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Bell-GHZ nonclassicality of many-observer interwoven frustrated down conversions

Marek Żukowski, Paweł Cieśliński, Marcin Markiewicz, Konrad Schlichtholz

TL;DR

The work extends interwoven frustrated down-conversion to a multi-observer Bell-GHZ test, showing that path-identity interference between source and local PDCs can violate a lifted Clauser-Horne inequality in three- and four-party configurations. The analysis combines analytic probability amplitudes, a GHZ-Hardy-type paradox, and linear-programming verification to demonstrate nonclassical correlations under suitable phase settings, with a compact quantum-marker expression $CH_Q= -|g|^6[1+2 \cos(\phi_A+\phi_B+\phi_C)]$. It further generalizes to $N$ parties via a polygon-graph topology and discusses robustness to imperfections, suggesting broad implications for QKD, randomness generation, and foundational tests. The results illuminate how active measurement choices in interwoven PDC systems can reveal nonlocality beyond conventional two-party scenarios, and provide a general methodology for identifying non-classicality in interferometric processes.

Abstract

Frustrated down conversion is a process in which a quantum superposition of emissions from two separate parametric down-conversion processes gives rise to observable interference. Depending on the phase relation between the probability amplitudes associated with emissions by the first and second crystal, the process can be enhanced or suppressed. This is achieved by aligning the setup so that the signal and idler modes from the first crystal are fed into the second and constitute its signal-idler modes. In Sci. Adv. 11, 1794 (2025), two-observer interwoven frustrated PDC processes produced interference effects based on path identity [Phys. Rev. Lett. 118, 080401 (2017)]. The signal and idler modes of source crystals I and II are arranged to fully overlap with the emission modes of crystals A and B, which serve as elements of measurement stations controlled by Alice and Bob. In the interwoven configuration, crystal A (B) receives the signal mode of crystal I (II) and the idler mode of crystal II (I), enabling interference between joint emission processes at the sources and at the measurement stations. It was conjectured that such interference may lead to new non-classical phenomena. In arXiv:2508.19207 it was shown that the process violates the standard Clauser-Horne Bell inequality without additional assumptions, provided suitable measurement settings are used. Here we extend the interference scheme to more than two measurement stations and demonstrate a violation of one of the WWWZB inequalities. This indicates that the proposed approach may provide a general method for revealing non-classicality in a range of phenomena discussed in [Rev. Mod. Phys. 94, 025007 (2022)]. We also present a GHZ/Hardy-type argument that further highlights the paradoxical character of the interference.

Bell-GHZ nonclassicality of many-observer interwoven frustrated down conversions

TL;DR

The work extends interwoven frustrated down-conversion to a multi-observer Bell-GHZ test, showing that path-identity interference between source and local PDCs can violate a lifted Clauser-Horne inequality in three- and four-party configurations. The analysis combines analytic probability amplitudes, a GHZ-Hardy-type paradox, and linear-programming verification to demonstrate nonclassical correlations under suitable phase settings, with a compact quantum-marker expression . It further generalizes to parties via a polygon-graph topology and discusses robustness to imperfections, suggesting broad implications for QKD, randomness generation, and foundational tests. The results illuminate how active measurement choices in interwoven PDC systems can reveal nonlocality beyond conventional two-party scenarios, and provide a general methodology for identifying non-classicality in interferometric processes.

Abstract

Frustrated down conversion is a process in which a quantum superposition of emissions from two separate parametric down-conversion processes gives rise to observable interference. Depending on the phase relation between the probability amplitudes associated with emissions by the first and second crystal, the process can be enhanced or suppressed. This is achieved by aligning the setup so that the signal and idler modes from the first crystal are fed into the second and constitute its signal-idler modes. In Sci. Adv. 11, 1794 (2025), two-observer interwoven frustrated PDC processes produced interference effects based on path identity [Phys. Rev. Lett. 118, 080401 (2017)]. The signal and idler modes of source crystals I and II are arranged to fully overlap with the emission modes of crystals A and B, which serve as elements of measurement stations controlled by Alice and Bob. In the interwoven configuration, crystal A (B) receives the signal mode of crystal I (II) and the idler mode of crystal II (I), enabling interference between joint emission processes at the sources and at the measurement stations. It was conjectured that such interference may lead to new non-classical phenomena. In arXiv:2508.19207 it was shown that the process violates the standard Clauser-Horne Bell inequality without additional assumptions, provided suitable measurement settings are used. Here we extend the interference scheme to more than two measurement stations and demonstrate a violation of one of the WWWZB inequalities. This indicates that the proposed approach may provide a general method for revealing non-classicality in a range of phenomena discussed in [Rev. Mod. Phys. 94, 025007 (2022)]. We also present a GHZ/Hardy-type argument that further highlights the paradoxical character of the interference.
Paper Structure (13 sections, 20 equations, 1 figure)

This paper contains 13 sections, 20 equations, 1 figure.

Table of Contents

  1. introduction
  2. The process Hamiltonians
  3. Bell's theorem for the configuration
  4. Why this does work
  5. Calculated probabilities
  6. Bell Theorem via linear programming: results
  7. Generalization to four parties.
  8. Yes, it works also for four parties, thus it works for $N$ parties
  9. The reasoning
  10. A quantum paradox
  11. concluding remarks
  12. Acknowledgments
  13. Appendix

Figures (1)

  • Figure 1: Three observer interwoven frustrated parametric down conversion: configuration for a Bell-GHZ experiment. The crystals $I$, $II$ and $III$ are treated as joint source of the initial state in modes $a_1c_2,b_1a_2, c_1b_2$. The dotted boxes represent the measuring stations A, B, C operated by "Alice, Bob and Charlie". Each the measurement station before the detection performs a unitary transformation on the respective modes: $a_1a_2$, $b_1,b_2$ and $c_1c_2$. These transformations depend on the local phase shifts $\phi_A, \phi_B, \phi_C$ and on whether the local pumps are blocked by the switches or not (this is marked as on/off). If a local pump is blocked then the parametric down conversion process is switched off in the local crystal. A perfect alignment is required to observe interference in the probability of each of the six detectors to register single photons in coincidence. To make this interference possible the triple pair emissions in crystals $I,II, III$ have to be indistiguishable, at the positions of the detectors, from triple pair emission originating from A, B, C. To have a phase stability of the interference all pumps must be derived from a single laser source. This can be done by beam-splitting the laser field mode. Also, the lengths of all optical paths shown in the figure can differ only by much less than the coherence length of the central pump laser radiation. This applies also to the paths of all the signal and idler down conversion modes.