Problem 8.5 For a quantum system with basis { \k)), consider all unitary operators of the form U= ax lk)(k1, (8.80) k wh

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Problem 8 5 For A Quantum System With Basis K Consider All Unitary Operators Of The Form U Ax Lk K1 8 80 K Wh 1 (56 KiB) Viewed 35 times

Problem 8.5 For a quantum system with basis { \k)), consider all unitary operators of the form U= ax lk)(k1, (8.80) k where ak = +1. These operators flip some of the phases of the basis states \k) and leave the rest unchanged. There are 2 such phase-flipping operators, where d = dim H. We consider a physical process D that consists of choosing one of these unitary operators at random (with equal probability 2-d) and applying it to an initial state of the system. This process can be viewed as a map on density operators: P→ D(P), (8.81) where Dip) is the final density operator, averaged over the random choice of U. Write the density operator as a density matrix with respect to the basis { \k). Show that the effect of D is to wipe out the coherences of this matrix while leaving the populations unchanged. In other words, random phase-flipping produces complete decoherence.