No-Go Theorem Matrix

A no-go result excludes a conjunction of assumptions. Reading its conclusion without its hypothesis list can turn a precise theorem into an unsupported metaphysical slogan. Use this table for a quick comparison, and the No-Go Theorem Assumption Index to understand how the premises differ.

Impossibility and representation results

Result	Essential setting	Conclusion	Does not establish
Bell	Local factorization, setting independence, specified outcomes and statistics	Some quantum correlations have no model satisfying the assumptions	Controllable faster-than-light signaling
Kochen–Specker	Sharp projectors in dimension at least three with functional consistency	No global context-independent value assignment of the required kind	That every hidden-variable theory is impossible
Gleason	Additive probabilities on projections, standard dimension hypotheses	Density-operator representation of those probabilities	A probability interpretation without probability assumptions
PBR	Ontological model plus preparation independence and quantum predictions	Distinct pure states cannot overlap in the stipulated ontic representation	That all epistemic language or all alternatives are excluded
No-cloning	The same deterministic channel must make two exact copies of every input in the set	Two distinct nonorthogonal pure states cannot both be cloned	That known states, orthogonal labels, or every probabilistic restricted task are excluded
No-broadcasting	One deterministic channel preserves every input exactly in both marginals	A finite-dimensional family is broadcastable iff its states commute	That approximate copying is forbidden
Leggett–Garg	Macrorealist premises including the relevant noninvasiveness condition	Bounds on temporal correlations	That every violation excludes every notion of realism
Wigner	A bijection of rays preserving transition probabilities	Unitary or antiunitary implementation under the theorem’s conditions	Selection of a Hamiltonian or collapse rule
Stone–von Neumann	Finite canonical degrees of freedom, regular irreducible Weyl representation	Uniqueness up to unitary equivalence in the stated class	Uniqueness for quantum fields or arbitrary nonregular representations

Gleason, Wigner, and Stone–von Neumann are representation or structure theorems, not all “no-go theorems” in the same sense. They appear here because foundational arguments frequently invoke their restrictions.

Three kinds of assumption

Kinematic assumptions specify the mathematical objects, such as Hilbert spaces, rays, or projector events. Operational assumptions specify allowed procedures, independence, compatibility, and probability assignments. Interpretive assumptions connect mathematical variables to a complete physical description or definite properties.

These categories can interact. In Bell’s theorem, the factorization equation acquires its causal meaning through the interpretation of the conditioning variables. In contextuality, operational equivalences determine which differences a model is required to ignore.

A diagnostic use

When a model claims to evade a theorem, identify the failed premise and show how the model realizes that failure. Merely saying that the model is “nonclassical” does not locate the logical change. Conversely, violating a premise does not automatically make the resulting model coherent or empirically successful.

An experiment additionally needs a bridge from actual procedures to the ideal theorem. Sampling, detector events, sequential disturbance, and finite statistics belong to that bridge.

References

• Bell, John S., “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964). Paper.
• Budroni, Costantino, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, “Kochen–Specker contextuality,” Reviews of Modern Physics 94, 045007 (2022). Review.
• Pusey, Matthew F., Jonathan Barrett, and Terry Rudolph, “On the reality of the quantum state,” Nature Physics 8, 475–478 (2012). Paper.