Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair

We implemented a joint weak measurement of the trajectories of two photons in a photonic version of Hardy’s experiment. The joint weak measurement has been performed via an entangled meter state in polarization degrees of freedom of the two photons. Unlike Hardy’s original argument in which the contradiction is inferred by retrodiction, our experiment reveals its paradoxical nature as preposterous values actually read out from the meter. Such a direct observation of a paradox gives us new insights into the spooky action of quantum mechanics. Although it is natural to ask what the value of a physical quantity is in the middle of a time evolution, it is difficult to answer such a question in quantum mechanics, especially when post-selection is involved. Hardy’s thought experiment [1] is a typical example in which we encounter such a difficulty. Figure 1(a) shows a photonic version of the experiment, which was recently demonstrated by Irvine et al [2]. The scheme consists of two Mach–Zehnder (MZ) interferometers MZ1 and MZ2 with their inner arms (O1, O2) overlapping each other at the 50 : 50 beam splitter BS3. If photons 1 and 2 simultaneously arrive at BS3, due to a two-photon interference effect, they always emerge at the same port. This corresponds to the positron–electron annihilation in the original thought experiment [1]. The path lengths of MZ1 are adjusted so that photon 1 should never appear at C1 by destructive interference, when photon 2 passes the outer arm N O2 and thus has no disturbance on MZ1. The path lengths 3 Author to whom any correspondence should be addressed. New Journal of Physics 11 (2009) 033011 1367-2630/09/033011+09$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 of MZ2 are adjusted similarly. Then, a coincidence detection at C1 and C2 gives a paradoxical statement on which paths the detected photons have taken. The detection at C1 (C2) implies that MZ1 (MZ2) has been disturbed by photon 2 (1) traveling along O2 (O1). We may thus infer that the conditional probabilities satisfy P(O1|C1C2)= P(O2|C1C2)= 1. (1) On the other hand, if both photons had taken the inner arms, the coincidence detection would have never happened due to the two-photon interference. Hence, we infer that P(O1O2|C1C2)= 0. (2) The two inferred statements are apparently contradictory to each other, which is the well-known Hardy’s paradox. One may argue that we should abandon the attempt to address the question itself on the grounds that the trajectory of photons cannot be measured without utterly changing the time evolution. But this reasoning is not necessarily true if we are allowed to repeat the same experiment many times. Aharonov et al have proposed weak measurement [3, 4], in which a measurement apparatus (meter) interacts with the system to be measured so weakly that the state of the system is not significantly disturbed. The readout of the meter from a single run of experiment may be subtle and noisy, but by taking the average over many runs we can correctly estimate the expectation value of the measured observable, 〈ψ | Â|ψ〉, when the initial state of the measured system is |ψ〉. In this set-up, we may ask what is the averaged readout over the runs in which the system is finally found to be in a state |φ〉. In the limit of no disturbance, this gives an operational way of defining what the value of  is in the middle of a time evolution from |ψ〉 to |φ〉, and is found to be given by the real part of the following expression: Âw ≡ 〈φ| Â|ψ〉/〈φ|ψ〉, (3) which is called the weak value of Â. So far, related interesting features have been discussed [5]–[11] and experimental observations of weak values have been reported [12]–[16]. Suppose that weak measurements of trajectories are applied to Hardy’s experiment at the shaded regions in figure 1(a). The state of the photons entering these regions is |ψ〉 = (|N O1〉|O2〉+ |O1〉|N O2〉+ |N O1〉|N O2〉)/ √ 3, and the coincidence detection retrodicts the state leaving the regions to be |φ〉 = (|N O1〉− |O1〉)(|N O2〉− |O2〉)/2 [8]. Then the weak values can be calculated to be |O1, O2〉〈O1, O2|w = 0, |N O1, N O2〉〈N O1, N O2|w =−1, |O1, N O2〉〈O1, N O2|w = 1, |N O1, O2〉〈N O1, O2|w = 1. (4) The first equation implies that equation (2) holds. We also see that equation (1) holds since, for instance, |O1〉〈O1|w = |O1, O2〉〈O1, O2|w + |O1, N O2〉〈O1, N O2|w = 1. Hence, the readout of the meter is indeed consistent with both of the naively inferred conditions (1) and (2). The reason why these two contradictory conditions are satisfied at the same time can now be ascribed to the appearance of a negative value, |N O1, N O2〉〈N O1, N O2|w =−1. It implies that the average readout over post-selected events falls on a value that never appears if no post-selection is involved. New Journal of Physics 11 (2009) 033011 (http://www.njp.org/)