Question 2 (15 marks, learning outcome O1) Let A be a quantum system of dimension d, let B be a qubit, and let {U, }g=0' be a set of 2x 2 unitary matrices. 2 (a) Show that the matrix U = Lol Inihn| U, is unitary (5 marks). (b) Suppose that all the matrices {U, }g'zol are proportional up to a global phase, that is, there exist complex numbers A;,...,A, ; suchthat U = A, U,, U, = A, U, ..., U, ; = Aq. 1 Uy. Show that in this case the matrix U is a product gate (5 marks). (c) Suppose that d =2, U, =1 and U; = Z, with Z := [0ihOl - | 1ih1l. Show that in this case the matrix U is not a product gate