Suppose Alice’s message X has N possible values that are equally likely, so that H(X) = log N. She represents these as pure states of a quantum system with a Hilbert space dimension d. Bob makes a...

Suppose Alice’s message X has N possible values that are equally likely, so that H(X) = log N. She represents these as pure states of a quantum system with a Hilbert space dimension d. Bob makes a measurement on the system, obtaining the result Y, from which he constructs his estimate Xˆ .

(a) Use Holevo’s theorem (Eq. 19.110) to show that H(X : Y) ≤ log d.

(b) Use Fano’s inequality (Eq. 20.12) to derive a lower bound on PE when N = 4 and d = 2.

(c) How does your result compare with the basic decoding theorem of Section 4.1?