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

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?

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 $$|\langle\Delta|MV|\psi\rangle| \le \||\Delta\rangle\|$$.