What No-Go Theorems Do and Do Not Prove

A no-go theorem excludes an explicitly defined mathematical possibility. Its force depends on the objects quantified over, the constraints imposed on them, and the required accuracy. It can decisively rule out a model class without deciding which alternative describes nature. The practical skill is to translate a claim back into those mathematical choices before drawing physical or philosophical conclusions.

Helpful background. Foundations Theorems Overview separates representation theorems, transformation obstructions, and probability-model obstructions.

A no-go statement excludes a conjunction

Let $\mathcal M(H)$ be the class of models satisfying hypotheses $H$, and let $T$ be a target prediction or task. A typical theorem says

$$\text{there is no }m\in\mathcal M(H) \text{ that realizes }T.$$

If the target is empirically successful, at least one defining condition of that model class must be relinquished or revised. The theorem alone does not identify a unique condition to abandon. In Bell’s theorem the class includes an explicit source law and a local response factorization; the word “realism” cannot replace those equations without changing the statement. In the PBR argument, product preparation independence is a different assumption about model composition.

Finding a model outside $\mathcal M(H)$ is not a counterexample to the theorem. It can be a scientifically interesting alternative, but its altered hypotheses should be declared. Conversely, merely naming a different interpretation does not establish that its mathematical predictions and composition laws evade a particular theorem.

Quantifier order can change the physical task

The no-cloning claim concerns one channel working for every member of a promised family:

$$\exists\mathcal C\;\forall\psi\in S: \quad\mathcal C(\rho_\psi)=\rho_\psi^{\otimes2}.$$

This is very different from allowing a separate channel for every known input. For each fixed $\psi$, the replacement channel $\mathcal C_\psi(\rho)=\operatorname{Tr}(\rho)\rho_\psi^{\otimes2}$ is physical. It prepares a known target and does not solve the common-device task. The canonical theorem keeps that quantifier order explicit.

Likewise, a separate value choice within each orthonormal basis is easy. Kochen–Specker forbids one assignment that agrees across all shared projectors. Allowing the value to depend on the context changes the global consistency requirement. This is why a collection of individually consistent measurements can still have a global obstruction.

Exact impossibility and approximate performance

An exact no-go statement does not automatically provide a useful error bound. Excluding exact equality might still allow a sequence of admissible models approaching the target arbitrarily well. To rule out approximation one needs a topology, a distance, and a positive separation margin.

For a fixed finite Bell scenario, compactness of the local polytope ensures that every exterior probability vector has positive distance from it. A separating witness then supplies an explicit bound. The statement changes if the scenario, allowed dimension, or measurement family changes along an approximating sequence.

For cloning, trace-distance contraction gives a positive two-state error bound under a declared joint-output metric. A heralded exact cloner solves another task because it can fail. Its success probability and flag are part of the instrument, as shown in the quantitative cloning extension. Dropping failure events from an average is not an exact deterministic implementation.

Relating an exact theorem to an experiment

An exact theorem uses exact probability or operator relations. An experiment supplies finite data under an apparatus and sampling model. Statistical uncertainty, loss, operational equivalences, and unintended context dependence affect how those data test the theorem’s model class. A reported confidence level is a property of a specified statistical test, not a modification of the deductive proof.

The converse separation matters too: an experimental imperfection does not invalidate a correct mathematical theorem. It can weaken that experiment’s ability to distinguish its competing physical models. A careful account states both the ideal implication and the observed data, then explains the link between them.

What remains open after an exclusion

A no-go result can leave many positive tasks unresolved: constructing an alternative model, identifying its empirical differences, analyzing its computational cost, or proving stability under noise. It need not explain why one outcome occurs, derive every part of the quantum formalism, or select a unique ontology. These are separate questions, not weaknesses in the theorem’s actual conclusion.

When reading a proposed violation, first identify its input family, permitted transformations, accessible records, and approximation metric. Then compare those objects with the original hypotheses. This turns a debate over slogans into a checkable mathematical comparison.

Exercise: impossibility without an error gap

Let the admissible outputs be $S=(0,1)$ and the target be $0$, with ordinary absolute distance. Contrast this with an exterior point of a compact finite-dimensional local polytope.

Solution

No element of $S$ equals zero, yet $1/n\in S$ for $n>1$ converges to it. The exact task is impossible while the infimum error is zero. A compact polytope is closed, so an exterior point has a strictly positive distance: a minimizing sequence has a convergent subsequence in the polytope, and zero distance would put the target inside. The closure assumption is what turns exclusion into a nonzero approximation gap in the second case.

References

• Bell, John S., Speakable and Unspeakable in Quantum Mechanics, 2nd edition, Cambridge University Press, 2004, doi:10.1017/CBO9780511815676 — precise model premises and their physical interpretation.
• Pusey, Matthew F., Jonathan Barrett, and Terry Rudolph, “On the reality of the quantum state,” Nature Physics 8, 475–478 (2012), doi:10.1038/nphys2309 — a composition-dependent no-go theorem.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005), doi:10.1103/PhysRevA.71.052108 — operationally explicit model-class distinctions.