Tests of No-Signaling

An operational no-signaling test asks whether one laboratory’s locally available outcome distribution depends on a remote freely varied setting. This is different from a Bell-inequality test: quantum correlations can violate a Bell bound while their marginals obey no-signaling. The theorem is treated in No-Signaling; the experimental task is to compare appropriately defined marginals.

The null hypothesis for a bipartite experiment

Let $x,y$ label Alice’s and Bob’s settings and $a,b$ their recorded outcomes. No-signaling from Bob to Alice requires

$$p_A(a\mid x,y)=\sum_b p(a,b\mid x,y) =p_A(a\mid x,y')$$

for all relevant $a,x,y,y'$. There is an analogous condition in the other direction. Use the actual accepted trial definition, including nondetections if those are valid outputs.

For a binary Alice output, one can estimate $\Delta_x=p_A(1\mid x,0)-p_A(1\mid x,1)$. With independent binomial samples, an approximate standard error is

$$\sigma_{\Delta_x}= \sqrt{\frac{\hat p_0(1-\hat p_0)}{N_0} +\frac{\hat p_1(1-\hat p_1)}{N_1}}.$$

If both probabilities are near $1/2$ and $N_0=N_1=10^4$, this is about $0.0071$. Small apparent differences are therefore expected in finite data. A joint test across many settings and outcomes must account for the multiple comparisons and any serial dependence.

Why coincidences can mislead

Suppose a hidden variable $\lambda=\pm1$ is equally likely and Alice always records $a=\lambda$. Bob’s apparatus accepts only $\lambda=+1$ for $y=0$ and only $\lambda=-1$ for $y=1$. Alice’s full data remain unbiased, but the subset selected by Bob is all $+1$ for one setting and all $-1$ for the other.

This apparent setting dependence in coincidences is a selection effect, not a usable remote signal. Alice cannot identify the selected subset without Bob’s acceptance labels. Setting-dependent losses, time-window pairing, and changing background rates can cause analogous problems in real data.

A no-signaling analysis should specify trial boundaries independently of outcomes where possible, model losses, and distinguish unconditional marginals from postselected marginals. Event-ready Bell experiments illustrate careful trial definition, but Bell significance and marginal no-signaling significance remain separate analyses.

From an anomaly to a causal claim

Random interleaving helps separate setting effects from drift. Electromagnetic pickup, shared electronics, timing errors, and ordinary communication must be bounded. Spacetime separation is relevant when the claim concerns faster-than-light influence; a dependence measured after a light-speed signal could arrive is not evidence against relativistic causality.

A failure to reject the null sets a sensitivity-dependent bound. It does not prove exact independence for all possible future experiments. A significant anomaly first challenges the joint model of devices, sampling, and statistical analysis. Establishing a controllable signal additionally requires that the receiver decode a chosen message from locally accessible data.

Exercise

In the selection example, calculate $p_A(a=+1\mid y)$ before Bob communicates acceptance, and compare it with $p_A(a=+1\mid y,\mathrm{accepted})$.

Solution

The unconditional probability is $1/2$ for both settings. Conditioned on acceptance it is one for $y=0$ and zero for $y=1$. Only the latter requires remote information to define the subset, so the construction offers no signaling channel.

References

• Hensen, B., et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015), doi:10.1038/nature15759 — event-ready trial structure and locality controls.
• Hensen, B., et al., “Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis,” Scientific Reports 6, 30289 (2016), arXiv:1603.05705 — additional statistical and setting analyses.