Foundations of quantum mechanics · Theory
Spurious quantum correlations
arXiv:2609.04157v1 ↗PDF ↗Submitted 3 September 2026
- Question
- Can correlations that appear nonclassical acquire a classical explanation when the assumed causal structure changes?
- Result
- The authors identify correlations with classical explanations in alternative causal structures without fine tuning, and other correlations that resist such explanations.
The paper in detail
A causal structure is a directed acyclic graph \(H\) whose nodes are observed variables and latent common causes. A distribution \(P\) over the observed nodes is classically compatible with \(H\) if it is the marginal of a joint distribution that factorizes along the graph, \[Q(x_1,\dots,x_m)=\prod_iQ\bigl(x_i\,\big|\,\mathrm{pa}(x_i)\bigr),\] where \(\mathrm{pa}(x_i)\) are the parents of \(x_i\). It is quantum compatible if latent nodes are quantum states shared among their children and observed nodes are measurements. Every compatible \(P\) obeys the conditional independences that d-separation reads off the graph: if \(Z\) d-separates \(X\) from \(Y\), then \(P(X|Y,Z)=P(X|Z)\). Writing \(\mathcal C_H\), \(\mathcal Q_H\) and \(\mathcal I_H\) for the classical, quantum and independence-respecting sets, \[\mathcal C_H\subseteq\mathcal Q_H\subseteq\mathcal I_H,\] and \(H\) is called non-classical when \(\mathcal C_H\subsetneq\mathcal I_H\).
A violated causal inequality shows \(P\notin\mathcal C_H\). But is \(H\) the right graph? The authors call \(P\) non-classical and not fine-tuned (NCNFT) in \(H\) if it lies in \(\mathcal I_H\setminus\mathcal C_H\) and every independence it satisfies is implied by d-separation in \(H\). Such a \(P\) is spurious if some other graph \(G\) explains it classically, again without fine tuning. A spurious quantum correlation is evidence against the classical model on \(H\), but not against classical physics.
Results. Consider a structure \(H_1\) with latents \(A,B\) and observed \(C,D,E,F\), factorizing as \(Q(a)Q(b)Q(d)Q(c|ab)Q(e|bd)Q(f|ace)\). Every NCNFT correlation in \(H_1\) is spurious, with an alternative graph that respects the same causal order. A quantum example with angles \(7\pi/32\), \(6\pi/32\) and \(\pi/32\) reaches about \(2.128\) on the instrumental-type bound \[P(E{=}F|D{=}0)+P(F{=}0|D{=}1)+P(E{=}0,F{=}1|D{=}2)\le2,\] so it is genuinely NCNFT in \(H_1\). The authors report spurious quantum correlations for further structures, including the triangle and instrumental scenarios. In contrast, NCNFT correlations are not spurious for a structure \(H_4\) or for bilocality. For the Bell structure, the paper relies on Wood and Spekkens, who showed that it admits no spurious quantum correlations.
The analysis assumes the exact distribution is known (i.i.d. sampling with no statistical error). It restricts candidate explanations in the same way as the IC* algorithm: latent common causes have at most two children and no parents.
Transferable methods
Search over causal explanations
Starting from the observed conditional independences, the authors use the IC* causal-discovery algorithm to list classical graphs with the same independence constraints. A short set-inclusion argument then decides whether every non-classical correlation of \(H\) is spurious.
The key argument is the paper’s Theorem 1. Suppose a second graph \(G\) satisfies \[\mathcal I_G=\mathcal I_H,\qquad\mathcal C_H\subsetneq\mathcal I_H,\qquad\mathcal C_G=\mathcal I_G.\] Take any \(P\) that is NCNFT in \(H\). Then \(P\in\mathcal I_H=\mathcal I_G=\mathcal C_G\), so \(G\) explains \(P\) classically. Because \(H\) and \(G\) imply the same independences, an independence of \(P\) is implied by d-separation in \(G\) exactly when it is implied in \(H\), so faithfulness carries over. Every NCNFT correlation of \(H\) is therefore spurious.
The work lies in finding \(G\). Independence data determine a graph only up to an equivalence class, and IC* returns its members: graphs whose d-separation relations among the observed nodes match exactly the independences of \(P\). The condition \(\mathcal C_G=\mathcal I_G\) says that the independences alone characterize the classical set of \(G\), so no Bell-type inequality constrains \(G\). The paper proves this for its graph \(G_1\) (Lemma 1).
How to reuse it
Treat a nonclassicality witness as a statement about a pair \((P,H)\), not about \(P\) alone. Before claiming a quantum advantage from \(P\notin\mathcal C_H\), list graphs \(G\) with the same d-separation relations and check whether \(\mathcal C_G=\mathcal I_G\). If such a \(G\) exists, the violation refutes the causal hypothesis but not classicality. Structures with no such \(G\) (the paper’s \(H_4\) and bilocality) are those where nonclassicality is robust.
Separate statistical fit from physical admissibility
A candidate explanation must do more than reproduce the data. It must not rely on finely tuned parameters, and, when locations are known, its arrows must respect spacetime order.
Faithfulness. \(P\) is fine-tuned with respect to a graph if it satisfies an independence \(P(X|Y,Z)=P(X|Z)\) that d-separation does not imply. In a classical model, such an independence holds only if the conditionals \(Q(x_i|\mathrm{pa}(x_i))\) satisfy extra polynomial equations, which pick out a measure-zero subset of parameter space. This is the standard reason causal discovery excludes fine tuning. Requiring \(G\) to explain \(P\) faithfully prevents \(G\) from reproducing an independence by cancellation.
Spacetime. If each variable has a location, a graph is admissible only when every edge points forward in a partial order \(\prec_{\mathrm{st}}\) compatible with light cones: \(X\to Y\in G\Rightarrow X\prec_{\mathrm{st}}Y\). The strongest results in the paper require \(G\) to have the same causal partial order as \(H\), so the classical explanation needs no influence that \(H\) itself forbids.
How to reuse it
Use independence tests and experimental timing together to assess alternative explanations. Matching independences alone does not prove that a graph reproduces the full distribution. The paper’s analysis also assumes exact distributions, unlike finite experimental samples.
Context & limitations
Why it matters. Clarifies which causal assumptions underpin claims of quantum nonclassicality.
Limitations. A preliminary theoretical draft; this does not invalidate Bell’s theorem under its usual assumptions.