Binarisation-loophole-free observation of high-dimensional quantum nonlocality

Abstract

Bell inequality tests based on high-dimensional entanglement usually require measurements that can resolve multiple possible outcomes. However, the implementation of high-dimensional multi-outcome measurements is often only emulated via a collection of ``click or no-click'' measurements. This reduction of multi-outcome measurements to binary-outcome measurements opens a loophole in high-dimensional tests Bell inequalities which can be exploited by local hidden variable models [Tavakoli et al., Phys. Rev. A 111, 042433 (2025)]. Here, we close this loophole by using four-dimensional photonic path-mode entanglement and multi-outcome detection. We test both the well-known Collins-Gisin-Linden-Massar-Popescu inequality and a related Bell inequality tailored for maximally entangled states in high-dimension. We observe violations that are large enough to also rule out any quantum model based on entanglement of lower dimension, thereby demonstrating genuinely high-dimensional nonlocality free of the binarisation loophole.

Figures (6)

• Figure 1: Multi-outcome vs binarised CGLMP test. Alice and Bob perform local measurements, $x$ and $y$, on a shared $d$-dimensional state $\rho$. (a) Multi-outcome measurements are used to resolve any one of the $d$ possible outcomes per party and round. (b) The multi-outcome measurements are emulated by a sequence of single-detector measurements, each oriented at the $a$'th ($b$'th) projection of the $x$'th ($y$'th) measurement. By post-processing the relative frequency of successful projections, one estimates the statistics of the multi-outcome measurement used in the Bell test.
• Figure 2: White noise tolerance of Bell nonlocality in the tests associated to $\mathcal{I}_4$ (blue) and $\mathcal{S}_4$ (red). Multi-outcome measurements (circular marker) have stronger robustness in larger dimension whereas binarised measurements (rectangular marker) have the opposite trend.
• Figure 3: Experimental setup. (a) Photon Source. A continuous-wave laser operates at 404 nm is split into parallel paths with 4 mm spacing in the horizontal direction after passing through BD1 and BD2, and 4 mm spacing vertically. When the laser array passes through the BBO crystal, SPDC process generates photon pairs (signal and idler) entangled in path DOF. Then the photon pairs are split by PBS2, sent to Alice and Bob. (b),(c) Multi-outcome measurements of Alice and Bob. Depending on their inputs $x\in\{1, 2\}$ (Alice) and $y\in\{1, 2\}$ (Bob), each party performs a local four-outcome measurement. First, HWPA2—an array of 808 nm half wave plates set at $0^\circ$ or $45^\circ$ in sequence—adjusts the polarisation in each path. The initial 2×2 path arrangement is then combined by BD3 and BD4 into two paths, converting path encoding into polarisation-path hybrid encoding. The measurement finally projects onto four different outcomes, collected separately by couplers $O_1$ to $O_4$. Coincidence events between all of Alice’s and Bob’s couplers are registered by a time to digital converter (UQDevice) with a coincidence window of 3 ns. HWP: Half wave plate. QWP: Quarter wave plate. HWPA: Half wave plate array. BD: Beam displacer. PBS: polarising beam splitter. BBO: $\beta$-barium-borate crystal.
• Figure 4: Experiment results for $\mathcal{I}_4$ (blue) and $\mathcal{S}_4$ (red), and upper bounds on quantum nonlocality for different dimensions. In both cases, we violate the Bell inequality and the limitations of three-dimensional entanglement. The statistical error for each violation is displayed at the end of each bar.
• Figure 5: Measurement setup for Alice and Bob. Input state is encoded in $2\times2$ path d.o.f., $\ket{0}$, $\ket{1}$ for upper two paths and $\ket{2}$$\ket{3}$ for lower two, converted into path-polarization hybrid state with polarization adjustment and BD3-5. The BD3 refracts horizontally polarized beams downwards by $4~mm$, while BD4 and BD5 refract vertically polarized beams horizontally by $4~mm$. Projection is achieved by HWP1-3 and QWP1-3 with following PBS, resulting in photon counts for $O_1 - O_4$, each of which corresponds to a different 4-dimensional projective measurement as listed in the box.