Revealing Bell Nonlocality Does Not Require the Same Physical Measurement to be Performed Twice

The paper in detail

In a CHSH test Alice chooses setting \(k\in\{1,2\}\), Bob chooses \(l\in\{1,2\}\), and they combine the correlations into \(\mathcal B=E_{11}+E_{12}+E_{21}-E_{22}\), which local hidden-variable (LHV) models keep within \(\lvert\mathcal B\rvert\le2\) and quantum mechanics within \(2\sqrt2\). The usual analysis identifies setting \(k\) with one Bloch direction \(\vec a_k\). Here setting \(k\) only fixes a probability density \(\alpha_k(\vec a)\) over directions; each run draws one direction, measures one photon along it and never returns to it.

The first point is that the local bound is untouched. An LHV model assigns an outcome \(A(\vec a,k,\lambda)=\pm1\) to every direction; averaging over a locally drawn \(\vec a\) produces another local response for setting \(k\), even if the density of directions depends on \(\lambda\), provided the setting itself is chosen freely. Non-repeatability therefore cannot help a local model. It can only weaken the quantum correlations, and the question becomes how much spread the entangled state can absorb.

Because the quantum correlation \(\mathrm{Tr}[\rho(\vec a\cdot\vec\sigma\otimes\vec b\cdot\vec\sigma)]\) is linear in each direction, averaging over directions is the same as measuring along the averaged Bloch vector, an unsharp observable of length \(\mu\le1\). For a uniform cap of polar width \(\Theta\), which is the patch shape that maximises \(\mu\) at fixed area, \(\mu=\cos^2(\Theta/2)\), and for a maximally entangled state the averaged CHSH value is \(2\sqrt2\,\mu^2\). Violation requires \(\Theta<47^\circ\), or \(65^\circ\) if only one party fluctuates. For the von Mises–Fisher distribution, the spherical analogue of a Gaussian, \(\mu=\coth\kappa-1/\kappa\) and the threshold is \(\kappa\approx6.285\). To leading order \(\mu\approx1-\sigma^2/2\) for either distribution, so the angular spread \(\sigma\) alone decides whether a violation is possible.

When directions are drawn uniformly from the whole sphere and are outside the hidden variables' control, they can be grouped after the experiment. The octahedral, dodecahedral and icosahedral tilings give tile-averaged directions of length 0.866, 0.916 and 0.943, all enough for a CHSH violation. With the 20 icosahedral tiles as settings, a ten-term inequality reached \(23.3\pm0.3\) against 20, and a combination of 1728 CHSH expressions, with critical visibility 0.781 for perfectly repeatable settings, reached 3941 against the local 3456 and the predicted 3932.9.

The paper also marks the limit of the approach. If the directions are entirely unknown and many pairs are measured per direction, the joint distribution of local averages and correlations always has an explicit local model; the same holds if only the polar angle of each direction is known. Some partial knowledge that groups nearby directions is necessary. The authors suggest that a resource theory of reference frames and measurement control would formalise this.

Transferable methods

1. Non-repeatability as an unsharp measurement

   The analysis needed a way to compare quantum predictions for a cloud of directions with the unchanged local bound. Linearity of the correlation function in each Bloch vector reduces any cloud to a single shortened vector, so all the physics is in one number per setting, the shrinking factor \(\mu\).

   With independent local densities \(\alpha_k\) and \(\beta_l\), \[E_{kl}=\int d\vec a\,d\vec b\,\alpha_k(\vec a)\beta_l(\vec b)\,\mathrm{Tr}[\rho(\vec a\cdot\vec\sigma\otimes\vec b\cdot\vec\sigma)]=\mathrm{Tr}[\rho(\vec\alpha_k\cdot\vec\sigma\otimes\vec\beta_l\cdot\vec\sigma)],\] where \(\vec\alpha_k=\int\alpha_k(\vec a)\,\vec a\,d\vec a\). For a cap of polar width \(\Theta\) about \(\vec a_k\), \[\mu=\frac{1}{4\pi\sin^2(\Theta/2)}\int_0^{2\pi}\!d\phi\int_0^\Theta\!\sin\theta\cos\theta\,d\theta=\cos^2\frac{\Theta}{2},\] so the optimal CHSH value becomes \(2\sqrt2\cos^4(\Theta/2)\), which exceeds 2 exactly when \(\cos(\Theta/2)>2^{-1/8}\).

   Why a cap is optimal: for a region \(S\) of fixed area, \(\mu=\frac1{\mathcal A}\int_S\cos\theta\,ds\) measured from the mean direction. Keeping the points with the largest \(\cos\theta\) maximises the integral, and those points form a cap.

   Editorial checks: the thresholds come out as \(\Theta_{\mathrm{LHV}}=47.02^\circ\) and \(65.53^\circ\) for one-sided spread; the von Mises–Fisher condition \((\coth\kappa-1/\kappa)^2=1/\sqrt2\) gives \(\kappa=6.2849\), with 86.5% of that distribution within \(47^\circ\); Monte Carlo averages over \(2\times10^6\) uniform directions give tile lengths 0.866, 0.916 and 0.943. All agree with the paper.

   How to reuse it

   Any expectation-value witness measured with a drifting reference frame can be analysed the same way: replace each nominal direction by the averaged Bloch vector, and the witness tolerates drift until the product of shrinking factors reaches its critical visibility.

2. Sharpness against statistics when settings are selected afterwards

   If directions are uniformly random, a setting can be defined afterwards by keeping only runs whose directions fall in an acceptance patch. A small patch gives sharp measurements but few runs; a large one gives many runs but blurred correlations. The paper finds the optimum and the minimum number of runs needed.

   A run lands in a given pair of caps with probability \(\sin^4(\Theta/2)\), so each correlation is estimated from about \(N_{\mathrm{tot}}\sin^4(\Theta/2)\) runs with variance \((1-\bar E^2)/[N_{\mathrm{tot}}\sin^4(\Theta/2)]\), where \(\bar E=\mu^2/\sqrt2\). Four independent correlations enter, giving \[\Delta=\sqrt2\left(\frac{2-\cos^8(\Theta/2)}{N_{\mathrm{tot}}\sin^4(\Theta/2)}\right)^{1/2},\qquad V=\frac{2\sqrt2\cos^4(\Theta/2)-2}{\Delta}.\] Since \(V\propto\sqrt{N_{\mathrm{tot}}}\), the width that maximises it does not depend on the number of runs.

   Editorial check: maximising \(V\) numerically gives \(\Theta_{\mathrm{opt}}=31.53^\circ\); \(V=1\) at the optimum needs \(N_{\mathrm{tot}}\approx2554\) runs; and for \(10^4\), \(10^5\), \(10^6\) and \(10^9\) runs the window with \(V\ge1\) is \(16.51\)–\(42.72^\circ\), \(8.64\)–\(45.83^\circ\), \(4.77\)–\(46.65^\circ\) and \(0.84\)–\(47.00^\circ\), matching Table 2.

   Tiling removes the waste: every direction belongs to one tile, and each tile is a setting with shrinking factor \(\mu_{20}=0.943\). For the 1728-term inequality the ideal quantum value is about 4424 and the tiled prediction \(\mu_{20}^2\times4424\approx3933\); the observed 3941 lies within 0.2% of it. Editorial note: \((3941-3456)/19\approx25.5\), not the 19.5 standard deviations stated in the text.

   How to reuse it

   Post-selecting settings from random data is a general trade between resolution and sample size. Writing the significance as an effect size times \(\sqrt N\) shows that the best resolution is independent of \(N\); only the minimum usable \(N\) depends on it.

Source: Eqs. (1)–(6); Methods: spherical cap as the optimal patch; trade-off in the acceptance-patch approach (Eqs. (19)–(20), Table 2); Eqs. (7)–(11). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Device-independent protocols that rest on Bell violations do not need exactly repeatable settings: imperfect repeatability acts as an unsharp measurement, and the paper gives the width that can be tolerated. It also separates what randomised measurements can certify. Distributions of correlation values from unknown random directions reveal entanglement but never nonlocality; some knowledge of where each direction lies is required.

Limitations. Using directions drawn from the whole sphere requires that the hidden variables cannot influence which direction is realised; with acceptance patches, which discard most runs, a local model could otherwise exploit the selection, as in the detection loophole. The experiment records coincidences only and does not address the locality or detection loopholes. The text calls the 1728-term violation 19.5 standard deviations, but the quoted \(3941\pm19\) against 3456 is about 25; one of the two figures is misprinted or uses a different error estimate. The paper has no AI-use statement.

Description based on the full arXiv HTML of v1, read 2026-11-14: the whole paper, including the Methods sections on the optimal patch, the von Mises–Fisher model, the acceptance-patch trade-off, the icosahedral directions and the local models for correlation distributions. The critical widths, the von Mises–Fisher threshold, the tile shrinking factors, the acceptance-patch optimum and the run counts in Table 2 were recomputed.