Bell nonlocality in quantum networks with unreliable sources: Loophole-free postelection via self-testing

We discuss Bell nonlocality in quantum networks with unreliable sources. Our main result is a condition on the observed data which ensures that inconclusive events can be safely discarded, without introducing any loophole. More formally, we characterize the fair-sampling property for measurements in a network. When all measurements are fair-sampling, we show that the post-selection of conclusive outcomes does not compromise the assumption of source independence, hence avoiding the detection loophole. Furthermore, we show that in some cases, the fair-sampling property can in fact be guaranteed based only on observed data. To show this, we prove that saturation of the Finner inequality provides a self-test of the underlying quantum model. We illustrate the relevance of our results by demonstrating an improvement in device-independent randomness generation for a photonic Bell test with a probabilistic source and for the triangle network.

💡 Research Summary

The paper addresses the challenge of demonstrating Bell nonlocality in quantum networks where each source may fail with a certain probability, producing inconclusive (“no‑click”) outcomes. In standard Bell tests, discarding such events can open the detection loophole because it may break the causal independence between sources and measurements. To overcome this, the authors first formalize a notion of fair‑sampling for measurements in a network setting. A measurement is fair‑sampling if it can be decomposed into (i) a local quantum filter applied independently to each incoming subsystem and (ii) an ideal, always‑conclusive measurement performed only when all filters succeed. If any filter fails, the outcome is declared inconclusive (∅).

Two key propositions are proved. Proposition 1 shows that a fair‑sampling measurement admits a decomposition where the coarse‑grained POVM element for all conclusive outcomes is a tensor product of positive operators (the filters) on each incoming system. Proposition 2 demonstrates that when all measurements in the network satisfy the fair‑sampling property, post‑selecting on the conclusive events yields a conditional distribution that can be realized by the same network topology with the same quantum states, after moving the filters to the sources. Consequently, post‑selection does not compromise source independence and does not introduce a detection loophole.

The authors then tackle the practical problem of verifying the fair‑sampling assumption without trusting the devices. They exploit the Finner inequality, a multipartite generalization of the Cauchy–Schwarz inequality that bounds the probability of all parties obtaining conclusive outcomes by the product of the individual conclusive probabilities. For a network of independent bipartite sources, the inequality reads

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