# Question 5

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.