How can I prove inequality from 4.66 to 4.67 in Nielson and Chuang's book?

Question

I am reading chapter 4 of Nielson and Chuang's QCQI book.

I cannot prove the inequality from (4.66) to (4.67) in page 195.

That inequality is the following:

$$ |\langle\psi|U^\dagger M|\Delta\rangle|+|\langle\Delta|MV|\psi\rangle| \leq \|{|\Delta\rangle}\| + \| |\Delta\rangle \|$$

$U,V$ are arbitrary unitary operators, $|\psi\rangle$ is an arbitrary state, $M$ is an POVM element, and $|\Delta\rangle = (U-V)|\psi\rangle$.

How can I prove this inequality?

Thanks.

Answer 1

From Cauchy-Schwarz inequality $|\langle u|v\rangle| \le \|u\|\|v\|$, we have

$$ |\langle\psi|U^\dagger M|\Delta\rangle| \le \|MU|\psi\rangle\|\||\Delta\rangle\|. $$

But $\|MU|\psi\rangle\| \le 1$, because $U$ is unitary and $M$ a POVM element. Therefore,

$$ |\langle\psi|U^\dagger M|\Delta\rangle| \le \||\Delta\rangle\|. $$

Similar reasoning shows that $|\langle\Delta|MV|\psi\rangle| \le \||\Delta\rangle\|$.