I was trying to find what does happen physically according to you. For the time being, you've only explained what does not happen ("measurement does not cause the collapse of the wave function",
"When you measure a particle, nothing happens to that particle as a result of the measurement.").

Reading your paper I thought something happened, somewhere in the transition from page 6 to page 7, because I understood that you were getting a well-defined value out of the measurement. The paper you linked now makes clear that in this model nothing happens at all, ever. Anything less than the wave function for the whole universe is just an illusion, there is no classical world.

It's not an accident that your description looked like the standard measurement scheme from von Neumann, as they are just reproducing it. They seem to think that when von Neumann described the measurement process as an interaction of two quantum systems he didn't notice that this interaction was beyond the classical concept of correlations, and this is why he needed a second stage to perform the measurement and collapse the state vector.

In the pre-measurement step, the 'target' (system to be measured, with states |u> and |d>) and the 'probe' (quantum detector, with states |+> and |->) are combined. This system will be a superposition of the states |u,+> and |d,->. The density matrix includes the diagonal terms |u,+><u,+| and |d,-><d,-| and also the diagonal terms |u,+><d,-| and |d,-><u,+|.

Von Neumann introduces a projection step which keeps only the diagonal terms. The resulting density matrix is no longer a pure state but a mixed state (for a closed system, pure states correspond to eigenvalues of the density matrix and then a diagonal matrix has to represent a classical mixture of the corresponding pure states and cannot be a superposition). After the measurement following von Neumann we have a statistical ensemble of the possible outcomes and choosing one of them doesn't pose any conceptual problem.

Cerf and Adami avoid the second step by taking the entangled system, ignoring the state of the target and looking only at the probe. Mathematically, one obtains a reduced density matrix by tracing out one subsystem and the result is a diagonal density matrix which contains only |+><+| and |-><-| without any interference terms. However, this is an improper mixed state and cannot be interpreted as before (because it's not the description of a full system but the result of looking at a subsystem).

Their reduced density matrix does not represent a defined (but unknown) state because the subsystem remains entangled with the rest of the system (target+probe). They say that the full system continues to evolve in a unitary way without any kind of collapse, but then how is any measurement done at all? Their description of the detector is an (improper) mixture of |+> and |->, at what point do we get one or the other as a result?

Note that Cerf and Adami consider their model completely unrelated to Zurek's. Decoherence and einselection are interesting, provide a physical mechanism to explain how the interference terms become (close to) zero in the "correct" basis in a (practically) irreversible way, and have some experimental support. But it doesn't completely solve the problem of measurement: how does the system settle on one particular outcome?