Honoring Epimenides of Crete \(±Δx\): From Quantum Paradoxes, through Weak Measurements, to the Nature of Time

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We now post-select only the cases in which a single photon reached detector D, which 

means that one of its paths was surely disrupted: 

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thereby entangling the two atoms into a full-blown EPR state: 

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EPR

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In other words, tests of Bell’s inequality performed on the two atoms will show the same 

violations observed in the EPR case, indicating that the spin value of each atom depends on 

the choice of spin direction measured on the other atom, no matter how distant. 

Unlike the ordinary EPR generation, where the two particles have interacted earlier, here the 

only common event lies in the particles’ future.

One might argue that the atoms are measured only after the photon’s interference, hence the 

entangling event still resides in the measurements’ past. However, all three events, namely, 

the photon’s interference and the two atoms’ measurements, can be performed in a space-

like separation, hence the entangling event may be seen as residing in the measurements’ 

either past or future. 

2.7 Nature Caught Contradicting Herself 

– The Quantum Liar Paradox 

A closer inspection of the abovementioned inverse EPR reveals something truly remarkable. 

Beyond the apparent time-reversal lies a paradox that in a way is even more acute than the 

well-known EPR or Schrodinger’s cat paradoxes. It stems not from a conflict between QM 

and classical physics or between relativity theory; rather, it seems to defy logic itself. 

The idea underlying the experiment is very simple: In order to prove nonlocality, one has to 

test for Bell’s inequality by repeatedly subjecting each pair of entangled particles to one out 

of three random measurements. Then, the overall statistics indicates that the result of each 

particle’s measurement was determined by the choice of the measurement performed on its 

counterpart. A paradox inevitably ensues when one of the three measurements amounts to 

the question “Are you nonlocally affected by the other particle?” Let us, then, recall the gist 

of Bell’s nonlocality proof

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for the ordinary EPR experiment. A series of EPR particles is 

created, thereby having identical polarizations. Now consider three spin directions, x, y, and 

z

. On each pair of particles, a measurement of one out of these directions should be 

performed, at random, on each particle. 

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Fig. 3. Entangling two atoms. 

Let many pairs be measured this way, such that all possible pairs of x, y, and z

measurements are performed. Then let the incidence of correlations and anti-correlations be 

counted. By quantum mechanics, all same-spin pairs will yield correlations, while all 

different-spin pairs will yield 50%-50% correlations and anti-correlation. Indeed, this is the 

result obtained by numerous experiments to this day. By Bell’s proof, no such result could 

have been pre-established in any local-realist way. Hence, the spin direction (up or down) of 

each particle is determined by the choice of spin angle (x, y, or z) measured on the other 

particle, no matter how distant. 

Let us apply this method to the abovementioned time-reversed EPR. Each Hardy atom’s 

position, namely, whether it resides in one box or the other, constitutes a spin measurement 

in the z directions (as it has been split according to its spin in this direction). To perform the 

z measurement, then, one has to simply open the two boxes and check where the atom is. To 

perform x and y spin measurements, one has to re-unite the two boxes under the inverse 

magnetic field, and then measure the atom’s spin in the desired direction. Having randomly 

performed all nine possible pairs of measurements on the pairs, many times, and using Bell’s 

theorem, one can prove that the two atoms affect one another non-locally, just as in the 

ordinary Bell’s test. A puzzling situation now emerges. In 44% of the cases (assuming 

random choice of measurement directions), one of the atoms will be subjected to a z

measurement – namely, checking in which box it resides. Suppose, then, that the first atom 

was found in the intersecting box. This seems to imply that no photon has 

ever crossed that path, since it is obstructed by the atom. Indeed, as the atom remains in the 

ground state, we know that it did not absorb any photon. But then, by Bell’s proof, the other 

atom is still affected nonlocally by the measurement of the first atom. But then again, if no 

photon has interacted with the first atom, the two atoms share no causal connection, in either 

past or future! 

The same puzzle appears when the atom is found in the non-intersecting box. In this case, we 

have a 100% certainty that the other atom is in the intersecting box, meaning, again, that no 

photon could have taken the other path. But here again, if we do not perform the which-box 

measurement (even though we are certain of its result) and subject the other atom to an x or 

y measurement, Bell-inequality violations will occur, indicating that the result was affected 

by the measurement performed on the first atom (Fig. 3). The situation boils down to: 

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