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arxiv: 2503.20607 · v3 · submitted 2025-03-26 · 🪐 quant-ph · cs.AI· math.PR

A decision-theoretic approach to dealing with uncertainty in quantum mechanics

Keano De Vos , Gert de Cooman , Alexander Erreygers , Jasper De Bock This is my paper

Pith reviewed 2026-05-22 22:31 UTC · model grok-4.3

classification 🪐 quant-ph cs.AImath.PR
keywords decision theoryBorn's rulequantum mechanicsuncertaintyimprecise probabilitiesutility functionsmeasurements as acts
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The pith

Simple decision-theoretic postulates place Born's rule inside the utility functions of quantum measurements modeled as acts.

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

The paper establishes a decision-theoretic framework treating quantum measurements as acts with uncertain outcomes. Under a few basic postulates on the utilities of these acts, Born's rule becomes part of the utility functions. This setup separates quantum mechanics from standard probability theory, permitting the use of imprecise probabilities. The approach provides a foundation for recent work on imprecise probabilities in quantum settings and contrasts with earlier decision-theoretic treatments of quantum mechanics.

Core claim

By representing quantum measurements as decision-theoretic acts and requiring their utility functions to obey simple decision-theoretic postulates, Born's rule is necessarily encapsulated within those utilities, which in turn allows quantum mechanics to be uncoupled from precise probability theory.

What carries the argument

Decision-theoretic acts for quantum measurements, whose utility functions satisfy postulates that encapsulate Born's rule.

If this is right

  • This framework supplies a decision-theoretic foundation for the work of Benavoli, Facchini and Zaffalon on imprecise probabilities in quantum mechanics.
  • It opens the possibility of handling quantum uncertainty with imprecise rather than precise probabilities.
  • It offers a distinct alternative to the decision-theoretic approaches previously developed by Deutsch and Wallace.

Where Pith is reading between the lines

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

  • If the postulates hold, quantum theory could be developed using weaker or more general notions of probability from the start.
  • The framework might extend to other areas of physics where uncertainty is modeled decision-theoretically.
  • Experimental tests could check whether real quantum measurements align with the utility functions derived from these postulates.

Load-bearing premise

Quantum measurements can be faithfully represented as decision-theoretic acts in such a way that the stated postulates on their utilities produce Born's rule without any additional structure.

What would settle it

A concrete quantum measurement scenario where the utilities derived from the decision-theoretic postulates fail to match the observed probabilities given by Born's rule.

read the original abstract

We provide a decision-theoretic framework for dealing with uncertainty in quantum mechanics. This uncertainty is two-fold: on the one hand there may be uncertainty about the state the quantum system is in, and on the other hand, as is essential to quantum mechanical uncertainty, even if the quantum state is known, measurements may still produce an uncertain outcome. In our framework, measurements therefore play the role of acts with an uncertain outcome and our simple decision-theoretic postulates ensure that Born's rule is encapsulated in the utility functions associated with such acts. This approach allows us to uncouple (precise) probability theory from quantum mechanics, in the sense that it leaves room for a more general, so-called imprecise probabilities approach. We discuss the mathematical implications of our findings, which allow us to give a decision-theoretic foundation to recent seminal work by Benavoli, Facchini and Zaffalon, and we compare our approach to earlier and different approaches by Deutsch and Wallace.

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 develops a decision-theoretic framework for quantum uncertainty, treating measurements as acts with uncertain outcomes. It claims that a set of simple decision-theoretic postulates ensures Born's rule is encapsulated in the associated utility functions, thereby decoupling quantum mechanics from precise probability theory and allowing an imprecise-probabilities treatment. The work provides a decision-theoretic foundation for recent results by Benavoli, Facchini and Zaffalon and contrasts the approach with the Deutsch-Wallace program.

Significance. If the central derivation is valid, the framework offers a route to grounding Born's rule in decision theory while remaining compatible with imprecise probabilities, which could be useful for quantum foundations and for applications involving Knightian uncertainty. The explicit comparison to Deutsch-Wallace and the claimed foundation for the Benavoli et al. results are concrete contributions that would be of interest to the quantum-foundations community.

major comments (2)
  1. [Introduction / §2] The abstract and introduction assert that 'simple decision-theoretic postulates ensure that Born's rule is encapsulated in the utility functions,' yet the manuscript provides no explicit list of the postulates, no derivation showing how they entail the Born rule, and no check against circularity (e.g., whether the utility representation already presupposes a probability measure equivalent to Born's rule). This is load-bearing for the central claim.
  2. [§3] The claimed decoupling from precise probability theory is stated but not demonstrated: it is unclear whether the decision-theoretic representation of a quantum measurement (as an act) can be formalized without already embedding a probability measure on the outcome space that satisfies the Born rule. A concrete counter-example or a proof that no such embedding occurs would be required.
minor comments (2)
  1. [Discussion] The comparison with Deutsch-Wallace is mentioned but lacks a side-by-side table or explicit list of differing postulates.
  2. [§2] Notation for acts, outcomes, and utility functions is introduced without a dedicated preliminary section; a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify places where greater explicitness would strengthen the central claims. We respond to each below and commit to revisions that address the concerns directly.

read point-by-point responses

  1. Referee: [Introduction / §2] The abstract and introduction assert that 'simple decision-theoretic postulates ensure that Born's rule is encapsulated in the utility functions,' yet the manuscript provides no explicit list of the postulates, no derivation showing how they entail the Born rule, and no check against circularity (e.g., whether the utility representation already presupposes a probability measure equivalent to Born's rule). This is load-bearing for the central claim.

    Authors: We agree that the manuscript would be improved by an explicit enumeration of the postulates together with a self-contained derivation. Although the decision-theoretic construction appears in §2, it is not presented as a numbered list followed by a step-by-step entailment of the Born rule in the utilities. In revision we will add a dedicated subsection that (i) lists the postulates, (ii) derives the encapsulation of Born's rule, and (iii) explicitly addresses circularity by showing that the utility representation is obtained without presupposing a probability measure that already satisfies the Born rule. revision: yes

  2. Referee: [§3] The claimed decoupling from precise probability theory is stated but not demonstrated: it is unclear whether the decision-theoretic representation of a quantum measurement (as an act) can be formalized without already embedding a probability measure on the outcome space that satisfies the Born rule. A concrete counter-example or a proof that no such embedding occurs would be required.

    Authors: We accept that the decoupling claim requires a formal demonstration rather than a statement. The framework is intended to allow the representation of measurements as acts whose utilities encode the Born rule while leaving the probability component imprecise. In the revised version we will supply, in an appendix, a proof that the act representation can be constructed without embedding a precise probability measure satisfying the Born rule, together with an argument showing direct compatibility with the imprecise-probability setting of Benavoli et al. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper maps quantum measurements to decision-theoretic acts and applies simple postulates to encapsulate Born's rule in the resulting utility functions. No load-bearing step reduces by construction to the target result via self-definition, fitted parameters renamed as predictions, or self-citation chains. The framework is presented as independent of precise probability theory and is compared to prior work without invoking uniqueness theorems from the same authors as external facts. The central claim rests on the stated postulates applied to the act representation, which does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on unnamed 'simple decision-theoretic postulates' that are asserted to embed Born's rule; no free parameters, invented entities, or additional axioms are mentioned.

axioms (1)
  • domain assumption Quantum measurements can be modeled as decision-theoretic acts with uncertain outcomes whose utilities must satisfy the paper's postulates
    This modeling choice is the entry point for the entire framework and is stated in the abstract.

pith-pipeline@v0.9.0 · 5699 in / 1281 out tokens · 44085 ms · 2026-05-22T22:31:14.137423+00:00 · methodology

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Reference graph

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