Experimental tests of multiplicative Bell inequalities and the fundamental role of local correlations

Bell inequalities are mathematical constructs that demarcate the boundary between quantum and classical physics. A new class of multiplicative Bell inequalities originating from a volume maximization game (based on products of correlators within bipartite systems) has been recently proposed. For these new Bell parameters, it is relatively easy to find the classical and quantum, i.e., Tsirelson, limits. Here, we experimentally test the Tsirelson bounds of these inequalities using polarization-entangled photons for a different number of measurements ($n$), each party can perform. For $n=2,3,4$, we report the experimental violation of local hidden variable theories. In addition, we experimentally compare the results with the parameters obtained from a fully deterministic strategy, and observe the conjectured nature of the ratio. Finally, utilizing the principle of “relativistic independence” encapsulating the locality of uncertainty relations, we theoretically derive and experimentally test new and richer quantum bounds for both the multiplicative and the additive Bell parameters for $n=2$. Our findings strengthen the deep correspondence between local and nonlocal correlations, and pave the way for both theoretical (e.g., better understanding of nonlocal correlations) and practical (e.g., Bell tests and quantum technologies with inefficient detectors) applications.

Figure 1

Experimental scheme and the chosen strategy to perform measurement of multiplicative Bell parameters. (a) Sketch of the setup used for generating polarization entangled photons and projecting them onto the states chosen by Alice and Bob's strategies. Entangled photon pairs are generated after pumping paired BiBO (bismuth triborate) crystals, and then separated by a 50:50 beamsplitter (BS) into two arms (Alice and Bob). The polarization measurement stage on each side consists of a half-wave plate (HWP) and polarizing beamsplitter (PBS). The photons are filtered by a 710-nm interference bandpass filter (IF), and coupled into single mode fibers, and then detected using single photon avalanche diodes whose signals are sent to a coincidence module from which coincidence events can be observed. (b) Real and imaginary parts of the experimentally reconstructed density matrix of the generated entangled state are shown in the ${\left|H\right.〉}_A{\left|H\right.〉}_B$, ${\left|H\right.〉}_A{\left|V\right.〉}_B$, ${\left|V\right.〉}_A{\left|H\right.〉}_B$, and ${\left|V\right.〉}_A{\left|V\right.〉}_B$ basis. The generated state fidelity is $\simeq 0.977$. (c) The projective measurement strategy, i.e., $\{a_1,a_2,...\}$ and $\{b_1,b_2,...\}$, chosen by Alice and Bob, are shown for $n=2$, $n=3$, and $n=4$ on the polarization Poincaré sphere.

Figure 2

Experimentally calculated multiplicative Bell parameters (MBell). (a) Logarithmic plot of the experimentally calculated Bell parameters (blue), and theoretical quantum limit of the Bell parameter, i.e., n!, (red). (b) Ratio of the parameters generated from the fully deterministic strategy ${\text{FD}}_n$ taken with the Tsirelson bound $n!$ (red), and with the experimentally observed Bell parameter MBell (blue).

Figure 3

Distribution of both (a) CHSH parameters (upper limit = 2.618) as well as (b) Bell parameters ${\mathcal{B}}_2$ (upper limit =1.428) for ${\eta }_A=0.7$.

Figure 4

Distribution of the correlation vectors for (a) ${\eta }_A=0.7$ and (b) ${\eta }_A=0.2$. The solid line indicates the region within which the correlation vectors should fall, and the points are experimentally measured vectors.