Consider the following experiment with pairs of down-conversion photons: the idler-photon is detected by a detector I, while the signal-photon enters an apparatus that splits it into three copies that impinge on the same detector S - see picture, and see explanation on the bottom of the question. So the wave-function before detection may be written as

(1) Ψ = -(i/2){ψ1 - ψ2 + sqrt(2) ψ3}.

As the wave-function indicates, in some trials of the experiment the wave-packet ψ1 produces a recording in the detector, in other trials ψ2, and in other trials ψ3, the probabilities being P1, P2 and P3, respectively, with P1 = 2P2 = 2P3.

Consider now a trial in which ψ3 produces a detection. The question is, the detector S, which is ideal, doesn't see the wave-packets ψ1 and ψ2? It is known from other experiments (Vaidman and Elitzur "Interaction-free measurements") that the detector S stops the wave-packets ψ1 and ψ2 , therefore it sees them. Then, why does it not report them?

If the initial wave-packet of the signal-photon is not split as described here, the ideal detector always reports it. Then, why does it not report, in a particular trial, all three wave-packets?

Please don't answer that the wave-function contains a single particle, not three - this answer is trivial. Also, the "many worlds" idea that would seem appealing here, is removed by the fact that, as I said, the non-reported wave-packets are stopped by the detector.

DESCRIPTION OF THE APPARATUS. The beam-splitters are 50-50%, the mirrors m are fixed, and the mirrors M1 and M2 are rotating. After the wave-packet ψ1 is reflected by the mirror M1, this mirror is rotated for not impeding the wave-packets ψ2 and ψ3 to pass. Similarly, after the wave-packet ψ2 is reflected by the mirror M2, this mirror is rotated for not impeding the wave-packet ψ3 to pass. The detectors are considered ideal.

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