Hi Eric, I’ve been getting myself a bit confused with the 2014 exam, Q3.iii; could you please tell me the best way to approach it?
This question is about the “completely dephasing map,” \[ \Lambda(\rho) = \sum_y | y \rangle\langle y | \rho | y \rangle\langle y |. \](1) In part (ii) of the question, you’re asked to prove it is a CPTP map. Then part (iii) asks you to find an expression for the von Neumann entropy of a state \(\sigma =\Lambda(\rho)\). We want to find the von Neumann entropy of a state, which we know is just the Shannon entropy of the eigenvalues of the state. So we just need to find the eigenvalues of \(\Lambda(\rho)\), using the expression (1). Try that, and ask again if you want more.
Hint: no extensive calculations are needed! If you don’t see it, it might help to write \(\Lambda(\rho)\) in a matrix representation.