A single particle [the example is discussed in references 4, 2, 1]
at time t_0 is (preselected) in the state
|psi_0> = 3 ^ (-1/2) ( |a> + |b> + |c> )
and at a later time t_f is (postselected)
in the state |psi_f> = 3 ^ (-1/2) ( |a> + |b> - |c> )
where |a>, |b> and |c> correspond to the particle being found
in 3 boxes: A, B and C, respectively. (The N boxes case
is discussed in reference 3.)
At the intermediate time t_i, where t_0 < t_i < t_f,
a measurement is performed on the system.
The ABL rule [see reference 5] states that if a measurement
is performed, at time t_f, on this system, with the above
preselection and postselection of states, the probability
for an outcome of either a or b (eigenvalues corresponding
to find the particle in box A or in box B, respectively) is 100%.
That is to say, the intermediate measurement cannot project
the initial state |psi_0> onto the state 2 ^ (-1/2) ( |b> + |c> ) --
particle not found in A -- or onto the state 2 ^ (-1/2) ( |a> + |c> )
-- particle not found in B --. That's because both states
are othogonal to the final state |psi_f>. Both states are
then impossible.
As long as we keep the QM formalism and the ABL rule,
in each case any particles (which end up postselected)
are ones which could not have been in any box except
the one which was opened, be it A or B.
Possible solutions? There are some. In example....
1. QM formalism is right. There is no paradox. That's real.
2. QM formalism is right. That's not real. QM does not speak
of reality.
3. Counterfactuals. To make a claim about the elements
of reality of an individual system we have to consider the *physical*
situation involved in an individual run of the experiment. But here,
in each run, we have to make a *choice* to measure A or B.
If we choose A, all postselected particles had to be found
in box A. If we choose B, all postselected particles had to be found
in box B. But the property of being, with certainty, in any one
of those 2 boxes (depending on wich one is opened) cannot apply
to the *same* *individual* particle in *any* given run of the
experiment.
4. We cannot use the ABL rule here [see reference 6], because
of the counterfactuals.
[Humour? What humour?]
[1] David Z. Albert, Yakir Aharonov, Susan D'Amato,
Physical Review Letters, vol. 54, pages 5 - 7,
(1985)
[2] David Z. Albert, Yakir Aharonov, Susan D'Amato,
Physical Review Letters, vol. 56, p. 2457, (1986)
[3] Yakir Aharonov, Lev Vaidman
J. Phys, A-24, pages 2315 - 2328, (1991)
[4] Lev Vaidman
Foundations of Physics, 26, pages 895 - 906, (1996)
[5] Yakir Aharonov, P.G. Begmann, J.L. Lebowitz,
Physical Review, 134-B, pages 1410 - 1416, (1964)
[6] R. E. Kastner
Foundations of Physics, 29, pages 851 - 863, (1999)
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