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

Sadra Boreiri, Nicolas Brunner, Pavel Sekatski

TL;DR

This work addresses Bell nonlocality in quantum networks with unreliable sources, where sources may fail and produce inconclusive events. It introduces fair-sampling as a network-level requirement and proves that post-selecting conclusive outcomes is harmless if measurements are fair-sampling, via a rigidity result based on the saturation of the quantum Finner inequality. The authors establish a general self-testing statement: equality in the Finner bound forces the underlying model to be the failing-source one, thereby preserving source independence in post-selected data, and they apply this to both a photonic CHSH setup with SPDC sources and to triangle networks to bound randomness. The results enhance device-independent randomness generation and Bell testing in realistic, imperfect networks and suggest new directions for loss-tolerant and topology-flexible quantum networks.

Abstract

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.

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

TL;DR

This work addresses Bell nonlocality in quantum networks with unreliable sources, where sources may fail and produce inconclusive events. It introduces fair-sampling as a network-level requirement and proves that post-selecting conclusive outcomes is harmless if measurements are fair-sampling, via a rigidity result based on the saturation of the quantum Finner inequality. The authors establish a general self-testing statement: equality in the Finner bound forces the underlying model to be the failing-source one, thereby preserving source independence in post-selected data, and they apply this to both a photonic CHSH setup with SPDC sources and to triangle networks to bound randomness. The results enhance device-independent randomness generation and Bell testing in realistic, imperfect networks and suggest new directions for loss-tolerant and topology-flexible quantum networks.

Abstract

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.
Paper Structure (19 sections, 8 theorems, 81 equations, 5 figures, 1 table)

This paper contains 19 sections, 8 theorems, 81 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unreliable quantum networks and the fair sampling assumption
  3. Loophole free post-selection via Finner self-testing
  4. CHSH test with SPDC source
  5. Nonlocality and Randomness in the triangle network
  6. Discussion
  7. Proof of Propositions 1 and 2
  8. Proof of Theorem 3
  9. Classical rigidity from the Finner equality condition
  10. Notation and a priori description of quantum models
  11. Proof of the quantum Finner inequality (binary sources) renou2019limits
  12. Quantum Finner rigidity
  13. Rephrasing the theorem
  14. Photonic entanglement with an SPDC source.
  15. Computing the output distribution
  16. ...and 4 more sections

Key Result

Proposition 1

The measurement $M_{A_j}^{a_j}$ is fair-sampling if and only if the (coarse-grained) POVM element corresponding to all conclusive outcomes admits the following decomposition with Hermitian $0 \preceq T^{(i)}_{Q_j^{(i)}} \preceq \mathds{1}$, i.e. it is a product on all the systems it acts upon.

Figures (5)

  • Figure 1: A measurement $M_{A_j}^{a_j}$ is fair-sampling if it can be decomposed as shown here. First, a filter is applied on each input system, and the outcome is inconclusive $\varnothing$ if any of the filters is unsuccessful (flag output $f_i=0$). When all filters are successful, a (always) conclusive measurement $\widetilde{M}_{A_j}^{a_j}$ is performed on the filter's output $(\bigotimes_i \mathcal{T}^{(i)})[\bullet]$.
  • Figure 2: (a) A quantum network with bipartite sources. Each party produces an output $a_j$ by measuring the quantum systems received from the connected sources. (b) In the failing-source model, each source may fail with a certain probability, in which case the parties connected to it will output an inconclusive outcome $\varnothing$.
  • Figure 3: (Top left) Sketch of an experimental setup for testing the CHSH Bell inequality with an SPDC source of polarization-entangled photons and single-photon detectors. (Bottom left) The three-source network describing the experiment. (Right) CHSH score (left scale, dashed) and randomness rate (right scale, full) obtained from the bare (red) and post-selected (blue) data, via the Corollary \ref{['corrolary']}, as the function of the pump parameters $T=T_1=T_2$ of the source.
  • Figure 4: Bell Scenario: Comparison of CHSH score and Randomness plots for Standard and post selected statistics
  • Figure 5: triangle network

Theorems & Definitions (12)

  • Definition
  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Corollary 3.1
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • ...and 2 more