POVM - Post-measurement State

Post-measurement State

Consider the case where the ancilla is initially a pure state . We entangle the ancilla with the system, taking

and perform a projective measurement on the ancilla in the basis. The operators of the resulting POVM are given by

.

Since the are not required to be positive, there are an infinite number of solutions to this equation. This means that there are infinite different experimental apparatuses that give the same probabilities for the outcomes. Since the post-measurement state of the system

depends on the, in general it cannot be inferred from the POVM alone.

Another difference from the projective measurements is that a POVM is not repeatable. If is subjected to the same measurement, the new state is

which is equal to iff that is, if the POVM reduces to a projective measurement.

This gives rises to many interesting effects, amongst them the quantum anti-Zeno effect.