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arXiv:2601.13875

On spooky action at a distance and conditional probabilities

Henryk Gzyl

quant-phSubmitted Jan 20, 2026• Announced Jan 21, 2026• 0 citations
v1 · Jan 20, 2026v2 · May 15, 2026
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Abstract

The aim of this exposé is to make explicit the analogy between the classical notion of non-independent probability distribution and the quantum notion of entangled state. To bring that analogy forth, we consider a classical systems with two dependent random variables and a quantum system with two components. In the classical case, afet observing one of the random variables, the underlying sample space and the probability distribution change. In the quantum case, when and event pertaining to one of the components is observed, the post-measurement state captures, both, the change in the state of the system and implicitly the new probability distribution. The predictions after a measurement in the classical case and in the quantum case, have to be computed with the conditional distribution given the value of the observed variable.

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The aim of this exposé is to make explicit the analogy between the classical notion of non-independent probability distribution and the quantum notion of entangled state.
To bring that analogy forth, we consider a classical systems with two dependent random variables and a quantum system with two components. In the classical case, afet observing one of the random variables, the underlying sample space and the probability distribution change. In the quantum case, when and event pertaining to one of the components is observed, the post-measurement state captures, both, the change in the state of the system and implicitly the new probability distribution.
The predictions after a measurement in the classical case and in the quantum case, have to be computed with the conditional distribution given the value of the observed variable.
Abstract Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B.
Abstract This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it is meaningless to speak about entanglement without pointing to the fixed observables A and B, so this is AB-entanglement. Dependence of quantum observables is formalized as non-coincidence of conditional probabilities. Starting with this probabilistic definition, we achieve the Hilbert space characterization of the AB-entangled states as amplitude non-factorisable states. In the tensor product case, AB-entanglement implies standard entanglement, but not vise verse. AB-entanglement for dichotomous observables is equivalent to their correlation, i.e., ⟨AB⟩ψ≠⟨A⟩ψ⟨B⟩ψ.\langle AB\rangle _{\psi} \not = \langle A\rangle _{\psi} \langle B\rangle _{\psi} .⟨AB⟩ψ​=⟨A⟩ψ​⟨B⟩ψ​. ⟨ A B ⟩ ψ ≠ ⟨ A ⟩ ψ ⟨ B ⟩ ψ . We describe the class of quantum states that are AuBuA_{u} B_{u}Au​Bu​ A u B u -entangled for a family of unitary operators (u). Finally, observables entanglement is compared with dependence of random variables in classical probability theory.
By comparing Schrödinger's cat with its classical counterpart, I show that a quantum superposition should be understood as an expectation over possible eigenstates weighted by wave-like probabilities. Upon the occurrence of a certain event, the quantum system is randomly realized into one of the possible eigenstates due to its intrinsic stochasticity. While the randomness of a single realization cannot be controlled or predicted, the overall distribution can be regulated via experimental setup and converges as the number of events increases. A measurement is indeed an activity employing a certain event to convert a quantum effect into a macroscopic outcome. Consequently, the puzzling concepts of wavefunction collapse, many worlds, and decoherence become unnecessary for understanding quantum superposition. This expectation-realization interpretation, which integrates probability theory with wave mechanics, can also be extended to quantum pathways. Moreover, it reframes tests of Bell's inequalities as validating the wave-like probability nature of quantum mechanics, with no need to invoke the mysterious notions of quantum non-locality and "spooky action at a distance".
We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability theory. In the contextualization approach each random variable is "automatically" labeled by all conditions under which it is recorded, and the random variables across a set of mutually exclusive conditions are probabilistically coupled (imposed a joint distribution upon). Analysis of all possible probabilistic couplings for a given set of random variables allows one to characterize various relations between their separate distributions (such as Bell-type inequalities or quantum-mechanical constraints). In the conditionalization approach one considers the conditions under which the random variables are recorded as if they were values of another random variable, so that the observed distributions are interpreted as conditional ones. This approach is uninformative with respect to relations between the distributions observed under different conditions because any set of such distributions is compatible with any distribution assigned to the conditions.
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available, whereas if information consisting in the observation of another random variable is available, then the best predictor is is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. we shall see that when the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle
Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding probabilistic weights (probability density or respectively the wave function). Such a model ensures theoretical estimations for predicted errors specific to the mean values as well as to the fluctuations of both types of observables. The model stands out of ourdays prevalent opinion that measurement theories must be depicted in entirely different manners in the classical respectively quantum cases.