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arxiv: 2606.31097 · v1 · pith:KWLI6KA5new · submitted 2026-06-30 · 🪐 quant-ph

Generalised Probabilistic Theories

Zujin Wen , Zihan Wang , Xiangjing Liu , Oscar Dahlsten This is my paper

Pith reviewed 2026-07-01 05:51 UTC · model grok-4.3

classification 🪐 quant-ph
keywords generalised probabilistic theoriesconvex frameworkquantum foundationsBell non-localitybox worldgeneralised Hamiltonian mechanicsphase space negativity
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The pith

States in physical theories are represented as real vectors of probabilities, transformations as matrices, and measurements as linear functionals.

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

The paper presents the generalised probabilistic theories framework as an operational description that applies to classical probability theory, quantum theory and theories beyond quantum mechanics. States appear as real vectors whose entries are the probabilities of different measurement outcomes. Transformations act as real matrices that map one state vector to another while preserving the probabilistic structure. Measurement outcomes correspond to real vectors that extract probabilities via inner products with the state vectors. This representation lets the same mathematical language describe local hidden variable models, quantum systems and hypothetical super-quantum correlations such as those in box world.

Core claim

Any physical theory of interest is modelled by a convex set of states given as real vectors, transformations given by real matrices that map the set to itself, and measurement outcomes given by real vectors whose inner product with a state vector equals the probability of that outcome.

What carries the argument

The convex framework in which states form a convex cone inside a real vector space, transformations are linear maps on that space, and effects are dual vectors that yield probabilities.

If this is right

  • Classical probability theory is recovered by taking states to be probability distributions over deterministic outcomes.
  • Quantum theory is recovered by embedding density operators into a real vector space with the Hilbert-Schmidt inner product.
  • Bell non-locality appears as a constraint on the joint probability vectors obtainable from composite systems.
  • Box world realises correlations stronger than quantum theory while remaining inside the same convex framework.
  • Generalised Hamiltonian dynamics can be defined on the discrete state space or on continuous phase space, with negativity of the phase-space function signalling contextuality.

Where Pith is reading between the lines

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

  • The framework supplies a uniform language in which to search for theories that lie between quantum mechanics and box world.
  • Phase-space negativity may serve as a diagnostic for contextuality in theories other than quantum mechanics.
  • Operational constraints such as no-signalling can be imposed directly on the vector representation without reference to Hilbert space.

Load-bearing premise

Every physical theory worth considering admits a representation in which states are convex combinations of real probability vectors and all transformations and measurements act linearly on those vectors.

What would settle it

An experiment on a physical system that produces outcome statistics which cannot be listed as the components of a real vector or whose allowed transformations cannot be expressed as linear maps on such vectors.

read the original abstract

We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.

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

0 major / 2 minor

Summary. The manuscript provides an introduction to the generalised probabilistic theories (GPT) framework. States are defined as real vectors representing lists of probabilities, transformations as real matrices, and measurement outcomes as dual functionals whose inner products yield probabilities. It explains the operational foundations via cones of states and effects, represents classical and quantum theories within the framework, discusses Bell non-locality and box world, and covers generalised Hamiltonian mechanics in both discrete and continuous phase space, including the significance of negativity for contextuality and tunnelling.

Significance. As a review of an established framework with no new theorems or predictions, the paper's value lies in its potential to serve as an accessible entry point for researchers entering quantum foundations. The coverage of standard examples (classical/quantum representations, box world) and extensions (continuous phase space, negativity) consolidates known material; if the exposition is precise and well-structured, it could usefully support teaching or onboarding without advancing the research frontier.

minor comments (2)
  1. [Abstract] The abstract states that the framework 'generalises quantum theory' but does not explicitly note that quantum theory is recovered as a special case within the convex set; a brief clarification in the introduction would help readers new to the area.
  2. [Introduction (or relevant section on continuous mechanics)] The description of continuous phase space mentions negativity in the context of tunnelling but does not reference specific prior works on Wigner-function negativity or its relation to contextuality; adding one or two key citations would strengthen the review character.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript as an accessible introduction to the generalised probabilistic theories framework. We are pleased that the referee recommends acceptance and agrees that the paper consolidates known material in a manner suitable for teaching and onboarding new researchers.

Circularity Check

0 steps flagged

No significant circularity; introductory review of established framework

full rationale

The manuscript is an introductory review of the established GPT/convex framework. It defines states as real vectors of probabilities, transformations as real matrices, and measurement outcomes as dual functionals, then describes known examples including classical/quantum theories, box world, and generalised Hamiltonian mechanics. No new theorems, derivations, predictions, or empirical claims are advanced. All content consists of standard operational definitions and reviews of prior results, with no load-bearing steps that reduce by construction to fitted inputs or self-citation chains. The derivation chain is absent because the paper does not claim to derive new results from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is an expository introduction and does not introduce or derive new free parameters, axioms, or entities; it restates the standard operational assumptions of the GPT framework.

axioms (3)
  • domain assumption States are real vectors representing lists of probabilities of measurement outcomes.
    Stated directly in the abstract as the starting representation.
  • domain assumption Transformations are real matrices.
    Stated directly in the abstract as the representation of state changes.
  • domain assumption Measurement outcomes are represented by functionals of the states via inner products.
    Stated directly in the abstract.

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discussion (0)

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Reference graph

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