### Question 3.

Could you please help me with an exercise given in the lectures (Notes 13, eq 1.18): show that there exist a set of unitary operators \(\{U_j\}_j\) and a probability distribution \(\{p_j\}_j\) such that \[ \rho_A \otimes \sigma_B = \sum\limits_j p_jU_j\rho_{AB}U_j^\dagger, \](1) where \(\sigma_B = I/d\) is a completely mixed state.

The exercise in (1.18) is a bit tricky— I wouldn’t know how to do it without having seen it before. The unitaries needed are described in Mark Wilde’s book. In Section 3.7.2 (p. 110), he defines qudit generalizations of the Pauli \(X\) and \(Z\) operators, which he calls the Heisenberg-Weyl operators. Then in Exercise 4.7.6 on p. 176 of the same book, he has an exercise to show that these operators “do the job” by averaging out to the completely mixed state (see equation (4.349) in the book). Note you have a doubly-indexed family of unitaries \(\{ X(i) Z(j) \}_{i,j}\), but that could always be reindexed to have only one index, like in the previous question. Actually, there is one more subtlety: in Exercise 4.7.6 the operators only act on one system; here you have two systems, but only want to map the \(B\) system to the completely mixed state. So you would use \(I\otimes X(i) Z(j)\) instead.