![]() 3, so if we were to make the postselection now this even would not be selected this was our main proof of the quantum pigeonhole effect. This state is orthogonal to the postselected state, Eq. In this case the state collapses to | Ψ ′ 〉 = Π 1,2 s a m e | Ψ 〉. 2, we can find particles 1 and 2 in the same box. Indeed, for example starting with the initial state, Eq. Now it is actually possible to find 1 and 2, or 1 and 3, or both 1 and 2 and 1 and 3 in the same box. Indeed, suppose we first measure if particles 1 and 2 are in the same box, then particles 1 and 3, and then we make the postselection. But, if we were to try to measure two or all pairs in the same experiment, the measurements would disturb one another and we would not see the effect. We certified this by showing that whichever pair of particles we measured we always find them in different boxes. We know that when considering three noninteracting particles in two boxes, given the appropriate pre- and postselection, no two particles are in the same box. The requirement that the interaction should not be too strong (which can be arranged, for example, by spacing the parallel beams further away or making the arms shorter) is more subtle. Ending up at D 1 is our desired postselection. We are interested in what happens in the cases when all three electrons end up at D 1 and when the interaction between the electrons is not too strong. This is the reason why postselection is essential to see the effect. In the case of a preselected-only ensemble, what happens to the state after the measurement does not matter, but if we are interested in following this measurement with a subsequent measurement and look at the different pre- and postselected ensembles, how much the intermediate time measurement disturbed the state is essential. On the other hand, if we measure each particle separately, we disturb the state, collapsing it on either | L 〉 1 | R 〉 2 or | R 〉 1 | L 〉 2. The global measurement will tell us that the particles are in different boxes and will not disturb the state at all, because it is an eigenstate of the measured operator. Indeed, suppose two particles are in an arbitrary superposition α | L 〉 1 | R 〉 2 + β | R 〉 1 | L 〉 2. ![]() ![]() The third thing to notice is that the global measurement which only asks about correlations but no other detailed information is, in some sense, better than the detailed measurement as it delivers the information about correlations while minimizing the disturbance that it produces to the state. Yet, as we show here, for quantum particles the principle does not hold. It seems therefore to be an abstract and immutable truth, beyond any doubt. Indeed, although on one hand it relates to physical properties of objects––it deals, say, with actual pigeons and pigeonholes––it also encapsulates abstract mathematical notions that go to the core of what numbers and counting are so it underlies, implicitly or explicitly, virtually the whole of mathematics. But, the pigeonhole principle that is the subject of our paper seems far less likely to be challenged. It all started with challenging the idea that particles can have, at the same time, both a well-defined position and a well-defined momentum, and went on and on to similar paradoxical facts. Arguably the most important lesson of quantum mechanics is that we need to critically revisit our most basic assumptions about nature.
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