Exploring the Limits of Quantum Nonlocality with Entangled Photons

Nonlocality is arguably among the most counterintuitive phenomena predicted by quantum theory. In recent years, the development of an abstract theory of nonlocality has brought a much deeper understanding of the subject, revealing a rich and complex phenomenon. In the current work, we present a systematic experimental exploration of the limits of quantum nonlocality. Using a versatile and high-fidelity source of pairs of polarization-entangled photons, we explore the boundary of quantum correlations, demonstrate the counterintuitive effect of more nonlocality with less entanglement, present the most nonlocal correlations ever reported, and achieve quantum correlations requiring the use of complex qubits. All of our results are in remarkable agreement with quantum predictions, and thus represent a thorough test of quantum theory. Pursuing such an approach is nevertheless highly desirable, as any deviation may provide evidence of new physics beyond the quantum model.

Popular Summary

The concept of Bell locality appears naturally in classical physics, but it does not allow distant observers to share correlations as strong as those predicted by quantum mechanics. Experimental violations of Bell inequalities have excluded locally causal descriptions and therefore result in quantum nonlocality. However, recent years have heralded the development of a generalized theory of nonlocality including extensions to quantum mechanics (i.e., postquantum theories) constrained only by no-superluminal signaling. Far beyond previous Bell inequality tests, whose sole goal was to exclude locally causal alternatives to quantum mechanics, here we use an ultrahigh-quality source of entangled photons to test quantum nonlocality, including many facets that previously were out of experimental reach.

We report a detailed scan of the boundary between quantum correlations and those from some postquantum extension. In particular, we produce pairs of polarization-entangled photons, and we examine the correlations (of the polarization) between the two 710-nanometer photons. With our source of photons, we achieve the most nonlocal correlations ever reported, and we use these correlations to bound the predictive power (the probability of guessing an outcome) for the general class of no-superluminal-signaling theories (i.e., any further extension of quantum mechanics). Using very weakly entangled states, we measure Bell violations that could provably not be observed using maximally entangled states of any dimension, thus demonstrating “more nonlocality with less entanglement” for the first time. We also obtain results that can only be explained using complex numbers, corresponding to measurements and quantum states spanning the entire qubit space. All of our results are consistent with quantum predictions.

Given the growing interest in quantum nonlocality, our findings are an important first systematic investigation of the limits of nonlocal correlations in nature. We expect that our results will motivate searches for new physics distinct from the quantum regime.

Figure 1

(a) Bell test scenario. Alice and Bob perform “black box” measurements on a shared (quantum) state $\rho$. The experiment is characterized by the data $\{p(a,b|x,y)\}$, i.e., a set of conditional probabilities for each pair of measurement outputs ($a$ and $b$) given measurement settings $x$ and $y$. Based on the data $p(a,b|x,y)$, Bell inequalities [see Eq. (3)] can be tested. (b) Geometrical representation of nonsignaling correlations. The set of local ($\mathcal{L}$), quantum ($\mathcal{Q}$), and nonsignaling ($\mathcal{N}\mathcal{S}$) distributions are projected onto a plane, where the following inclusion relations are clear: $\mathcal{L}\subset \mathcal{Q}\subset \mathcal{N}\mathcal{S}$.

Figure 2

A diagram of the entanglement source. The high-power laser (L) is prepared in a specific polarization state (depending on the Bell test) by two half-wave plates (HWP1 and HWP2). We precompensate for the temporal decoherence (arising from the group velocity dispersion in the down-conversion crystals) by passing the laser through a crystal (TC) designed to have the opposite group velocity dispersion. Passing the pump through a pair of orthogonal nonlinear crystals (NLC) produces the entangled photons. The measurements are performed using a motorized half-wave plate (HWP3) and a polarizing beam splitter (PBS). We then spectrally filter (IF) the photons to limit the collected bandwidth to 20 nm, as well as spatially filter the photons using a single-mode fiber (SMF) to remove any spatial decoherence. Finally, the photons are detected using avalanche photodiodes (APD), the events of which are recorded on a time-to-digital converter (TDC) and saved on a computer for analysis.

Figure 3

Testing the boundary of quantum correlations in the simplest Bell test scenario. (a) Plot of the experimental measurements of the curve Eq. (6). Here, local correlations ($\mathcal{L}$) form the inner blue square, no-signaling correlations form the outer orange square (the vertices represent the PR box and its symmetries), and quantum correlations form the green circle; the black dots are the 180 measured data points, all of whose error bars lie within the thickness of the dot. (b) A comparison of the analyzed data with the quantum mechanical maximum ($2\sqrt{2}$). Here, $\theta$ is defined in Eq. (6) and corresponds to rotating around the circle in Fig. 3. The vertical axis is the distance from $2\sqrt{2}$ of the radius of curve formed by ${\mathcal{S}}_{\mathrm{CHSH}}^{\prime }$ and ${\mathcal{S}}_{\mathrm{CHSH}}$ (i.e., the radius of the data point at a given $\theta$). Plotted values greater than zero correspond to measured values less than $2\sqrt{2}$. Here, the uncertainty is smaller for measurements around the vertices of the local correlations (at 45°, 135°, 225°, and 315°) due to large correlations between ${\mathcal{S}}_{\mathrm{CHSH}}^{\prime }$ and ${\mathcal{S}}_{\mathrm{CHSH}}$. We observe 8 data points beyond the quantum limit, which we expect to happen with a probability of 0.66.

Figure 4

Testing the boundary of quantum correlations with the tilted Bell inequality. (a) A plot of the measured values for a projection where the quantum boundary can only be attained using partially entangled states. The orange line is the boundary for no-signaling correlations (the PR box sitting at the top), and the red line is the boundary of the set of correlations achievable by a maximally entangled state (see Appendix pp2), whereas the horizontal axis at ${\mathcal{S}}_{\mathrm{CHSH}}=2$ coincides with the boundary of the local set $\mathcal{L}$. The green curve represents the quantum boundary, with the black points corresponding to measured data points. For large values of $-E_1^A-E_1^B$ (corresponding to less entangled states), the system becomes increasingly sensitive to system noise (i.e., slight state creation and measurement imperfections), resulting in the measured values deviating slightly from the quantum curve. (b) A plot of the measured values for the tilted Bell inequalities. The red line is the bound of maximally entangled sates, the blue line is the local bound, and the black points are the analyzed data. The red and blue lines cross at $1/\sqrt{2}+1/2$, where maximally entangled states can no longer violate a tilted Bell inequality. Here, for the measured points up to $\tau =1.323$, we see a value of ${\mathcal{S}}_{\tau }$ at least 3 standard deviations above the local bound; notably, we have violations for $\tau =1.223$, 1.250, 1.265, 1.296, and 1.323 (circled data points in both plots), as well as $\tau =1.300$ (see text and Appendix pp3), none of which are possible for maximally entangled states in any dimension, implying that with less entanglement we have more nonlocality.

Figure 5

A plot of the measured chained Bell inequality values for $n=2$ to $n=45$. Here, the local limit is $I_n=1$ and the no-signaling limit is $I_n=0$. The quantum boundary in this case is the green line; our measured Bell inequality values are connected by the black line, with the error bars lying within the thickness of the line. The local content for a given $n$ is represented by distance from 0 to the measured $I_n$ value (black line), which is colored blue, and the nonlocal content is the distance from the measured value to 1, colored orange. As the value of $I_n$ approaches 0, the correlations present in the system match those of a PR box—if $I_n=0$ were measured, the system would require the use of a PR box for every measurement. Our measured points deviate from the quantum boundary due to the 0.3% noise from imperfect state preparation, which becomes more noticeable with larger number of measurements (e.g., $I_{45}$ requires 360 specific measurements along the Bloch sphere).