Exercise 3

from Example Sheet 3

Exercise 3.

Show that for any two states \(\rho\) and \(\sigma\) and any unitary \(U\), \[ F(U \rho U^\dagger,U \sigma U^\dagger) = F(\rho,\sigma) \] by using the fact that for any positive semi-definite operator \(A\) and unitary \(U\), we have \(\sqrt{U A U^\dagger} = U \sqrt{A} U^\dagger\).

We have \[\begin{aligned} F(U\rho U^\dagger, U\sigma U^\dagger) &= \| \sqrt{U \rho U^\dagger}\sqrt{U \sigma U^\dagger}\|_1 \\ &= \| U \sqrt{\rho} U^\dagger U \sqrt{\sigma}U^\dagger\|_1\\ &= \| U \sqrt{\rho}\sqrt{\sigma}U^\dagger\|_1\\ &= \max_{V} | \operatorname{tr}[ V U \sqrt{\rho}\sqrt{\sigma}U^\dagger ]| \\ &= \max_{V} | \operatorname{tr}[ U^\dagger V U \sqrt{\rho}\sqrt{\sigma} ]| \\ &= \max_{UVU^\dagger} | \operatorname{tr}[ U^\dagger U V U^\dagger U \sqrt{\rho}\sqrt{\sigma} ]| \\ &= \max_{UVU^\dagger} | \operatorname{tr}[\sqrt{\rho}\sqrt{\sigma} ]| \\ &= \|\sqrt{\rho}\sqrt{\sigma}\|_1 \\ &= F(\rho,\sigma). \end{aligned}\] We use the characterization of the trace distance as a maximum over unitary operators \(V\), and along the way proved that the trace norm was invariant under unitary conjugation. We also used that maximizing over all \(V\) was the same as maximizing over all \(U V U^\dagger\), since \[ \{U V U^\dagger: V \text{ unitary}\} = \{ V : V \text{ unitary}\}. \]