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 $\{U V U^\dagger: V \text{ unitary}\} = \{ V : V \text{ unitary}\}.$