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 \(n\).