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The different branches of the observer correspond to the different projections of the quantum computer in the |0>, |1> basis of the qubits. Suppose that this observer measures the state of a qubit in the |0>, |1> basis. Let's call this qubit a "spin" to avoid confusion with the qubits that are part of the observer.

Then what we can achieve is the following.

1) We start with the spin in state |0>, then we rotate it to

1/sqrt(2) [|0> + |1>]

2) The observer then does a measurement in the |0>, |1> basis, which causes a qubit (that was initiallized to |0>) of his memory to be entangled with the state of the spin. This is performed using the controlled NOT gate. Also another qubit of his memory that was initialized to |0> is flipped to |1>. That qubit detects that a measurement has taken place (but not the result of the measurement).

3) The observer then applies the controlled NOT gate again, reversing the measurement. Then he flips another qubit that was initialized to |0> to |1>, which records the fact that the memory qubit that registered the spin has been erased.

4) At this stage the spin is back in the state 1/sqrt(2) [|0> + |1>]. The observer can verify this by applying the inverse rotation that he appied to the spin at the start, rotating it back to the state |0>. A measurement of the spin by the observer (or some other observer) will yield zero with 100% probability.

Now, the fact that the observer knows that he measured the spin in the |0>, |1> basis when it was rotated to 1/sqrt(2) [|0> + |1>] means that in the CI interpretation, the spin's state should have collapsed to either |0> or |1>. Only one of the branches really exists. Then, applying the inverse rotation won't bring the spin back to the state |0>, instead it will be a mixed state of

1/sqrt(2) [|0> + |1>]

and

1/sqrt(2) [|0> - |1>]

Measuring the spin again in the |0>, |1> basis must thus yield a 50% probability of finding it to be |0>.

So, the CI interpretation makes a different prediction than the MWI. Moreover, since the spin can be measured by an external observer, the CI interpretation predicts non-unitary time evolution for an isolated system that can be verified by an external observer.