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arxiv: 2601.13875 · v2 · pith:E6DKWDEDnew · submitted 2026-01-20 · 🪐 quant-ph

On spooky action at a distance and conditional probabilities

Henryk Gzyl This is my paper

Pith reviewed 2026-05-21 15:29 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum entanglementconditional probabilityspooky action at a distancemeasurement updateclassical quantum analogydependent random variablespost-measurement state
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The pith

The post-measurement predictions in both classical and quantum cases are computed using conditional distributions given the observed value.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper makes explicit the analogy between classical non-independent probability distributions and quantum entangled states. It considers a classical system with two dependent random variables and a quantum system with two components. After observing one random variable classically, the sample space and probability distribution are updated. In the quantum case, the post-measurement state captures the change in the system and the new probability distribution. Predictions after measurement in either case use the conditional distribution based on the observed value.

Core claim

The aim is to show that just as in classical probability, where observing one of two dependent random variables changes the sample space and the probability distribution to the conditional one, in quantum mechanics, when an event pertaining to one component of an entangled system is observed, the post-measurement state captures both the change in the state of the system and implicitly the new probability distribution. Thus, 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.

What carries the argument

The conditional distribution given the observed variable, which updates the classical sample space or the quantum post-measurement state.

If this is right

  • Predictions for the unobserved component must use the conditional distribution after the observation.
  • The post-measurement state in quantum mechanics encodes the updated probabilities similarly to classical conditioning.
  • This approach frames the correlations in entangled states as conditional probability effects rather than non-local influences.
  • The analogy holds for systems with two components or variables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If correct, this suggests that 'spooky action at a distance' can be reinterpreted as standard conditional probability updating in both domains.
  • This view might be extended to more complex quantum systems or other interpretations of quantum mechanics.
  • One could test by deriving specific numerical predictions for Bell inequality violations using only conditional distributions.

Load-bearing premise

The post-measurement quantum state captures the change in the system and the new probability distribution in precisely the same manner as the classical update of sample space and distribution.

What would settle it

Finding a quantum entangled system where the probabilities for the second particle after measuring the first cannot be matched by applying classical conditional probability rules to the initial joint distribution.

read the original abstract

The aim of this expos\'e 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents an expository analogy between classical non-independent random variables (where observing one updates the sample space and probability distribution) and quantum entangled states (where the post-measurement state captures the change in the system and implies a new probability distribution). The central claim is that predictions after a measurement must be computed using the conditional distribution given the value of the observed variable in both the classical and quantum cases.

Significance. If the analogy holds without omitting key quantum features, the work could offer a pedagogical clarification of entanglement in terms of conditional probabilities. The manuscript is conceptual rather than deriving new quantitative results, reproducible code, or falsifiable predictions, so its significance rests on the accuracy and completeness of the presented parallel.

major comments (2)
  1. [Abstract] Abstract, final sentence: the claim that predictions 'have to be computed with the conditional distribution given the value of the observed variable' in both cases is load-bearing for the analogy but is not supported for entangled quantum states, since no single joint probability distribution over underlying variables reproduces all quantum statistics (e.g., the CHSH correlator for the singlet state reaches 2√2, exceeding the classical bound of 2).
  2. [Quantum case section] Quantum case discussion (inferred from abstract and reader's weakest assumption): the assertion that the post-measurement state captures the change 'in precisely the same manner' as classical conditioning does not address how non-commutativity or contextuality in quantum mechanics can alter conditional probabilities across different measurement choices, unlike the fixed joint distribution available classically.
minor comments (3)
  1. [Abstract] Abstract: 'afet observing' appears to be a typo for 'after observing'.
  2. [Abstract] Abstract: 'when and event' appears to be a typo for 'when an event'.
  3. [Abstract] The abstract would benefit from explicitly naming the classical example (e.g., specific dependent variables) and quantum example (e.g., singlet state) used to develop the analogy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying important limitations in the scope of the analogy presented. We respond to each major comment below, clarifying the manuscript's intent and indicating planned revisions to address the concerns raised.

read point-by-point responses

  1. Referee: [Abstract] Abstract, final sentence: the claim that predictions 'have to be computed with the conditional distribution given the value of the observed variable' in both cases is load-bearing for the analogy but is not supported for entangled quantum states, since no single joint probability distribution over underlying variables reproduces all quantum statistics (e.g., the CHSH correlator for the singlet state reaches 2√2, exceeding the classical bound of 2).

    Authors: We agree that no single joint probability distribution can reproduce all quantum statistics, as shown by the CHSH violation for the singlet state. The manuscript draws an analogy specifically between classical conditioning on dependent random variables (updating the distribution given an observed value) and the quantum post-measurement state for a fixed measurement on one subsystem of an entangled pair. It does not claim or require the existence of a non-contextual hidden-variable model that assigns joint probabilities to all observables simultaneously. To prevent misinterpretation, we will revise the final sentence of the abstract to qualify the claim as applying to conditional predictions following a specific measurement outcome in both frameworks. revision: partial

  2. Referee: [Quantum case section] Quantum case discussion (inferred from abstract and reader's weakest assumption): the assertion that the post-measurement state captures the change 'in precisely the same manner' as classical conditioning does not address how non-commutativity or contextuality in quantum mechanics can alter conditional probabilities across different measurement choices, unlike the fixed joint distribution available classically.

    Authors: The referee correctly identifies that non-commutativity and contextuality imply that conditional probabilities in quantum mechanics depend on the measurement context chosen, without a fixed underlying joint distribution. Our exposition considers a concrete entangled state and a particular observable measured on one component, illustrating how the post-measurement state encodes the updated probabilities for the other component in a manner parallel to classical conditioning on that observed value. We do not address or claim equivalence across incompatible measurement choices. We will add a clarifying paragraph in the quantum case section acknowledging this limitation and noting that the analogy is restricted to the update for a given measurement rather than a global classical joint distribution. revision: partial

Circularity Check

0 steps flagged

Expository analogy without circular derivations or self-referential claims

full rationale

The paper is an expository work whose central aim is to draw a conceptual parallel between classical conditioning on dependent random variables and the use of post-measurement states in quantum mechanics. The provided abstract and description state directly that predictions in both cases are to be computed via the conditional distribution given the observed value; this is presented as the explicit analogy to be made, not as a derived result obtained from equations, fitted parameters, or prior self-citations within the manuscript. No load-bearing steps reduce by construction to inputs, no uniqueness theorems are imported, and no ansatzes or renamings of known results appear. The work remains self-contained as an interpretive exercise and does not rely on any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions from probability theory and quantum mechanics without introducing new fitted parameters or postulated entities.

axioms (2)
  • standard math Standard definition of conditional probability for dependent random variables.
    Invoked when describing the classical case after observation.
  • domain assumption Quantum measurement postulate that produces a post-measurement state.
    Used to describe the update in the entangled quantum system.

pith-pipeline@v0.9.0 · 5644 in / 1191 out tokens · 59868 ms · 2026-05-21T15:29:45.964072+00:00 · methodology

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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