information Bob can infer about Alice’s information is exactly $$I(X:Y)$$. The Holevo bound gives that $I(X:Y) \leq \chi(\{p_x, \rho_x\})$ where $$\{p_x,\rho_x\}$$ is the ensemble used by Alice to encode her information. We don’t know how the encoding is done or if the channel is noisy, but we know only $$n$$ photons were used, which we assume are qubits (information encoded only in the polarization). Thus, the output state $$\rho = \sum_x p_x \rho_x$$ is $$2^n$$-dimensional, and its entropy is at most $$n$$. Since $$\chi(\{p_x, \rho_x\}) \leq S(\rho) \leq n$$, we obtain an upper bound of $$n$$.