Experimental test of the irreducible four-qubit Greenberger-Horne-Zeilinger paradox

The paradox of Greenberger-Horne-Zeilinger (GHZ) disproves directly the concept of EPR elements of reality, based on the EPR correlations, in an all-versus-nothing way. A three-qubit experimental demonstration of the GHZ paradox was achieved nearly 20 years ago, followed by demonstrations for more qubits. Still, the GHZ contradictions underlying the tests can be reduced to a three-qubit one. We show an irreducible four-qubit GHZ paradox, and report its experimental demonstration. The bound of a three-setting-per-party Bell-GHZ inequality is violated by $7\sigma$. The fidelity of the GHZ state was around $81\%$, and an entanglement witness reveals a violation of the separability threshold by $19\sigma$.

Figure 1

Examples of GHZ paradoxes. $X$ and $Y$ are possible local Pauli measurements $\widehat{X}$ and $\widehat{Y}$ on qubits in a state leading to a GHZ contradiction, while $x$ and $y$ are corresponding elements of reality. The symbols in each horizontal line are as follows: In (a) they form left to right, for party 1, 2, and 3, and in (b) for party 1, 2, 3, and 4. The predictions in red [last line of (a) and of (b)] show an algebraic or logical contradiction. In both cases the quantum prediction is the property of the considered state, while the local realistic prediction is a direct consequence of the first three equations (which are consistent with the corresponding quantum predictions). We have, in (a), an irreducible GHZ paradox for a three-qubit GHZ state [31, 32]. In (b), we have a reducible GHZ paradox for a four-qubit GHZ state [8, 9, 29]. As in the case of party 1, the observable in each condition is the same, and the paradox can be divided into two separate parts by the vertical line; the observable and element of reality for the first party does not contribute to the contradiction. This means that the paradox can be reproduced with a biseparable state.

Figure 2

Experimental setup. UV laser pulses pump two beta-barium-borate (BBO) crystals. The instances in which the top two pairs of entangled photons emerge, each from a different source, are used to produce the required four-photon interference (this is signaled by detections in detectors D1 and D3). The UV laser which we use has a central wavelength of 394 nm and pulse duration of 120 fs, with a repetition rate of 76 MHz. The average pump power is 450 mW. Fine adjustments of the delays between paths 2 and 4 are made by translation stages $\mathrm{\Delta }d$. We insert a compensator (Comp.) in front of the polarizing beam splitter (PBS) to counter the additional phase shifts of the PBS. To make the photons from the two sources indistinguishable at detectors D2 and D4 behind the PBS, all photons are spectrally filtered (bandwidth 3.2 nm), in accordance with Refs. [39, 40]. The detection is made by fiber-coupled silicon single-photon detectors, whose total collection and detection efficiency is about 14%. The four-photon coincidence events are registered by a programmable multichannel coincidence unit. Quarter-wave plates (QWP) and polarizers are used to set the basis for the measurements of polarizations of the photons.

Figure 3

Measurement of witness and fidelity of the four-photon GHZ state. (a) The results of Pauli measurement ${\widehat{Z}}^{\otimes 4}$ are listed. From left to right: ${\left|H\right.〉}_1{\left|H\right.〉}_2{\left|H\right.〉}_3{\left|H\right.〉}_4,{\left|H\right.〉}_1{\left|H\right.〉}_2{\left|H\right.〉}_3{\left|V\right.〉}_4$, etc. (b) Expectation values of the other four observables are listed. The error bars denote standard deviation, deduced from propagated Poissonian counting statistics of the raw detection events.

Figure 4

Experimental estimates of probability distributions observed in the measurement of (a) ${\widehat{X}}^{\mathrm{I}}$, (b) ${\widehat{X}}^{\mathrm{II}}$, (c) ${\widehat{X}}^{\mathrm{III}}$, (d) ${\widehat{X}}^{\mathrm{IV}}$, (e) ${\widehat{X}}^{\mathrm{V}}$, (f) ${\widehat{X}}^{\mathrm{VI}}$. In (h), we show the ideal quantum prediction for measurement arrangement (f), while (g) represents the $100\%$ flipped situation which must arise in the case of local realism, if the first five situations (a)–(e) follow an ideal quantum prediction (recall the $1=-1$ contradiction). In the figure, $\pm$ stands for corresponding one-qubit eigenstates with eigenvalues $\pm 1$. The interferometric visibilities in situations (a)–(e) are $0.733\pm 0.039, 0.730\pm 0.043, 0.704\pm 0.042, 0.711\pm 0.041$, and $0.727\pm 0.043$, respectively. The experimental results in (f) are in agreement with the quantum mechanics predictions (h) while in a strong conflict with local realism (g), and this is with a visibility of $0.733\pm 0.042$ or the interference in the situation (f). Visibility is defined as $(N_{max}-N_{min})/(N_{max}+N_{min})$, where $N_{max}$ is the sum of eight “tall” probabilities, and $N_{min}$ is the sum of the other eight “short” probabilities. The integration time of each fourfold coincidence is 3 min.