I used to believe that the wavefunction collapse came from the interaction of the system we want to measure {S} with the measurement apparatus {M} : {S} undergoing a non unitary transformation, but {S+M} undergoing a perfectly unitary transformation, given by the Schrödinger equation but with too many degrees of freedom to be calculated.
In that picture there is no unitarity problem for the measurement since the whole system undergoes a unitary transformation.
But that picture must be wrong in some way: let's say we want to measure a spin along $z$, and that the initial state of the measurement apparatus is $ \langle I |_{M} $ : we must find a unitary transformation $U$ for {S+M} such that :
-Spin eigenstates remain unchanged under $U$ (up to a phase): $$ \langle \uparrow |_{S} \langle I |_{M} \xrightarrow{U} \langle \uparrow |_{S} \langle F_{1} |_{M}$$ $$ \langle \downarrow |_{S} \langle I |_{M} \xrightarrow{U} \langle \downarrow |_{S} \langle F_{2} |_{M}$$
($\langle F_{i}|$ being any measurement apparatus final state)
-A superposition can be projected, on $\langle \uparrow |$ or $\langle \downarrow |$ depending on $\langle I |$ and/or $U$: $$ \frac{1}{\sqrt2}(\langle \uparrow |_{S}+\langle \downarrow |_{S}) \langle I |_{M} \xrightarrow{U} \langle \uparrow |_{S} \langle F_{3} |_{M}$$
But this is not possible : $$ (\langle \uparrow |+\langle \downarrow |) \langle I | = \langle \uparrow |\langle I |+\langle \downarrow | \langle I | \xrightarrow{U} \langle \uparrow | \langle F_{1} | + \langle \downarrow | \langle F_{2} |$$
$$ \langle \uparrow | \langle F_{1} | + \langle \downarrow | \langle F_{2} | = \langle \uparrow | \langle F_{3} |$$
except if $\langle \downarrow | \langle F_{2} |=0$, which can't be since $U$ is unitary.
Does it mean that we expect the measurement apparatus initial state to be correlated to the measured system initial state, or even {S+M} to be in an initial non factorizable state ? In any measurement experiment this is true since we prepare the system in some KNOWN state. I cannot find any other explanation (maybe many world interpretation).