No-Go Theorem Assumption Index

A no-go result has the form “no model in this specified class reproduces these predictions.” To use one reliably, identify the class before quoting the conclusion. This index emphasizes the assumptions that distinguish the results; the theorem matrix offers compact cross-comparison and the linked explanations develop the arguments.

Which object is constrained?

Result	Mathematical object	Premise that needs particular attention	Conclusion
Bell	Joint conditional outcome distribution	Bell factorization and setting independence	Some quantum correlations admit no such local model
Kochen–Specker	Global sharp-event valuation	Same value across contexts, functional consistency, dimension at least three	No global valuation satisfying these rules
PBR	Preparation measures on ontic states	Product preparation independence	Distinct pure states have no common ontic probability mass
Gleason	Probability measure on projections	Context-independent assignment, additivity and dimension	Probabilities have a density-operator trace form
No-broadcasting	One physical channel on a state family	Exact preservation of both output marginals	The input family must commute
Leggett–Garg	Pairwise temporal correlations	Definite values and noninvasive measurability	Correlations fit a common joint-distribution bound

Gleason is a representation theorem, though it can imply no-go corollaries. Wigner and Stone–von Neumann are also positive classification results. Calling every foundational theorem a prohibition obscures what it actually establishes.

Three meanings of independence

Measurement independence in a Bell model is $\mu(\lambda\mid x,y)=\mu(\lambda)$: settings do not alter the hidden-variable distribution being averaged. Preparation independence in PBR factorizes the underlying state distribution of separately prepared systems. Noncontextuality reuses a representation for operationally identical events or procedures across their allowed contexts. None is a synonym for the others.

As a diagnostic, a model can assign the wavefunction as an ontic variable and still violate Bell factorization. Conversely, an ordinary classical hidden-variable distribution may satisfy setting independence while allowing invasive temporal measurements. Its Bell and Leggett–Garg properties must be assessed separately.

Exact theorem, noisy witness, and experiment

First specify an exact mathematical prediction. Next choose a robust witness with an explicit noise tolerance. Finally justify why the observed procedure estimates that witness. This progression adds assumptions about sampling, compatibility, losses, interventions and uncertainty.

For example, the CHSH theorem bounds an expectation $S\le2$, not every finite observed score. A statistical rejection quantifies the probability of a score at least as extreme under a declared local-model null and trial protocol. A small p-value is not the probability that a particular interpretation is true. Bell Loopholes explains how protocol defects can change the relevant null class.

How to assess an escape route

Rejecting a premise is logically legitimate, but the resulting theory must say what replaces it. A contextual theory needs a rule for context-dependent outcomes. A preparation-correlated theory needs a composition law. A collapse theory needs stochastic dynamics and probabilities. The next question is whether the replacement is internally consistent and empirically adequate, not whether it retains a familiar label such as realism.

To see why an assumption matters, construct a counterexample when it is removed. The projective qubit valuation in Kochen–Specker, commuting broadcasters, and invasive temporal classical models show that the qualifications are doing work.

Exercise: locate the missing premise

Someone argues: “The two laboratories prepare a product density matrix, therefore PBR proves their underlying states are independent.” Identify the reversed implication.

Solution

A product quantum state gives factorized quantum statistics for local measurements. Preparation independence asserts a product law for underlying variables. A hidden correlation may be invisible to the allowed measurements. PBR takes the ontic factorization as a premise; it does not derive that premise from the product density matrix.

References

• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014), doi:10.1103/RevModPhys.86.419 — correlation-model assumptions.
• Leifer, Matthew S., “Is the Quantum State Real? An Extended Review of $\psi$-Ontology Theorems,” Quanta 3, 67–155 (2014), doi:10.12743/quanta.v3i1.22 — ontological models and composition.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005), doi:10.1103/PhysRevA.71.052108 — operational noncontextuality.