Photon Bell tests keep the trials where detectors stay silent

An efficient detector changed the question

A photon can be lost in the optics or missed by a detector. If a Bell test keeps only pairs that were seen at both ends, those losses can select an unrepresentative sample. In 2013 Giustina and colleagues combined efficient light collection with superconducting transition-edge sensors and violated a Bell inequality without the fair-sampling assumption. The result addressed detection; fast, spacelike-separated setting choices were still a later step.

Every undetected photon has to stay in the analysis. If only coincidences are kept, whether a photon is detected can depend on hidden instructions and on the settings, and a local explanation might survive even when the remaining pairs trace an impressive polarization curve. Better detection matters because this inequality counts failed events against the result.

Read the evidence: M. Giustina et al. (2013), Bell violation using entangled photons without the fair-sampling assumption, Nature 497, 227–230 — author manuscript

Every trial has an outcome, including no click

In the formula, plus means a click in the chosen channel and zero means no click there. Each probability is conditional on the numbered local settings. Both 2015 teams used this Clauser–Horne–Eberhard form: the favourable double click must outweigh three defined penalty terms, including events where one side clicks and the other does not.

The local bound can be checked without quantum theory. Give each side a fixed click or no-click answer for each of its settings. Listing all sixteen such instructions gives eight scores of zero and eight of minus one, so no random mixture can score above zero. Trials with no clicks add nothing to the score but still count in the denominators of the probabilities.

\begin{aligned}J={}&p_{++}(1,1)-p_{+0}(1,2)\\ &-p_{0+}(2,1)-p_{++}(2,2)\\ J_{\mathrm{local}}&\leq0\end{aligned}

Read the evidence: M. Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons, PRL 115, 250401 — author manuscript L. K. Shalm et al. (2015), Strong Loophole-Free Test of Local Realism, PRL 115, 250402 — author manuscript, revised 2016

A less entangled state can give stronger evidence

The best state for this lossy experiment need not be the maximally entangled one. The two polarization-pair possibilities can be given unequal amplitudes, meaning unequal weights before the alternatives are combined. Eberhard-type optimization adjusts those weights and the analyzer angles to reduce the events that incur penalties. In the ideal symmetric limit without background, the required efficiency approaches two thirds; background and state imperfections raise it.

Efficiency here means transmission, collection and detection combined, so a detector specification alone does not qualify the experiment. Tuning the state also does not permit subtracting inconvenient single clicks: the state and the analysis angles have to improve the score over all events. The two 2015 setups each used their own optimized states and had their own efficiencies.

Read the evidence: P. H. Eberhard (1993), Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment — APS abstract M. Giustina et al. (2013), Bell violation using entangled photons without the fair-sampling assumption, Nature 497, 227–230 — author manuscript L. K. Shalm et al. (2015), Strong Loophole-Free Test of Local Realism, PRL 115, 250402 — author manuscript, revised 2016

Pulses define the trials, not coincidences

The Vienna experiment locked its source pulses and fast setting changes to a one-megahertz clock, and outcomes fell in locally defined time slots. Shalm’s team synchronized at about ninety-nine thousand trials per second, with several possible pump-pulse slots in each switching interval. The two schedules differ, so their trial counts cannot be compared one for one.

In Shalm’s geometry, some slots did not meet the spacetime requirements, so the reported totals used suitable central slots. Adding more pulses raises the counts but can spoil the separation. With measured delays and their uncertainty included, each local choice must lie outside the relevant emission light cone and be unable to reach the remote outcome in time.

Each team’s statistics belong to its own experiment

Vienna reported a p-value of at most 3.74 times ten to the minus thirty-one, allowing for device memory and a quantified excess predictability of the settings. Shalm’s five-pulse analysis reported about 5.8 times ten to the minus nine; a conservative allowance for predictability raised that bound to 2.3 times ten to the minus seven.

These numbers come from separate data sets and analyses and cannot be merged into one imaginary experiment. A p-value bounds how likely a result this extreme would be if the local model and its input assumptions held. It is not the probability that locality is true, and a tiny statistical bound does not cover every calibration or human error.

Local timing rules stop the coincidence analysis from choosing pairs by comparing detection times from both distant sides. Fresh settings and a justified bound on their predictability still matter. Spacetime placement limits particular causal connections, but no experiment can exclude arbitrary correlations reaching back into a common past. Deciding when an outcome is final also needs a stated physical convention.

The achievement was to bring efficient detection and the timing requirements into a single optical experiment. The sixteen-instruction check explains the inequality’s event logic; the actual experimental claim also depends on measured efficiencies, calibrated timing and the statistical analysis of all defined trials.