Higher-order interference and single-system postulates characterizing quantum theory

Cozmin Ududec; Howard Barnum; Markus P. Mueller

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Higher-order interference and single-system postulates characterizing quantum theory

Howard Barnum, Markus P. Mueller, Cozmin Ududec

Quantum Physics · Submitted 17 Mar 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension.

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arXiv ID1403.4147 (version 4)

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CitationCozmin Ududec, Howard Barnum, Markus P. Mueller. "Higher-order interference and single-system postulates characterizing quantum theory." arXiv:1403.4147 (2014). https://pith.science/paper/G2HHUU2Z

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What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.58 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.90 · mentions code or data availability 0.02.

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We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation. We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension. Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics. We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation.	contribution 1.00	0.87	0.18	asserted 1.00
s1	We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension.	result 0.73	0.95	0.57	asserted 1.00
s2	Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics.	result 0.96	0.95	0.26	asserted 1.00
s3	We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.	result 0.93	0.91	0.05	hedged possibility 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation. We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension. Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics. We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Why three? A two-level system with four mutually unbiased questions
quant-ph 2026-09 accept novelty 6.0 of 10

A two-level system with four mutually unbiased yes/no questions exists as the four-dimensional Bloch ball, is realizable inside two qubits, and is the odd sector of two fermionic modes that parity superselection removes.
A change of perspective: switching quantum reference frames via a perspective-neutral framework
quant-ph 2018-09 accept novelty 6.0 of 10

Choosing a quantum reference frame is a gauge fixing and switching frames is a symmetry transformation, with Dirac quantization as the frame-neutral bookkeeping and reduced quantization as a single frame's view.
General probabilistic theories: An introduction
quant-ph 2021-03 accept novelty 1.0 of 10

The paper provides a self-contained convex-geometric introduction to general probabilistic theories, including state spaces, effect algebras, tensor products, channels, and compatibility, with classical, quantum, and ...

How later papers used it

65 later papers cite it

Evidence: substantial. 57 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 222 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 cited it as support. 1 more paper by the same authors is listed: 1 built on it.

Builds on it 1 paper · 1 by the same authors

By the same authors

Extended this paper's results showing that under Postulates 1 and 2 the existence of higher-order interference is equivalent to each of three other statements.

Thermodynamics and the structure of quantum theory (2016) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 paper

Used Postulate 2 from this paper.

Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory (2015) · 0.94 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 11 papers with this label

Said that Hamiltonian dynamics consistent with principles of branch locality may be a signifier of quantum theory, as it would expect from this paper's results.

On defining the Hamiltonian beyond quantum theory (2018) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

24 arXiv papers cite this in 129 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 88 sentences without a confident label

Contests it 1 paper

The relation of energy observability to dynamical correspondence is, roughly, that a dynamical correspondence is a linear map (but not necessarily an injection[In [ref] (on p. 29) dynamical correspondences are mistakenly described as injections; they are injective in the simple case, which is the one under consideration there, but not in general.]) of observables into the Lie algebra, i.e. in the opposite direction from the map required by energy observability, and also that dynamical correspondence imposes some conditions of compatibility with the Jordan structure, while the notion of energy observability uses only the convex structure of the state space and does not need Jordan structure for its definition.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces · From Jordan state spaces to complex quantum theory via relations between continuous symmetries and observables · contest 0.88

Builds on it 1 paper

We extend the results of Ref. [ref] , where it was shown that in the context of Postulates 1 and 2 the existence of higher-order interference is equivalent to each of three other statements.

Thermodynamics and the structure of quantum theory · Conclusions · 1.00

Uses its method or tool 1 paper

We will simply use Postulate 2 from [ref] .

Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory · Tool 2: Reversible pure state conversions · 0.94

Compares with it 2 papers

As mentioned in the introduction, Theorem [ref] improves on the main result of [ref] , which characterized the same class of theories, but in addition to spectrality and strong symmetry, used the additional assumption of no higher-order interference.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces · Higher order interference · 0.96
With all the assumptions made along the way, the landscape of inference theories devised here is perhaps not quite as general as the commonly employed landscape of generalized probability theories [ref] .

Toolbox for reconstructing quantum theory from rules on information acquisition · The landscape $$ · 0.85

Cites it as support 8 papers

The second sentence holds because Postulates 1 and 2 of [ref] imply both P and S.

Entropy, majorization and thermodynamics in general probabilistic theories · Measurements, measurement entropy, and majorization · 0.97
However, the conjunction of spectrality and strong symmetry, even with respect to a restricted set of effects, is shown in [ref] to imply no-restriction.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces · Background and framework · 0.97
Indeed, many authors emphasize this point [ref] , e.g., Koberinski & Müller write [ref] : We suggest that (continuous) reversibility may be the postulate which comes closest to being a candidate for a glimpse on the genuinely physical kernel of “quantum reality”.

The Relativity Principle at the Foundation of Quantum Mechanics · Introduction · 0.97

Cites it as background 15 papers

Postulate 2 of [ref] is Strong Symmetry (SS): that every set of mutually distinguishable pure states can be taken to any other such set of the same size by a reversible transformation (affine automorphism of $Ω$).

Entropy, majorization and thermodynamics in general probabilistic theories · Spectrality · 0.81
A series of recent papers [ref] have shown that any of various packages of probabilistic or information-theoretic axioms force the state spaces of a finite-dimensional probabilistic theory to be those of formally real, or euclidean, Jordan algebras.

Some Nearly Quantum Theories · Introduction · 0.94
Section [ref] defines perfect distinguishability, spectrality and strong symmetry, and reviews important consequences (mostly from [ref] ) of the latter two properties that will be needed in proving the main theorem.

Strongly symmetric spectral convex bodies are Jordan algebra state spaces · Introduction · 0.96

What it cites · 37 sentences without a confident label

Uses the method or tool of 2 papers

Following the definition of [ref] , expressed in the language of [ref] , every system satisfying Postulates 1 and 2 defines a contextuality scenario given by a hypergraph $H$, where the vertices of $H$ are the projective units $u_F$ ($F≠{0}$ any face of $A_+$), and the edges are collections of effects $u_F_1,…, u_F_n$ with $∑_i=1^n u_F_i=u_A$.

(Non-)Contextuality of Physical Theories as an Axiom · Consequences of Postulates 1+2 · 0.85
Following the definition of [ref] , expressed in the language of [ref] , every system satisfying Postulates 1 and 2 defines a contextuality scenario given by a hypergraph $H$, where the vertices of $H$ are the projective units $u_F$ ($F≠{0}$ any face of $A_+$), and the edges are collections of effects $u_F_1,…, u_F_n$ with $∑_i=1^n u_F_i=u_A$.

A Combinatorial Approach to Nonlocality and Contextuality · Consequences of Postulates 1+2 · 0.85

Compares with 11 papers

Sorkin's [ref] original definition, and the discussion in [ref] , used the $M=3$ case as their definition of third-order interference, but the two can straightforwardly if somewhat tediously be shown to be equivalent.

Three Slit Experiments and the Structure of Quantum Theory · Third-order interference · 0.82
Sorkin's [ref] original definition, and the discussion in [ref] , used the $M=3$ case as their definition of third-order interference, but the two can straightforwardly if somewhat tediously be shown to be equivalent.

Quantum Mechanics as Quantum Measure Theory · Third-order interference · 0.82
Niestegge has also used the absence of higher-order interference as one ingredient in deriving Jordan algebraic systems [ref] .

Conditional probability, three-slit experiments, and the Jordan algebra structure of quantum mechanics · Discussion and conclusions · 0.81

Cites as support 1 paper

Ref. [ref] shows that bit symmetry and the fact that all effects are allowed imply this proposition.

The structure of reversible computation determines the self-duality of quantum theory · Consequences of Postulates 1+2 · 0.98

Cites as background 25 papers

Rafael Sorkin defined a notion of $k$-th order interference [ref] , which can be manifested in analogues of the two-slit experiment involving $k$ or more slits.

Quantum Mechanics as Quantum Measure Theory · Third-order interference · 0.98
More details on the framework can be found in e.g. [ref] , [ref] , [ref] , [ref] , [ref] ; also, [ref] offer accessible introductions.

Teleportation in General Probabilistic Theories · Operational Probabilistic Theories · 1.00
More details on the framework can be found in e.g. [ref] , [ref] , [ref] , [ref] , [ref] ; also, [ref] offer accessible introductions.

Cloning and Broadcasting in Generic Probabilistic Theories · Operational Probabilistic Theories · 1.00

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