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Experimental Test of Generalized Hardy's Paradox

Since the pillars of quantum theory were established, it was already noted that quantum physics may allow certain correlations defying any local realistic picture of nature, as first recognized by Einstein, Podolsky and Rosen. These quantum correlations, now termed quantum nonlocality and tested by violation of Bell's inequality that consists of statistical correlations fulfilling local realism, have found loophole-free experimental confirmation. A more striking way to demonstrate the conflict exists, and can be extended to the multipartite scenario. Here we report experimental confirmation of such a striking way, the multipartite generalized Hardy's paradoxes, in which no inequality is used and the conflict is stronger than that within just two parties. The paradoxes we are considering here belong to a general framework [S.-H. Jiang \emph{et al.}, Phys. Rev. Lett. 120, 050403 (2018)], including previously known multipartite extensions of Hardy's original paradox as special cases. The conflict shown here is stronger than in previous multipartite Hardy's paradox. Thus, the demonstration of Hardy-typed quantum nonlocality becomes sharper than ever.

Since the pillars of quantum theory were established, it was already noted that quantum physics may allow certain correlations defying any local realistic picture of nature, as first recognized by Einstein, Podolsky and Rosen. These quantum correlations, now termed quantum nonlocality and tested by violation of Bell's inequality that consists of statistical correlations fulfilling local realism, have found loophole-free experimental confirmation. A more striking way to demonstrate the conflict exists, and can be extended to the multipartite scenario.
Here we report experimental confirmation of such a striking way, the multipartite generalized Hardy's paradoxes, in which no inequality is used and the conflict is stronger than that within just two parties. The paradoxes we are considering here belong to a general framework [S. Introduction.-Quantum mechanics, according to Bell's theorem [1], violates the notion of local realism that is always assumed true in the so called "classical theories". The conflict was first raised by Einstein, Podolsky and Rosen (EPR) back to 1935 [2], then put down to physical ground by Bell in 1964, who showed it through quantum violation of an inequality bounded by the local hidden variable (LHV) theories [1]. Bell's inequality, however, uses statistics of joint measurements, while Hardy's paradox [3,4], a logical quantumversus-classical contradiction without the need of inequalities, provides a more striking conflict than that with inequalities. In particular, Hardy's paradox enables us to prove Bell's theorem under the same condition, proposed by EPR that one party's measurement outcome allows this party to predict the other party's measurement outcome, with certainty. Thus, Hardy's paradox, in any of its high-dimensional [5] or multipartite extensions [6], giving a simple way to show that quantum correlation cannot be explained with classical theories, is arguably considered as "the simplest version of Bell's theorem" and "one of the strangest and most beautiful gems yet to be found in the extraordinary soil of quantum mechanics" [7].
Hardy's original proof of nonlocality involved two particles, a positron and an electron in two overlapping Mach-Zehnder interferometers. The paths of particles are arranged such that the average probability of positron-electron annihilation does not equal one [8]. In fact, there are some remarkable merits in generalizing Hardy's paradox to the multipartite scenarios. First, differing from the GHZ paradox [9], Hardy's paradox considers joint correlations in the form of probabilities, in a way that when it transforms to a group of correlation functions, the Hardy-typed Bell's inequality [10], which is derived from the paradox, has not only fully correlated functions but also partially correlated ones -i.e., the latter involve only a subset of parties, instead of involving all of them as in the former -a vital property that makes the inequality capable of detecting more quantum states [11] (see also the results in [12] for the Mermin-Ardehali-Belinskii-Klyshko inequality [13][14][15]). Second, with the GHZ state prepared, the quantumclassical conflict is sharper in the three-and four-party scenarios than in the two-party scenario [6]. This feature increases the noise resistance in practical experiments, despite the difficulties of preparing multipartite entangled states. Third, only when n ≥ 3 is it possible to create the Hardy-typed paradox for mixed states [16], extending the study of quantum paradoxes well beyond pure states in theory itself.
So far, a number of experiments have been carried out to confirm the paradox in two-particle systems [17][18][19][20][21][22][23][24][25]. However, Hardy's paradox for multipartite systems have not been experimentally demonstrated, even for the three-qubit system. In this Letter, we perform the first experimental test of quantum nonlocality via a generalized Hardy's paradox proposed in [26]. This paradox belongs to a general framework for nparticle Hardy's paradoxes, including Hardy's original one for two-qubit and Cereceda's extension for arbitrary n-qubit as special cases. Theoretically, it is proved that for any n ≥ 3, the generalized paradox is always stronger than previously known ones. Furthermore, its corresponding Hardy-typed inequalities have a lower visibility with the GHZ state corrupted by white noise. The inequalities are tight in that they coincide with the facets of LHV polytopes [27].
Consider an n-qubit system, in which qubits are labeled with the index set I n = {1, 2, · · · , n}. For each qubit, e.g., the k-th, one can choose a projector from set {a k ,ā k , b k ,b k } to perform the measurement, where a and b denote two dif- (a). For simplicity, we employ subscripts representing subsets of I n , e.g.,ā α = k∈αā k , where α ⊆ I n is a subset of all the n qubits; we also employᾱ = I n /α. Moreover, we denote |α| as the size of the subset α -i.e., the number of elements in α -and successful probablity p(a 1 = 1, a 2 = 1, · · · ) as p(a 1 a 2 · · · ). For any given α and β satisfying 2 ≤ |α| ≤ n, 1 ≤ |β| ≤ |α| and |α| + |β| ≤ n + 1, in the regime of the LHV theory, the conditions In quantum theory, however, by choosing appropriate a and b for the GHZ state, p(a 1 a 2 · · · ) can be greater than zero. Hence the quantum-classical conflict appears. This is the generalized Hardy's paradox [26].
Diagram of generalized Hardy's paradox experiment is shown in right panel of figure 1 (a). Similar to other nonlocality tests, the GHZ state is distributed to n spatially separated parties, with each particle being then measured along some direction represented on the Bloch sphere. To obtain polarized-entangled photons, we adopt beamlike type-II compound β-barium borate (c-BBO) crystal used in [29], where a half-wave plate (HWP) is sandwiched between two 2-mm-thick, identically cut BBOs, as shown in figure 1 (b). Pumped by ultraviolet (UV) laser pulses, photon pairs, emitted as two seperate circular beams of Gaussian intensity distribution, are generated through spontaneous parametric down conversion (SPDC) [30] from the left BBO crystal. The polarization state can be written as |H o |V e , where H (V ) denotes the horizontal (vertical) polarization state of photons and subscripts o (e) indicate two spatial modes on which photons are ordinary (extraordinary) light. Passing the HWP, above two-photon state is converted to |V o |H e , which is then superposed with photon pair |H o |V e generated by the right BBO. By careful spatial and temporal compensations, polarized-entangled photons in the state |ψ + = (|H o |V e + |V o |H e )/ √ 2 are prepared. We can convert |ψ + to desired |φ + by inserting HWP in spatial mode e. In our experimental setup, pump pulses pass through two c-BBOs successively. In this way two pairs of photons are prepared in state (|H o1 |H e1 + |V o1 |V e1 )/ √ 2 ⊗ (|H o2 |H e2 + |V o2 |V e2 )/ √ 2 with non-vanishing probability, here we use subscript 1 and 2 to distinguish the two c-BBOs from which photons generate.
The two photons in modes e 1 and e 2 are then combined on a PBS. The PBS transmits H and reflects V , leading to a coincidence registration of a single photon at each output. In this way the two terms |H ⊗4 and |V ⊗4 are post-selected, the two entangled-photon pairs are therefore effectively projected into the four-photon GHZ state: To achieve good spatial and temporal overlap, a movable prism is used to guarantee the delay between two spatial modes are equal. Based on state (1), it is straightforward to generate 3-photon GHZ state projecting one of the photon (in this experiment the photon in spatial mode o 1 ) into |+ = (|H + |V )/ √ 2. Subsequently, each photon of the GHZ states is led to a measurement module, which contains a quarter-wave plate (QWP), a half-wave plate (HWP) and a PBS. By setting QWP and HWP at (0 • , 0 • ) and (45 • , 22.5 • −θ/4), measurement under basis |H , |V and (|H ± e iθ |V )/ √ 2 can be performed. After the PBS, transmitted and reflected modes correspond to the two orthogonal projection of measurement basis. All the 16 four-photon coincidence events can be simultaneously recorded by a coincidence counting system. To guarantee indistinguishability, each photon is spectrally filtered before entering detectors.
In this setup, the four-photon coincidence count rate is about 24.1 Hz. To verify entanglement of the 3-and 4-photon GHZ state, we measured the entanglement witness [31,32] where population P N = |H H| ⊗N + |V V | ⊗N , M k = [cos(kπ/N )σ x + sin(kπ/N )σ y ] ⊗N . Expectations of P N and M k can be obtained by measurement under basis |H , |V and (|H ± e ikπ/N |V )/ √ 2. Experimentally, we yield witness of 3-and 4-photon GHZ states as −0.417(9) and −0.398(8), both are negative by more than 40 standard deviations, which prove presence of genuine 3-and 4-partite entanglement. From the witnesses, we can directly calculate fidelities of GHZ states as 0.917(9) and 0.898 (8), which is defined as To demonstrate generalized Hardy's paradox, two typical cases from generalized [n; |α|, |β|]-scenario are selected. For 3-and 4-photon GHZ states, we both test the optimal case |α| = 2, |β| = 2 and standard case |α| = n, |β| = 1 to reveal there exist optimal conditions in the regime of generalized Hardy's paradox, in which the probabilities p(a In ) surpass standard multipartite Hardy's paradox [28]. To make probabilities p(b α aᾱ) and p(b β aβ) be zero, direction of the measurement |a = a 0 |H + a 1 e iθa |V and |b = b 0 |H + b 1 e iθ b |V should be determined by the relations (n − |α|)θ a + |α|θ b = (2m 1 + 1)π, (n − |β|)θ a + |β|θ b + |β|π = (2m 2 + 1)π, where a i , b i are all reals, and we take m 1 = m 2 = 0 for simplicity. All the QWPs are set at 45 • , angles of HWP using in experiment for scenario selected are listed in Table I. In experiment, we measured expectations of projection operators b α aᾱ,b β aβ, and N k=1 a k (which are equal to probabilities of successful projection). The experimental results are illustrates in Table II, showing that the coincidence probabilities obey what quantum mechanics predicts. As we can see in the table, the measurements N k=1 a k obtain results 1 of explicitly non-zero probabilities. For n = 3, p std (a 1 a 2 a 3 ) = 0.132 (16), p opt (a 1 a 2 a 3 ) = 0.259 (22); for n = 4, p std (a 1 a 2 a 3 a 4 ) = 0.093 (12), p opt (a 1 a 2 a 3 ) = 0.127 (12), where subscripts std and opt stand for standard and generalized optimal case, all the results are within the 1 standard deviation of theoretical prediction. Probabilities of with x > 0, y > 0, here we take x = y = 1; F can be calculated as: F (n; |α|, |β|; x, y) = min 0≤m≤n x m |α| + y n − m |β| .
Conclusion.-In summary, we give the first experimental test of the Hardy-typed quantum nonlocality in the multipartite scenarios. We utilize 3-and 4-photon GHZ states based on high-brightness beamlike c-BBO to experimentally perform the test. Our results confirm that the conflicts between quantum mechanics and LHV theories, revealed by a generalized Hardy's paradox, are stronger than those in the standard extensions of Hardy's paradox. The 3-and the 4-qubit generalized paradoxes, in particular, are stronger than the original 2-qubit paradox. The overall efficiency of our detection device is limited, therefore, we need to assume that the detection efficiency is independent of measurement settings, known as fair sampling assumption [34]. To achieve higher detection efficiency, one can employ superconducting-nanowire single-photon detectors (SNSPD) [35,36]. As a further justification for the Hardy-typed quantum nonlocality, Bell's inequalities derived from the paradoxes are shown to be violated in quantum theory.
This work was supported by the National Natural Science Foundation of China (Grant No. 11475089), the Chinese Academy of Sciences, and the National Fundamental Research Program.