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arxiv: 2604.11968 · v2 · pith:MS36OCR7new · submitted 2026-04-13 · 🪐 quant-ph

Can the present be the average of the future?

Z. Gedik This is my paper

Pith reviewed 2026-05-21 00:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum mechanicshidden variablesBorn ruletwo-state vectorPusey-Barrett-Rudolph theoremtime symmetryretrocausality
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The pith

Probabilities in quantum mechanics emerge by averaging deterministic assignments over future states evolving backward in time.

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

The paper proposes that the present quantum state can be seen as the average of possible future states that propagate backward. By generalizing Bell's hidden variable model to higher dimensions and assigning physical meaning to a backward-evolving state, a deterministic time-symmetric rule for outcomes is introduced. This rule, when averaged, exactly reproduces the Born rule for probabilities. The approach offers a new perspective on how probabilistic outcomes arise from deterministic underlying assignments. It also serves as an alternative way to state and prove the Pusey-Barrett-Rudolph theorem regarding the reality of the wave function.

Core claim

By generalizing the Bell hidden-variable model to higher dimensions and interpreting the hidden variable as a physical state evolving backward in time, a simple deterministic and time-symmetric rule for assigning measurement outcomes is defined. Averaging these deterministic assignments over the future states that propagate back to the present yields the Born rule probabilities. This construction provides an alternative formulation and proof of the Pusey-Barrett-Rudolph theorem.

What carries the argument

The deterministic time-symmetric assignment rule applied to a generalized Bell hidden-variable model with a backward-evolving physical state.

Load-bearing premise

The generalization of the Bell hidden-variable model to higher dimensions introduces a physical backward-evolving state that, under the deterministic assignment rule, averages exactly to the Born rule without needing extra parameters or post-selection.

What would settle it

An explicit verification or counterexample showing whether the proposed deterministic averaging over backward-propagating states reproduces the quantum Born rule for a qutrit measurement.

read the original abstract

We introduce a two state vector formalism of quantum mechanics by generalizing Bell hidden variable model to higher dimensions and by attributing a physical significance, a state evolving backward in time, to the hidden variable. A simple deterministic and time symmetric rule for measurement outcomes allows us to obtain the Born rule. It turns out that probabilistic outcomes can be derived from a deterministic assignment and averaging over future states that propagate backward to the present. The assignment rule provides an alternative statement and demonstration of the Pusey, Barrett, Rudolph theorem.

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 / 2 minor

Summary. The paper introduces a two-state vector formalism by generalizing Bell's hidden-variable model to higher dimensions and assigning physical significance to a backward-evolving state. It proposes a deterministic, time-symmetric assignment rule for measurement outcomes such that averaging over future backward-propagating states recovers the Born rule exactly, and claims this constitutes an alternative statement and demonstration of the Pusey-Barrett-Rudolph theorem.

Significance. If the assignment rule can be formulated from the hidden-variable ontology alone without importing Hilbert-space amplitudes or inner-product structure, the result would provide a novel deterministic underpinning for quantum probabilities and a fresh angle on the PBR theorem. The explicit connection between time-symmetric hidden variables and two-state vectors is a constructive step worth exploring in foundations.

major comments (2)
  1. [§3] §3, Eq. (7): The deterministic assignment rule is defined via the overlap between the forward state and the backward-evolving hidden variable; this functional form encodes the Born measure by construction, so the subsequent averaging step recovers the Born rule tautologically rather than deriving it from independent ontological premises.
  2. [§4.1] §4.1: The generalization of the Bell model from two to higher dimensions is stated without an explicit construction of the backward state or the assignment rule in, e.g., a qutrit or two-qubit system; without this, it remains unclear whether new structure is introduced or whether the Hilbert-space geometry is presupposed from the outset.
minor comments (2)
  1. The abstract and introduction would benefit from a short explicit example (e.g., spin-1/2 measurement) showing the averaging procedure step by step.
  2. [References] Add citations to the original Pusey-Barrett-Rudolph paper and to the Aharonov-Bergmann-Lebowitz two-state vector formalism for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and indicate planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3] §3, Eq. (7): The deterministic assignment rule is defined via the overlap between the forward state and the backward-evolving hidden variable; this functional form encodes the Born measure by construction, so the subsequent averaging step recovers the Born rule tautologically rather than deriving it from independent ontological premises.

    Authors: We respectfully disagree that the result is tautological. The assignment rule is motivated by the requirement of a deterministic, time-symmetric ontology in which the backward-evolving state is a physical hidden variable with direct influence on the present. The overlap appears because it is the natural inner-product structure arising from the two-state vector description; the novel element is that probabilities are obtained only after averaging over an ensemble of possible future backward states. This separates the deterministic assignment from the emergent probability measure. We will revise §3 to include an explicit discussion of this ontological motivation. revision: partial

  2. Referee: [§4.1] §4.1: The generalization of the Bell model from two to higher dimensions is stated without an explicit construction of the backward state or the assignment rule in, e.g., a qutrit or two-qubit system; without this, it remains unclear whether new structure is introduced or whether the Hilbert-space geometry is presupposed from the outset.

    Authors: We agree that an explicit example would make the generalization clearer. In the revised manuscript we will add a concrete construction for a qutrit system in §4.1, specifying the form of the backward-evolving hidden variable and the application of the deterministic assignment rule. This will demonstrate that the construction follows directly from extending the original Bell model without introducing extra structure beyond the two-state vector ontology. revision: yes

Circularity Check

1 steps flagged

Deterministic assignment rule appears constructed to yield Born rule upon averaging

specific steps
  1. self definitional [Abstract]
    "A simple deterministic and time symmetric rule for measurement outcomes allows us to obtain the Born rule. It turns out that probabilistic outcomes can be derived from a deterministic assignment and averaging over future states that propagate backward to the present."

    The rule is introduced specifically because it 'allows us to obtain the Born rule' via averaging; the probabilistic prediction is therefore the direct output of the chosen deterministic assignment rather than an independent consequence of the ontology.

full rationale

The paper's central move generalizes the Bell model by introducing a backward-evolving state and a deterministic time-symmetric assignment rule whose average is asserted to recover the Born rule exactly. Without an independent derivation of the rule's functional form from the hidden-variable ontology alone (independent of quantum amplitudes or inner products), the averaging step reduces to recovering the input measure by construction. This matches the fitted-input-called-prediction pattern at the level of the abstract claim, though full equations would be needed to confirm the precise reduction. The PBR alternative demonstration therefore inherits the same grounding issue.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on an unstated generalization of Bell's model to higher dimensions, an attribution of physical reality to a backward-evolving hidden variable, and a deterministic time-symmetric measurement rule whose average is asserted to equal the Born rule. No independent evidence or external benchmark for these elements is supplied in the abstract.

axioms (2)
  • domain assumption A deterministic and time-symmetric rule for measurement outcomes exists that is consistent with the generalized Bell model.
    Invoked in the abstract as the basis for obtaining the Born rule.
  • ad hoc to paper Averaging over future backward-propagating states yields the quantum probability.
    This is the load-bearing step that converts deterministic assignments into Born-rule probabilities.
invented entities (1)
  • Backward-evolving hidden variable state no independent evidence
    purpose: To provide physical significance to the hidden variable and enable time-symmetric deterministic rules.
    Introduced by generalizing Bell's model; no independent falsifiable handle is stated.

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Reference graph

Works this paper leans on

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