Closing the detection loophole in the triangle network with high-dimensional photonic states

TL;DR

The paper addresses closing the detection loophole in network nonlocality, specifically in the triangle configuration, by showing that high-dimensional photonic NOON states with $N=2$ can exhibit robust nonlocal correlations even under significant photon loss. It develops a generalized token-counting framework compatible with photon loss and detector types, certifies nonlocality under single-photon loss up to $10.3\%$ and minimal full-loss noise, and strengthens this with neural-network heuristics indicating substantial real-world robustness. The authors further show that heralded SPDC sources, with a failure-certification mechanism, can eliminate the need for global post-processing, thereby closing the detection loophole in practice. These results point to a practical path toward loophole-free network nonlocality experiments using high-dimensional photonic states and heralded sources.

Abstract

Bell nonlocality without input settings, e.g. in the triangle network, has been perceived to be particularly fragile, with low robustness to noise in physical implementations. Here we show to the contrary that nonlocality based on N00N states already for $N=2$ has an exceptionally high robustness to photon loss. For the dominant noise factor, single photon loss in the transmission channels, we can certify noise robustness up to $10\%$ loss, while for a realistic noise model we use neural network-based heuristics to observe $\sim 50\%$ robustness. Moreover we show that the robustness holds even for imperfect sources based on SPDC sources, where the heralding information of the sources can be used to avoid any global post-processing of the outcomes, such as discarding rounds when photons fail to arrive, and thus demonstrate how the detection loophole in the triangle network can be closed.

Figures (8)

• Figure 1: (a) In a triangle network each of the three sources $\alpha,\beta,\gamma$ distributes physical systems to two parties (either quantum or classical). The parties generate a classical output after locally manipulating their received systems. (b) In the proposed quantum experiment, each source distributes a N00N state and each party uses a beamsplitter and either photon number resolving or "click/no-click" detectors.
• Figure 2: Distance to the local set according to LHV-Net for different values of channel transmittivity $\eta$ and source impurity $Q$, for the full photon loss model with click/no-click detectors, for $t=0.75, \varphi=\frac{\pi}{2}.$
• Figure 3: Cube representation of any LHV model which could possibly reproduce p(a,b,c). (a) First, we rearrange $\beta$ to collect all $o^*$ outputs from Charlie in one region, defining $\beta_R := p(S_\beta^R)$. Consequently, Bob is prohibited from outputting $o^*$ within the region $\alpha_1$, and Alice is restricted from doing so within the region $\beta_2$, both of which are highlighted in purple Alice and Bob are prohibited from outputting $o^*$ within the striped purple regions. b) Next, we repeat the same procedure for $\gamma$ and Alice's $o^*$, defining $\gamma_R$$:=p(S_\gamma^R)$. For Bob and Charlie, regions where an $o^*$ output is not possible are identified and marked in yellow. (c) Finally, for Bob, we rearrange $\alpha$ to get all $o^*$ outcomes on the bottom and $\alpha_R:=p(S_\alpha^R)$, leading to the inequalities $\alpha_L \beta_R \geq p(b = o^*), \beta_L \gamma_R \geq p(b = o^*), \gamma_L \alpha_R \geq p(b = o^*)$, which lead to the inequality in Lemma 1.
• Figure 4: Cube representation of any LHV model which could possibly reproduce p(a,b,c). (a) The three cuboids $A_{o^*}, B_{o^*}, C_{o^*}$ are highlighted in yellow, green and blue, respectively. (b) For Constraint 3 we first notice that the $\chi_i$ responses on Alice's face define prisms (drawn approximately as cyclinders in this image for simplicity) within the cube, within which all the $a=\chi_i$ events must appear. We then consider the two prism parts which fall in $C_{o^*}$, and move down in both of them to reach $q(i,R)$ (roughly speaking) and a part in $B_{o^*}$. Then consider the two prisms parts in $B_{o^*}$ and move up in only one of them in order to reach $q(i,L)$. Substracting these quantities gives information on $q(i,L) - q(i,R)$ (up to some factors)while crucially canceling the prism part in the bottom right-most subcuboid.
• Figure 5: (a) Threshold $t$ values beyond which the distribution $p(a, b, c)$ does not admit a local model for a given $\lambda_0^2$. (b-c) Certified nonlocality of the (b) click/no-click and (c) photon number resolving distribution (events of 3 photons and higher coarse-grained) for the single photon loss model. For all plots $\varphi=\frac{\pi}{2}.$

Theorems & Definitions (2)

• proof
• proof