Experimental ladder proof of Hardy's nonlocality for high-dimensional quantum systems

Recent years have witnessed a rapidly growing interest in high-dimensional quantum entanglement for fundamental studies as well as towards novel applications. Therefore, the ability to verify entanglement between physical qudits, $d$-dimensional quantum systems, is of crucial importance. To show nonclassicality, Hardy's paradox represents “the best version of Bell's theorem” without using inequalities. However, so far it has only been tested experimentally for bidimensional vector spaces. Here, we formulate a theoretical framework to demonstrate the ladder proof of Hardy's paradox for arbitrary high-dimensional systems. Furthermore, we experimentally demonstrate the ladder proof by taking advantage of the orbital angular momentum of high-dimensionally entangled photon pairs. We perform the ladder proof of Hardy's paradox for dimensions 3 and 4, both with the ladder up to the third step. Our paper paves the way towards a deeper understanding of the nature of high-dimensionally entangled quantum states and may find applications in quantum information science.

Figure 1

The diagram of Hardy's ladder proof with $K=2$ in three-dimensional OAM subspace. Blue (dash) lines indicates zero probabilities while red (solid) lines corresponds to nonzero probabilities. According to the local hidden variable theory, we know that once we measured $P(A_2=1,B_2=1)>0$ then both $P(A_2=1,B_1=)=0$ and $P\left(A_2=1,B_1=0\right)=0$ will lead to $P\left(A_2=1,B_1=1\right)>0$. Similarly, we also have $P\left(A_1=1,B_2=1\right)>0$ . Thus we have $P\left(A_1=1,B_1=1\right)>0$. We repeat this to go down the ladder and finally obtain $P\left(A_0=1,B_0=1\right)>0$, being inconsistent with Eq. (1) of $P\left(A_0=1,B_0=1\right)=0$. In other words, if the assumptions of Eqs. (1, 2, 3) hold, the measurement result of $P\left(A_2=1,B_2=1\right)>0$ will support quantum mechanics but contradict any local hidden variable theory.

Figure 2

Optical system for the ladder test of Hardy's paradox based on high-dimensional entangled photons. A collimated 355-nm ultraviolet laser (JDSU) pumps a 5-mm-long β-barium borate (BBO) crystal, where a degenerate 710-nm signal and idler photons are produced in pairs via type-I collinear SPDC. Afterwards, we separate the two photons into two different paths by a nonpolarizing beam splitter (BS). A longpass filter (LF) is used to block the pump beam after the crystal. The output facet of the crystal is imaged onto both spatial light modulators ($\mathrm{SL}{\mathrm{M}}_1, {\mathrm{SLM}}_2$, Hamamatsu, X10486-1) by two lenses in a 4-$f$ configuration ($f_1=200\mathrm{mm}$,$f_2=400\mathrm{mm}$). Each SLM is then similarly reimaged by another pair of lenses ${\mathrm{L}}_3$ and ${\mathrm{L}}_4$ ($f_3=500\mathrm{mm}$ and $f_4=2\mathrm{mm}$) onto a single-mode fiber (SMF), which is connected to a single-photon detector (SPCM-AQRH-14). The SLM together with the SMF acts as a programmable mode filter. Bandpass filters (BF) of 10-nm width and centered at 710 nm are placed in front of SMFs to ensure degeneracy of the detected photon pairs. The detectors are connected to a coincidence counting circuit (&) with a 25-ns coincidence time window. The measured entangled OAM spectrum characterized by the peak-normalized coincidence counts $N_{\mathrm{CC}}$ is shown by the inset.

Figure 3

Experimental results for the ladder test of Hardy's paradox for ${|\mathrm{\Psi }\rangle }_{3\mathrm{D}}$. The empty bars (blue edges) are the Hardy fractions predicted theoretically by Eq. (9), while the solid bars (green) are those obtained experimentally, where $P_1=9.23\pm 0.34\%, P_2=18.00\pm 0.54\%$, and $P_3=21.85\pm 0.52\%$. In contrast, all the other 13 joint probabilities are almost zero, in good agreement with the theoretical prediction. Because local realistic models would only allow all probabilities to be zero, the results are nicely demonstrating that quantum mechanics contradicts local hidden variable models.

Figure 4

Experimental results for the ladder test of Hardy's paradox in a four-dimensional OAM subspace. The empty bars (blue edges) are theoretical predictions of the Hardy fractions based on Eq. (13), while the solid bars (green) are those obtained experimentally. The experimental ladder test results in $P_1=5.44\pm 0.14\%, P_2=5.62\pm 0.11\%$, and $P_3=4.00\pm 0.05\%$, which is within a good agreement with the theoretical predictions of 5.31, 6.89, and 6.16%. Since all the other 19 joint probabilities are close to zero, our measurements contradict local hidden variable models, which would require zero probabilities for all joint measurements.