Contextuality in Mathematical Form

A family of probability distributions can agree on every shared marginal without being the marginals of one global distribution. This extension problem gives a precise finite mathematical framework for contextuality. The canonical contextuality article separates traditional sharp-value contextuality from generalized operational noncontextuality. Here we study a fixed measurement cover and specify exactly what its global-extension test does and does not establish.

Required background. Contextuality explains why shared measurement labels require an operational identification; Measure Theory Minimum supplies marginal probability distributions.

A measurement cover and its empirical distributions

Let $X$ be a finite set of measurement labels, with finite outcome set $O_x$ for each $x$. A cover $\mathcal M$ is a collection of subsets $C\subseteq X$ whose union is $X$. Each $C$ is a context that can be measured jointly in the chosen experimental model. Its outcome assignments form

$$O_C=\prod_{x\in C}O_x.$$

An empirical model consists of normalized distributions $e_C$ on these sets. Compatibility of its marginals means

$$(e_C)|_{C\cap D}=(e_D)|_{C\cap D} \qquad(C,D\in\mathcal M),$$

where restriction sums over coordinates outside the intersection. This is often called no-disturbance. It identifies the measured statistics of a shared event across contexts, not a claim that every sequential laboratory implementation leaves an underlying state undisturbed.

Global distributions and deterministic assignments

A global assignment is a tuple $s\in O_X$. A nonnegative normalized distribution $q$ on $O_X$ extends the empirical model if

$$e_C(t)=\sum_{s:s|_C=t}q(s) \qquad(t\in O_C).$$

Such a $q$ is exactly a probabilistic mixture of deterministic global assignments: sample $s$ and answer any context by its restriction. Conversely, a mixture of those assignments gives this formula. This proves the equivalence between extension and that specified noncontextual hidden-variable class, without postulating an experimentally accessible simultaneous measurement of every label.

Conditional stochastic responses that factorize over individual labels also yield a global extension: integrate their product distribution over the hidden variable. Whether such factorization or deterministic responses are warranted is a question about the physical noncontextuality model. Generalized operational noncontextuality for preparations, transformations, and unsharp measurements is not exhausted by this one extension test.

A linear feasibility problem

Index rows by context-outcome pairs $(C,t)$ and columns by global assignments $s$. Set

$$A_{(C,t),s}=\mathbf1_{s|_C=t}.$$

Then extension asks for $Aq=e$ with $q\ge0$. Summing the rows of any one nonempty context block enforces $\sum_s q(s)=1$, because the empirical distribution in that block is normalized. The number of unknowns is $\prod_x|O_x|$, so brute-force enumeration grows rapidly.

A feasible vector provides a certificate by direct substitution. An infeasibility certificate can be a linear inequality satisfied by every column assignment and violated by $e$; the Bell separation argument explains why such inequalities exist for the finite convex hull. Numerical solver output should be accompanied by residual tolerances or an exact certificate. Tiny negative entries are not silently valid probabilities.

Compatible pairs without a global triple

Consider three binary labels $X=\{A,B,C\}$ and contexts $AB,BC,CA$. On each pair, put probability $1/2$ on each of its two opposite-bit outcomes and zero on equal bits. Every single-label marginal is uniform, so the three pair distributions agree on intersections.

A putative global distribution would have probability one on each event $A\ne B$, $B\ne C$, and $C\ne A$. The intersection of these three probability-one events must also have probability one. But for binary values the first two imply $A=C$, so the intersection is empty. No global distribution exists.

This is a valid abstract marginal obstruction. It is not automatically a quantum sharp-measurement realization. Three pairwise commuting PVMs commute jointly and supply a common joint spectral measure, which would provide the missing extension. Implementing an abstract scenario with unsharp joint measurements introduces further constraints and assumptions. An incidence pattern alone does not prove quantum realizability.

Support obstructions and probability obstructions

The support of $e_C$ is the set of context assignments with nonzero probability. If no global assignment restricts into every support, then no global probability extension can exist: any positive-weight global assignment in an extension would have to do so. This is a strong contextuality obstruction, exhibited by the binary triangle above.

The converse fails. Supports may permit global assignments while their required probability weights cannot be matched by any common mixture. For example, noisy CHSH correlations can have strictly positive entries in every pair distribution while still violating a Bell inequality. CHSH as a marginal problem makes this distinction explicit. Testing only which outcomes are possible loses constraints carried by their probabilities.

Exercise: a quantitative triangle inequality

Show that every global distribution on three bits satisfies $\Pr(A\ne B)+\Pr(B\ne C)+\Pr(C\ne A)\le2$. What does this say if each observed pair has anticorrelation probability at least $1-\varepsilon$?

Solution

At each deterministic triple, either all bits agree and no pair differs, or exactly one bit differs from the other two and two pairs differ. Averaging bounds the sum by two. The observed lower bound would give $3(1-\varepsilon)\le2$, so a global extension requires $\varepsilon\ge1/3$. A smaller value certifies an extension obstruction under the stated shared-label and joint-context assumptions; it still does not certify that a particular laboratory device realizes the abstract scenario exactly.

References

• Abramsky, Samson, and Adam Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New Journal of Physics 13, 113036 (2011), doi:10.1088/1367-2630/13/11/113036, arXiv:1102.0264 — measurement covers, global extensions, and the hierarchy of contextuality obstructions.
• Fine, Arthur, “Hidden Variables, Joint Probability, and the Bell Inequalities,” Physical Review Letters 48, 291–295 (1982), doi:10.1103/PhysRevLett.48.291 — the binary Bell marginal-extension theorem.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005), doi:10.1103/PhysRevA.71.052108 — the broader operational framework and its different premises.