Ex4 21

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.21}$

a) Most of the time the effect of performing two measurements, one right after the other, cannot be achieved by a single measurement. Find a sequence of two measurements whose effect cannot be achieved by a single measurement, and explain why this property is generally true for most pairs of measurements.

b) Describe a sequence of two distinct nontrivial measurements that can be achieved by a single measurement.

c) For each of the measurements specified by the operators $A$, $B$, $C$, and $M$ from Examples 4.3.3, 4.3.4, 4.3.5, 4.3.6, say whether the measurement can be achieved as a sequence of single-qubit measurements.

d) How does performing the sequence of measurements $Z\otimes I$ followed by $I\otimes Z$ compare with performing the single measurement $Z \otimes Z$?