Kolmogorovian Censorship, Predictive Incompleteness, and the Locality Loophole in Bell Experiments

TL;DR

The paper explores quantum probabilities through the lens of Kolmogorovian Censorship and predictive incompleteness, arguing for a framework that preserves locality while explaining Bell inequality violations.

Contribution

The paper introduces predictive incompleteness as a novel framework for quantum probabilities that preserves relativistic locality.

Findings

Kolmogorovian Censorship identifies classical probabilities within fixed measurement contexts but does not resolve conceptual tensions in Bell experiments.

Predictive incompleteness offers a minimal framework matching experimental practice while preserving locality.

The paper clarifies logical relations among quantum probability interpretations and justifies moving from Kolmogorov’s to Gleason’s framework.

Abstract

We revisit the status of quantum probabilities in light of Kolmogorovian Censorship (KC) and the Contexts, Systems, and Modalities (CSM) framework, and we discuss KC-based ideas with respect to superdeterminism, counterfactuality, and predictive incompleteness. After briefly recalling the technical content of KC and its scope, we show that KC correctly identifies that probabilities are classical within a fixed measurement context but does not by itself remove the conceptual tension that motivates nonlocal or conspiratorial explanations of Bell inequality violations. We argue that predictive incompleteness—the view that the quantum state is operationally incomplete until the measurement context is specified—provides a simple, minimal, and explanatory framework that preserves relativistic locality while matching experimental practice. Finally we clarify logical relations among these positions, highlight the assumptions behind them, and justify the move from Kolmogorov’s to Gleason’s framework for quantum probabilities.

Full text

1. Introduction

In the scientific literature there are still debates to decide whether quantum mechanics (QM) is contextual [1,2] or noncontextual [3] and whether quantum probabilities fit within the usual Kolmogorovian framework [4,5] or not [6,7,8,9]. For the sake of completeness, the classical Kolmogorov axioms are briefly presented in Appendix A.

To introduce the debate, it can be said that in quantum mechanics the standard Kolmogorov axioms of positivity, normalization, and countable additivity are still true but that the structure of the event space is no longer the same: technically, it is no longer a $[eqn]$ -algebra; see Appendix A for a definition. This can be interpreted in different ways, and a milestone in this debate is the so-called Kolmogorovian Censorship (KC) [4,10,11], stating that quantum probabilities can be identified with classical, Kolmogorovian probabilities when the measurement context has been specified or more generally when a classical probability distribution of measurement contexts has been given.

So at this introductory level, one may say that troubles appear when trying to put together several incompatible contexts—this is what is forbidden by the Censorship. The KC sounds rather obvious on a physical intuitive basis, and technically it has been demonstrated for a countable number of measurement contexts [10,11]. It also has some relationship with different views on QM, such as superdeterminism [12,13], as will appear below. In the present discussion we will assume usual relativistic (forward) causality and omit retrocausal models; see, e.g., [14,15] for reviews.

2. Kolmogorovian Censorship and Bell Experiments

Looking in more detail, there is some controversy regarding the physical meaning of the KC, as we will explain now. In particular, Szabó et al. [4,5] claim that QM probabilities are Kolmogorovian, not only in a single context, but also in a loophole-free Bell experiment, where a random choice among four different contexts is implemented [16]; this well-known scheme is detailed in Appendix B. The main point made by Szabó et al. is that each actual measurement context has probability 1/4 of occurring, so that Bell’s S parameter is reduced from $[eqn]$ down to $[eqn]$ (see Appendix B), which is below 2, as would be expected classically. Then they conclude that Bell’s inequalities (BIs) are either irrelevant (when considering the four contexts separately) or not violated, when considering the four contexts together.

More precisely, it is uncontroversial that the probabilities predicted by QM in a given context, or commuting subalgebra, are Kolmogorovian; the problem arises when trying to “gather” probabilities predicted by QM in different contexts. Said otherwise, in any given context it is possible to build a Kolmogorovian probability distribution, or equivalently a hidden variable theory; however this distribution must be matched to the actual context.

In a loophole-free Bell test [16], the remote random choices of measurements (polarizer orientations) are designed to forbid that this matching may be performed by a relativistically causal transmission between the source and the detectors. Despite these barriers, the matching might be obtained either by an instantaneous influence between the source and the detectors (explicit nonlocality) or by assuming that it is pre-established before the actual experiment takes place (superdeterminism). One has then the following options:

• 1.Szabó et al. [4,5] claim that the only relevant way to interpret QM probabilities in different contexts is to consider a classical probability distribution over these different contexts. This corresponds by construction to a feasible experiment, typically a loophole-free Bell experiment, including the random choice among four different contexts [16]. Then the global probability distribution is Kolmogorovian, and Bell’s inequalities (BIs) are not violated, because as written above the probabilities in each context are divided by four, as well as the resulting S value. However a drawback of this approach is that the pre-established matching between the source and the measurements is still required, through some kind of “global determinism” [17], redefining the proper probability distribution for each context. Therefore, though BIs are not violated, the basic problem of the origin of the matching is still present.
• 2.Rather than considering a probability distribution over the contexts, one may calculate the correlation function in each context, which is the very idea of a loophole-free Bell test, and gather them in Bell’s S value. However, one may argue that the four correlation functions correspond to four different incompatible experiments, and thus bringing their results together is counterfactual because they cannot be measured simultaneously; then BI cannot be demonstrated. This is a standard answer to the problem, but it remains debated; see below for further explanations.
• 3.Still an alternative way is to admit that the results in the four contexts can be combined, as would be the case classically since they apply to the same system; then one obtains Bell’s inequalities, which are experimentally violated, so one should explain why. We consider that this is a meaningful question, and there are basically three options, spelled out in detail in [18] (see also Appendix C):
• 3a, 3b—in the spirit of option 1 above, a way to violate Bell’s inequalities is by admitting that the system’s parameters and the orientations of the polarizers are not independent variables, despite the fact that these orientations are chosen randomly and independently at a large distance. This can be obtained either by (3a) admitting superdeterminism, i.e., denying the possibility of independent random choices [17], or by (3b) admitting a nonlocal influence between the source and the measurements. These two options agree with QM, and they have recently been shown to be equivalent [19]. However, they are in our opinion equally undesirable [18]—though they are matter of ontological choice and cannot be proven wrong.
• 3c—the third option, known as predictive incompleteness [18,20], is to recognize that the quantum state by itself is not enough to specify the measured probability distribution, as long as the context has not been specified. This contradicts predictive completeness (also called outcome independence), which is a required hypothesis for Bell’s theorem, and therefore BI cannot be demonstrated. We note that another hypothesis in BI proof, called elementary locality or parameter independence, is fully respected, in agreement with relativistic causality. Then telling whether or not predictive incompleteness is a form of nonlocality is a matter of definition; it may be called Bell nonlocality, not forgetting the above features. This conclusion corresponds to the detailed analysis presented in [18] (see also Appendix C) and also to the more general framework presented in [21] to “complete” the usual quantum state by specifying the measurement context. These two papers, as well as the arguments above, are consistent within the general quantum framework called CSM (Contexts, Systems, and Modalities) [22,23].

3. Discussion: From Kolmogorov’s Axioms to Gleason’s Theorem

3.1. Kolmogorovian Censorship vs. Predictive Incompleteness

From the above it should be clear that the ontologies underlying either Szabó’s position or CSM are quite different. Szabó et al. claim that QM probabilities are Kolmogorovian and that QM can ultimately be seen as a (super)deterministic theory. On the other hand, CSM stipulates that QM is fundamentally non-deterministic, due to the conjunction of quantization and contextuality [24,25], and that quantum probabilities are non-Kolmogorovian, unless restricted to a single context according to the KC.

This is why the (non-Kolmogorovian) predictive incompleteness of $[eqn]$ is useful: it leaves enough freedom so that the independent choice of the measurement can contribute to the determination of the observed probability distributions, avoiding both explicit nonlocality and superdeterminism. It is worth emphasizing again, as spelled out in [18], that predictive incompleteness makes no sense in classical physics and appears as a specific quantum feature, thereby able to consistently explain the violation of Bell’s inequalities.

3.2. What About Non-Measurable Subsets?

Let us note that the main proof in [5] is based on a theorem in Pitowsky’s book [26], which also speaks about the violation of Bell’s inequalities using non-measurable subsets of local hidden variables (Chapter 5 in [26]). These kinds of ideas were developed more recently [12,13] and are very controversial [27,28]. We do not want to enter into this debate here, but we note the following sentence by Palmer [13]: “Importantly, this means a non-conspiratorial interpretation of (our approach) implies that physical theory does not have the post-hoc property of counterfactual definiteness. (…) This important point seems to have been lost in more recent discussions of Bell’s Theorem.”

This gives a major hint about the meaning of Palmer’s approach: if counterfactual definiteness—that is, treating in statistical calculations the results of unperformed measurements on an equal footing with actual results—is rejected, then Bell’s inequalities do not hold any more (option 2. above); this is well known and has not “been lost in more recent discussions of Bell’s Theorem”. Then the work in [12,13] may be seen as a possible way to justify a rejection of counterfactual definiteness [29], but there is a much simpler one: counterfactual definiteness is not compatible with predictive incompleteness; see Appendices and [18]. Actually Ref. [12] gives a good definition of predictive incompleteness: it is “the lack of information about the measurement outcomes in the wave-function, combined with the fact that an observation in one location can tell us something about the measurement outcome in another location”. Then [12] tries to explain this by using supermeasurements, but here we obtain it without using any non-measurable set of hidden variables—as a consequence of the CSM contextual quantization postulates [22,23,24,25]. Again, acknowledging that the quantum state vector is predictively incomplete may appear disturbing, but it makes perfect sense and ensures that many “paradoxes” are removed in the CSM framework.

3.3. From σ-Algebras to Projection Lattices: Gleason’s Route and the Gluing of Contexts

A compact way to understand the formal passage from classical to quantum probability is to observe that the classical domain of events—a Boolean $[eqn]$ -algebra, see Appendix—is replaced in quantum theory by the lattice $[eqn]$ of orthogonal projections on a Hilbert space $[eqn]$ . In the classical Kolmogorov framework, additivity is required for disjoint sets; Gleason’s theorem implements the exact analog of this requirement in the quantum setting by demanding additivity for a countable set of mutually orthogonal projections.

Concretely, one considers a function $[eqn]$ satisfying the following:

• 1. $[eqn]$ (normalization for the identity operator I);
• 2.If $[eqn]$ for $[eqn]$ , then $[eqn]$ (orthogonal additivity).

Gleason’s theorem shows that, for $[eqn]$ , any such m is of the trace form

$$[eqn]$$

for a unique density operator $[eqn]$ . Thus the formal move of replacing a classical $[eqn]$ -algebra with the projection lattice and replacing disjoint additivity with orthogonal additivity is precisely the route Gleason follows. This perspective clarifies the sense in which contexts are “glued.” A single measurement context corresponds to a maximal commuting family of projections; restricted to that commuting subalgebra, the projection lattice is isomorphic to a classical Boolean algebra and the probabilities are Kolmogorovian.

The nontrivial content of Gleason’s theorem is to ask for a single assignment m that is consistent across all such commuting subalgebras simultaneously: the value assigned to a projection does not depend on which orthogonal decomposition (which context) it appears in. The essential fact that a given projector appears in a continuous infinity of context, provided that $[eqn]$ , is called intertwining of contexts by Gleason. In the CSM framework, a modality is defined as the association of a projector $[eqn]$ (a usual quantum state) and a context (a complete set of orthogonal projectors including $[eqn]$ ). Sharing a projector is then an equivalence relation for modalities; it is called extravalence [23], and corresponds physically to mutual certainty. This justifies that the probability depends only on the projector (the extravalence class) and not on the embedding context.

Two additional remarks are in order. First, the lattice $[eqn]$ is non-Boolean (non-distributive), and this structural difference is the source of contextuality: additivity on orthogonal families is a strictly different requirement from full $[eqn]$ -additivity on a Boolean algebra. Second, Gleason’s theorem requires $[eqn]$ ; nevertheless, the two-dimensional case is included, provided that it is embedded into higher dimensions [24,30]. It is actually well known that one can build a classical model for a single qubit, but not for two qubits.

Operationally, Kolmogorovian Censorship and Gleason’s hypotheses are therefore complementary: KC explains why probabilities look classical inside a fixed context, while Gleason characterizes when and how those context-by-context classical assignments can be coherently extended to a single quantum probability law on the whole projection lattice.

3.4. A Philosophical Detour Through Physical Realism

We have seen that the KC in a single context is easily integrated in the CSM framework, but one should not conclude that quantum probabilities are Kolmogorovian in a classical sense: this would be true in classical physics because there is only one universal context, but this fails in quantum physics because there is a continuous infinity of different, incompatible contexts. This point of view is implicit in textbook quantum mechanics, and it is made explicit in the CSM framework by using operator algebra and infinite tensor products. This framework allows a contextual unification of classical and quantum physics within a unique macroscopic physical world [2,31].

One may notice that Rédei [11] (see also [32]) considers that the KC is problematic, in particular because it implies that “probabilities are thus not features of quantum systems in and of themselves, they are features that only manifest themselves upon measurement. Philosophers (or physicists) with a robust realist conviction may find unattractive this strongly instrumentalist flavor of interpretation of quantum probability forced upon us by the KC.” In the CSM approach, quantum probabilities also obtain a meaning only upon measurement due to the predictive incompleteness of $[eqn]$ , and they are also genuinely non-classical probabilities. Nevertheless, CSM is based on physical realism, defined by the statement that the purpose of physics is to study entities of the natural world, existing independently from any particular observer’s perception, and obeying universal and intelligible rules [21,22,23]. We claim therefore that even with a “robust realist conviction” one can accept that $[eqn]$ is predictively incomplete—as the best way to make sense of this conundrum, within the framework of contextual objectivity [33].

4. Conclusions

The Kolmogorovian Censorship is a useful technical observation: quantum probabilities are Kolmogorovian when a measurement context is fixed, but this formal fact does not resolve the deeper interpretational choice exposed by Bell-type experiments. Superdeterminism and nonlocal hidden variable accounts restore a single global joint distribution only at the cost of questionable ontological commitments and scientific methodology. By contrast, predictive incompleteness accepts that the quantum state alone does not determine outcome statistics across incompatible contexts and thereby preserves locality and empirical adequacy without invoking conspiratorial correlations. It also leads straightforwardly to Gleason’s theorem and Born’s rule. Framing the debate in these terms clarifies which extra assumptions are being made in different reconstructions and points to concrete empirical and conceptual criteria that future work should address.

As a final remark, a significant challenge in quantum reconstructions is to state clearly what the hypotheses or postulates are and what their consequences are. It should also be made clear whether one looks for a fully deductive reasoning or a partially inductive one, that is an “Inference to the Best Explanation” [24,25]. In the CSM point of view, a fully deductive approach does not fit, since, quoting Landau [34], “quantum mechanics (…) contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation”. Then what is desired is not a deduction of the Laws of Nature from some postulates in a mathematical sense, but rather a fully consistent construction, including both classical and quantum physics from the beginning and clearly separating experimentally based evidence from its mathematical description [2,31].