Photon Bell tests: the trials with no clicks still count

An efficient detector changed the question

In 2013, Giustina and colleagues violated a photon Bell inequality without assuming that detected pairs fairly represented every emitted pair. Their transition-edge sensors and efficient collection made losses small enough for an Eberhard test. The achievement concerned detection; it did not simultaneously establish the rapid, spacelike-separated setting choices of the later experiments.

An undetected photon is not a missing row to discard. Keeping only coincidences lets detection depend on hidden instructions and settings. A local explanation might then survive even when the surviving pairs show an impressive polarization curve. Better detection matters because the inequality charges for unsuccessful events.

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

Write an outcome for every optical trial

For the displayed convention, plus means a click in the selected channel and zero means no such click. 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 explicitly defined penalties, including single-sided nondetections.

Its local bound can be checked without quantum theory. Assign each side a predetermined click or no-click answer for each setting. Enumerating all sixteen instructions gives eight scores of zero and eight of minus one. Random mixtures cannot become positive. Empty trials contribute zero but remain in the probability denominators.

\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

Less entanglement can buy more evidence

Maximizing ideal correlation is not the same task as maximizing a violation with loss. Eberhard-type optimization uses unequal polarization-pair amplitudes and adapted angles to suppress costly events. In the ideal symmetric, background-free limit, the efficiency threshold approaches two thirds; real background and imperfect states raise the requirement.

Efficiency here includes transmission, collection and detection together. A detector specification alone does not certify the experiment. Nor does tuning the state excuse subtracting inconvenient single clicks. State preparation and analysis angles must improve the complete event ledger. The two 2015 apparatuses used their own optimized states and 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

A pulse defines a trial before a coincidence does

The Vienna experiment synchronized source pulses and fast setting changes to a one-megahertz clock. Outcomes occupied locally defined time slots. Shalm’s team instead used synchronization at about ninety-nine thousand trials per second, with several possible pump-pulse slots inside a switching interval. These are different optical schedules, not interchangeable trial counts.

For Shalm’s geometry, some slots failed the spacetime constraints; the reported aggregates used suitable central slots. Including extra pulses can improve counts while spoiling separation. Each local choice must be outside the relevant emission light cone and unable to reach the remote outcome in time, with measured delays and uncertainty included.

Keep each team’s statistical statement attached

Vienna reported a p-value bounded by 3.74 times ten to the minus thirty-one, allowing device memory and quantified excess setting predictability. Shalm’s five-pulse analysis reported approximately 5.8 times ten to the minus nine; its conservative predictability allowance changed that bound to 2.3 times ten to the minus seven.

These numbers belong to separate records and analyses. They must not be combined into one purported experiment. A p-value bounds the chance of a specified extremeness under the local null and its input assumptions. It is not the probability that locality is true, and a tiny statistical bound does not measure every calibration or human error.

The remaining assumptions stay visible

Local timing rules prevent a coincidence finder from selecting pairs using both distant detection times. Fresh settings and a justified predictability bound still matter. Spacetime placement constrains particular causal connections; it cannot experimentally exclude arbitrary correlations extending into a common past. Outcome completion also needs a stated physical convention.

The linked Bell Lab illustrates correlation and sampling choices. It does not implement either photon source, pulse-slot record, superconducting detector or finite-sample test. Our enumeration verifies an inequality’s event logic, not the historical timing survey or reported significance. The evidence challenges the specified local models without providing a controllable faster-than-light signal.