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

Kazuhiro Yokota^1,2,3, Takashi Yamamoto^1,2, Masato Koashi^1,2 and Nobuyuki Imoto^1,2

¹ Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
² CREST Photonic Quantum Information Project, 4-1-8 Honmachi, Kawaguchi, Saitama 331-0012, Japan

³ Author to whom any correspondence should be addressed.

E-mail: yokota@qi.mp.es.osaka-u.ac.jp

Received 10 December 2008
Published 4 March 2009

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 NO₂ and thus has no disturbance on MZ1. The path lengths 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

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

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, , 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 | . 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 | , and is found to be given by the real part of the following expression:

which is called the weak value of . 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 , and the coincidence detection retrodicts the state leaving the regions to be | = (|NO₁–|O₁)(|NO₂–|O₂)/2 [8]. Then the weak values can be calculated to be

The first equation implies that equation (2) holds. We also see that equation (1) holds since, for instance, |O₁O₁|_w = |O₁, O₂O₁, O₂|_w + |O₁, NO₂O₁, NO₂|_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, |NO₁, NO₂NO₁, NO₂|_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.

In this paper, we report an experimental demonstration of weak measurements applied to Hardy's experiment. Since the observables to be measured are of path correlations between the two photons, we need to implement a joint weak measurement that can extract such correlations. Although a number of schemes for joint weak measurement have been proposed [18, 19], we propose here the most intuitive approach using meter qubits initially prepared in an entangled state. We conducted measurements with varied strength and confirmed that it works properly when each photon is injected to a fixed arm. We also measured the visibility of the interferometers to verify that the disturbance vanishes in the limit of weak measurement. Then we conducted Hardy's experiments and observed that, as the measurement became weaker, three readouts moved toward conditions (1) and (2) while another readout went negative.

Let us first introduce our scheme of joint weak measurements for two qubits in a general setting. Let {| 0, | 1} be the standard basis of a qubit. For weak measurement of the observable |00| of a qubit, one can prepare another qubit in a suitable state and apply a controlled-NOT (C-NOT) gate between the two qubits [14]. Here, we want to carry out weak measurements of observables |kl_s1s2kl|(k,l = 0, 1) for the two signal qubits, s₁ and s₂, and we may use two meter qubits, m₁ and m₂, and apply two C-NOT gates in parallel. The signal (meter) qubit corresponds to the control (target) qubit and the meter qubit is flipped when the signal qubit is in |1. Note that in our implementation of Hardy's experiment, the photon path corresponds to the signal qubit and its polarization to the meter qubit. We found that the desired weak measurements are achieved by a simple choice of the initial state of the meter qubits,

where δ² + 3ε² = 1 and δ≥ε≥0. When the signal qubits are in , the application of the parallel C-NOT operation U_sm results in

The meter qubits will then be measured in the basis {| kl_m1m2} to produce an outcome (k, l). If the signal qubits are in state |ij_s1s2 initially, the probability P_m(k, l) of outcome (k, l) becomes

We see that the outcome (i, j), corresponding to the state of the signal, has a larger probability than the others. The contrast δ²–ε² will be regarded as the measurement strength. When δ²–ε² = 1(δ = 1, ε = 0), this scheme gives a projection measurement (strong measurement) on the signal qubits. Decreasing δ²–ε² makes the measurement weaker, and at δ²–ε² = 0(δ = ε = 1/2), the operation of U_sm introduces no disturbance on the signal qubits (appendix A). Let us introduce a normalized readout R(k, l) by

Then, regardless of the value of δ²–ε² > 0, we have R(k,l) = 1 for (k,l) = (i,j) and R(k,l) = 0 otherwise. When the initial state of the signal qubits is , the normalized readout becomes R(k,l) = |c_{k, l}|², which coincides with the expectation value | kl_s1s2kl|. Of course, for the estimation of R(k, l) and hence of P_m(k, l), we need to repeat the preparation of the initial state | ψ_s and the measurement many times.

Now in a situation like Hardy's experiment, we are interested in the readout of the meter on the condition that the signal qubits are finally measured to be in the state | _s. From the definition (3), we have

The conditional probability of the outcome (k, l) is then given by

with . The normalized readout is found to be

We see that the normalized readout in the limit of no disturbance coincides with the real part of the weak value.

In our experiment, the signal qubit corresponds to the paths taken by the photon as | 0 = | NO and | 1 = | O, whereas the meter qubit is assigned to the polarization of the same photon as | 0 = | +  and | 1 = | –, where and H/V is the horizontal/vertical polarization. The interferometers in figure 1(a) are constructed so that there is no polarization dependence, except for the shaded region where wave plates are placed to realize U_sm. The polarization dependence of the coincidence events is then analyzed to determine P_m(k,l|).

The detail of the setup is shown in figure 1(b). We generate the photon pairs in a nonmaximally entangled state from PDC, where and η and are properly adjusted to be real numbers. The coefficients η and , which can be tuned by HWP3, are determined by the ratio between the coincidence counts of |HH and |VV. The state is transformed into |ξ_m by rotating the polarization of two photons by the half-wave plates HWP1 and HWP2. The measurement strength is simply calculated by the measured η and and the angle of HWP1 and HWP2. After the photons pass through BS1, BS2 and BS3, the two-photon state is represented as |ψ_s|ξ_m. The C-NOT operations between signals and meters are simply performed by flipping the polarization as |±→| via HWP4 and HWP5 in the overlapping arms, O₁ and O₂. The polarization basis (| + , | –) for the meter measurement is selected by HWP and polarizing beam splitters (PBS) just before detectors C₁ and C₂. The coincidence counts, typically a few thousands, are accumulated over 5–12 min per basis to determine each R(k,l|). All error bars assume Poisson statistics of the counts.

First we show the observed trade-off between the measurement strength and the visibilities of interference at MZ1 and MZ2. The measurements were done as follows: we blocked the arm O₂(O₁) and recorded the coincidence detection at C₁ and C₂ by changing the path length via piezo stage P₁ (P₂). In order to measure the polarization-independent visibilities, we measured the interference fringes in the states | + + , | + –, |– +  and |–– independently and calculated the average visibility. The visibilities for various measurement strengths are shown in figure 2. We can clearly see that weakening the measurement makes the visibility larger (appendix B).

We also show that our measurement gives proper values of the readouts R(k,l|) when the photons travel the fixed arms |O₁, NO₂, |NO₁, O₂ and |NO₁, NO₂. We blocked the arms of no interest and determined R(k,l|) from the coincidence detection on the basis {| + + , | + –, |– + , |––}. The experimental results are plotted in figure 3. In any measurement strength, the corresponding readout for the signal state is close to 1 and the other readouts are close to 0. This clearly shows that the apparatus properly measures the trajectories of the photons. The systematic deviation from the expected values at small values of δ²–ε² is believed to be the result of very tiny misalignments of the axes of the wave plates. Such an enhancement is a feature of weak measurement and can be used for super-sensitive measurements as shown in [16].

Figure 3. Experimental results when the initial states are in (a) |O1, NO2, (b) |NO1, O2 and (c) |NO1, NO2.

With this meter, we performed measurements of the trajectories of the photons in Hardy's thought experiment. The observed readouts are shown in figure 4. Let us see the results around 0.3 measurement strength. The visibility above 0.9 shown in figure 2 suggests that the measurement is fairly weak, and the systematic deviation in figure 3 is small at this strength. For each of the photons, the readouts suggest a high probability of taking the overlapped arm: R(O₁|)≡R(O₁, NO₂|) + R(O₁, O₂|) = 0.72±0.09 and R(O₂|) = 0.53±0.09. Nonetheless, the readout for the joint probability is around zero, namely, R(O₁, O₂|) = –0.18±0.05. We also see that R(NO₁, NO₂|) = –0.43±0.04 takes a large negative value, which cannot be interpreted as a probability. Although the statistical and systematic errors are large around zero measurement strength, we see that R(O₁, NO₂|), R(NO₁, O₂|), R(O₁, O₂|) and R(NO₁, NO₂|) approach the expected weak values in equation (4) when the measurement strength goes to 0.

Let us emphasize here that our experimental results by themselves can elucidate the paradoxical nature of Hardy's experiment, without any reference to the theory explaining how our measurement works. From the results of figure 3, we see that our measurement apparatus should properly give the probabilities for the trajectories of the photons, if such quantities ever exist. Then, in Hardy's set-up, the same measurement presents a contradictory statement in figure 4 that the probability of finding a photon in arm O₁ approaches 1 and so does the probability for arm O₂, whereas the joint probability of photons being in both of the arms stays at about 0. Moreover, the readout points to a negative value for the joint probability for arms NO₁ and NO₂. Unlike Hardy's original argument, our demonstration reveals the paradox by observation rather than inference.

We have experimentally demonstrated a joint weak measurement with an entangled photon pair and directly observed paradoxical results in Hardy's thought experiment. Our demonstration clearly reproduces Hardy's paradox in an experimentally accessible manner. Weak measurements have attracted attention due to the possibility of achieving small-backaction and high-sensitivity measurements by simple optical setups. We believe that the demonstrated joint weak measurement is useful not only for exploiting fundamental quantum physics, but also for various applications such as quantum metrology and quantum information technology.

Note added. During the preparation of this manuscript, we came to know of a work by Lundeen and Steinberg [20], which has demonstrated similar results by using, interestingly, different methods for joint weak measurement and for constructing Hardy's interferometer.

Acknowledgments

This work was partially supported by the JSPS (Grant-in-Aid for Scientific Research (C) number 20540389) and by the MEXT (Grant-in-Aid for Scientific Research on Innovative Areas number 20104003, Global COE Program and Young Scientists (B) number 20740232).

Appendix A. 

The positive operator valued measure (POVM) for the measurement on the signal qubits is given by

The same POVM can be realized also by a nonentangled initial state of the meter qubits, δ²|00_m1m200| + ε²(|01_m1m201| + |10_m1m210| + |11_m1m211|). This separable state of the meter qubits, however, fails to derive a joint weak value, because the signal qubits are utterly disturbed even if δ²–ε² = 0.

Appendix B. 

A general relation between the visibility V and the measurement strength was discussed in [21]. It was shown that V² + K²≤1 holds, where K≡1–2p_err stands for the measurement strength defined in terms of the error probability p_err in the which-path measurement. In our case, the initial marginal state of the meter qubit m₁ is a mixed state, which is, from equation (5), calculated to be a mixture of state with probability p₀≡δ² + ε² and with p₁≡2ε². The state | _0m1 leads to a measurement strength K₀ = (δ²–ε²)/(δ² + ε²) and visibility , whereas | _1m1 leads to K₁ = 0 and V₁ = 1. Hence, on average, the measurement strength becomes K_ave≡p₀K₀ + p₁K₁ = δ²–ε² and V_ave≡p₀V₀ + p₁V₁ = 2ε(δ + ε), which is shown as the solid curve in figure 3. Due to the mixture, the averaged values K_ave and V_ave do not saturate the general bound V_ave² + K_ave²≤1.

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