Indistinguishability in general probabilistic theories

Ana Bel\'en Sainz; John H. Selby; Marcin Karczewski; M\'at\'e Farkas; Victoria J. Wright

REVIEW 4 major objections 5 minor 28 references

Indistinguishability in general probabilistic theories

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper defines indistinguishable particle types in any general probabilistic theory by symmetry orbits or by splitting the symmetrisation process, and recovers bosons and fermions for quantum pairs.

desk verdict A genuinely new orbit-based framework for particle types in GPTs, but the diagrammatic half is a promising sketch whose central uniqueness theorem is explicitly deferred. read the letter →

arxiv 2412.20963 v1 pith:JX5XION2 submitted 2024-12-30 quant-ph

John H. Selby , Victoria J. Wright , Máté Farkas , Marcin Karczewski , Ana Belén Sainz This is my paper · ORCID

classification quant-ph MSC 81P0518M0581P16 PACS 03.65.Ta03.65.-w

keywords generalprobabilistictheoriesindistinguishableparticlesparticletypesexchangesymmetryoperationalsymmetrisationKaroubienvelopebiproductsprocess

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that indistinguishability of particles is a general feature that can be defined in any general probabilistic theory (GPT), not only in quantum theory. Its first definition types indistinguishable particles by the orbits of symmetric pure states under transformations that preserve the swap symmetry; its second, diagrammatic definition types them by the direct-sum sectors obtained when the operational symmetrisation process is split in a completed process theory. In both approaches a pair of identical quantum systems yields exactly two types, bosons and fermions, so the paper offers a theory-independent backdrop against which quantum indistinguishability can be compared with other theories. Applying the same definitions elsewhere gives one or two types in a classical GPT depending on a choice of pure-state convention, no new types in Boxworld, and three types in an epistemic toy theory.

What carries the argument

The orbit quotient $\mathcal{O}_{T^s_S} = S^2_{ex,s}/T^s_S$ (Eq. 13) is the central object of the first framework: it collects symmetric pure states into orbits under swap-preserving transformations, and each orbit is declared to be one type of indistinguishable particle. The central object of the second framework is the operational symmetrisation idempotent $\mathrm{Sym} = \frac12(\mathrm{id}+\mathrm{swap})$ (Eq. 23), an equal mixture of doing nothing and swapping; in the Karoubi envelope, a categorical completion in which every idempotent is forced to split into a system, $\mathrm{Sym}$ splits through a system $\mathrm{Sym}^n_A$, and the unique most refined decomposition of $\mathrm{Sym}$ into orthogonal idempotents turns that system into a biproduct (direct sum) of particle-type systems. The first object carries the state-space classification; the second carries the process-theoretic classification and also defines a non-disturbing 'particle type measurement' on symmetrised systems.

What would settle it

Exhibit a finite process theory in which the operational symmetrisation idempotent admits two different most-refined decompositions into orthogonal idempotents: the paper's claim that there is a unique canonical decomposition would then fail, and the diagrammatic particle-type sectors would no longer be well-defined.

Watch

Extended reading notes

Core claim

On the paper's own terms, a pair of indistinguishable particles is a pair whose every state is invariant under relabelling, $\omega = P\omega$ for the swap map $P$. Particle types are then the equivalence classes of symmetric pure states under the subgroup $T^s_S$ of transformations that commute with $P$ on the symmetric subspace, namely the orbit set $\mathcal{O}_{T^s_S} = S^2_{ex,s}/T^s_S$. In the diagrammatic formulation, the equal mixture of doing nothing and swapping, $\mathrm{Sym} = \frac12(\mathrm{id} + P)$, is an idempotent process; splitting it via the Karoubi envelope and decomposing it into a sum of orthogonal idempotents produces a biproduct decomposition $\mathrm{Sym}^n_A = \bigoplus_i (S^n_A)_i$, and each summand is one particle type. For two quantum systems the decomposition recovers the antisymmetric (fermion) and symmetric (boson) sectors, e.g. $B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$ symmetrises to $B(\mathbb{C}^3)\oplus B(\mathbb{C}^1)$. The paper presents these as the two different routes to the same intended notion of particle type, applicable to arbitrary GPTs.

Load-bearing premise

The load-bearing premise is that physical indistinguishability is fully captured by exchange symmetry of states: a pair of particles counts as indistinguishable exactly when swapping their labels leaves the state invariant, so only swap-invariant states and swap-preserving transformations enter the classification; if a GPT could distinguish particles by tracking trajectories or by any other means, this definition would classify the wrong objects.

Editorial extensions

If this is right

If the orbit definition is correct, the particle types of a GPT are fixed by the geometry of its symmetric pure states and by which transformations preserve swap symmetry; in any transitive GPT whose pure states form a single orbit, the types are counted by the splitting of that single orbit under $T^s_S$.
The diagrammatic definition makes particle type a measurable, non-disturbing property of a symmetrised system, because the idempotent decomposition $\mathrm{Sym} = \sum_i \mathrm{Part}_i$ is interpreted as the outcomes of a particle-type measurement.
For quantum systems of more than two particles, the diagrammatic approach predicts additional sectors beyond bosons and fermions, which the paper reads as paraparticles.
In Boxworld, imposing indistinguishability does not create new particle types: the product and entangled orbits already present are merely restricted to symmetric states.
In the toy model, the orbit definition predicts exactly three indistinguishable-particle types, distinguished by whether the epistemic state contains 0, 2, or 4 symmetric ontic states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An open question the paper does not settle is whether the orbit classification and the diagrammatic sector decomposition always agree; finding a GPT where the two counts differ would show that 'particle type' is formalism-dependent, and one would need an extra principle to select the physical notion.
The dependence of the orbit definition on choosing symmetric-extremal states versus extremal symmetric states (the two options in Sec. 2.2) suggests that a fully predictive theory of particle types needs an additional operational criterion to fix the pure-state convention; the classical GPT example, where the two choices give two types versus one, is a concrete place to test such a criterion.
Because the framework deliberately sets aside trajectory-based indistinguishability, an extension combining exchange symmetry with trajectory labels might be needed for GPTs that admit definite particle paths; the present definitions would then describe only the exchange-symmetric sector of that richer notion.
The sector decomposition suggests a way to define fusion rules for particle types in GPTs by composing symmetrised systems and reducing the resulting idempotents, which could connect indistinguishable-particle statistics in non-quantum theories to anyonic behaviour.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper does something actually new: it gives a theory-independent way to define indistinguishable particle types that works for arbitrary GPTs, not just those with a swap experiment. The orbit-based definition in Sec. 2.2 is simple and clear, and the quantum recovery of bosons and fermions is essentially right. The diagrammatic approach via Karoubi envelope and biproducts is a neat idea and well-motivated, even if it is not fully carried through. The examples—classical, Boxworld, Spekkens—are refreshingly concrete and honestly presented, and the paper is upfront about what it leaves open. I take the authors seriously as thinkers.

The soft spots are real but not fatal to the programme. The stress-test is right: the diagrammatic classification of particle types rests on the claimed uniqueness of the most refined decomposition of the symmetrisation idempotent, and the proof is deferred to a 'second version of this work.' That means Eq. (43) is not yet established, and 'particle type' in that framework is not yet a theorem but a conjecture. The orbit-based approach does not suffer from this, but it has its own unresolved ambiguity: the two choices for what counts as a symmetric pure state (extremal symmetric vs symmetric extremal) can give different particle types in classical theory, as the paper itself shows. That is not a mistake, but it means the definition is not yet uniquely pinned down. Also, the qubit example in Sec. 3.6 mislabels the sectors: the qutrit is the bosonic (symmetric) sector and the qubit is the fermionic (antisymmetric) sector, not the other way around. Minor in the grand scheme, but it should be fixed.

The Boxworld and Spekkens analyses are asserted more than proved, so I would want derivations before trusting those counts. Still, the central conceptual contribution—that particle types can be defined as orbits under symmetry-preserving transformations, and that this recovers quantum statistics—holds up for what it claims.

Who is this for? People working on GPTs and on the foundations of indistinguishability. It is a good reading-group paper because it will spark discussion about what 'indistinguishability' should mean. I would cite it as the starting point for a new line of work, but I would pair it with the caveat that the diagrammatic part is not yet complete.

Recommendation: yes, send it to peer review. It deserves refereeing, and the referees should push the authors to either supply the uniqueness proof or clearly mark it as a conjecture. The paper is not ready to be accepted as-is, but it is exactly the kind of nonstandard contribution a serious editor should engage with.

Referee Report

4 major / 5 minor

Summary. The paper proposes two frameworks for defining types of indistinguishable particles in general probabilistic theories (GPTs). In the first, orbit-based framework, particle types are equivalence classes of symmetric pure states under the subgroup of transformations that preserve the symmetric subspace; in quantum theory this recovers bosons and fermions for pairs. In the second, diagrammatic framework, particle types are obtained by splitting the operational symmetrisation idempotent into orthogonal idempotents in the Karoubi envelope and biproduct completion, again recovering bosons and fermions for pairs of quantum systems. The paper works through several examples (quantum theory, classical GPT, Boxworld, and Spekkens' toy model) and discusses limitations, deferring two key proofs to a future version.

Significance. If the deferred results can be supplied, the paper would provide a genuinely theory-independent language for particle types, extending earlier swap-experiment approaches that are limited to a narrow class of GPTs. The orbit-based treatment is fully worked out for the examples and the recovery of bosons and fermions in quantum theory is convincingly demonstrated via Schur-Weyl duality. The paper is clearly written and the examples are instructive. However, the diagrammatic half of the paper currently rests on an unproven uniqueness assertion, and the claimed generality is not yet supported for classical GPTs.

major comments (4)

[3.6, Eqs. (42)–(43)] The diagrammatic definition of particle types rests on the asserted existence and uniqueness of the "most refined" decomposition of the symmetrisation idempotent into orthogonal idempotents. This assertion is explicitly deferred to a second version ("This fact will be demonstrated in a forthcoming second version of this work" and "In a second version of this work, we will see that this decomposition cannot be refined further"). Because Eq. (43) and the interpretation of (S^n_A)_i as particle types depend on this uniqueness, the central construction of the diagrammatic framework is not yet established. Please provide the proof, or a precise statement of the conditions (e.g., centrality or invariance under the reversible transformations) under which the decomposition is unique, and verify that the quantum decomposition Sym^2_Q = B(H_s) ⊕ B(H_a) satisfies those conditions.
[3.6] The diagrammatic framework does not yet apply to classical GPTs: the text states that the definition "only really makes sense if the system A ⊗ ... ⊗ A cannot itself be (nontrivially) written as a direct sum decomposition," that classical theory is the key counterexample, and that formalising this case is left to a second version. Since classical GPTs are the most basic examples and the abstract promises a general notion of indistinguishable particles in GPTs, this gap needs to be addressed before the framework can be said to deliver a theory-independent classification.
[2.2 and 2.3.2] The orbit-based definition is parameterised by an unresolved choice between Option I (extremal symmetric states) and Option II (symmetric extremal states). The two options are not equivalent in general: for the classical GPT, Option I gives two particle types (diagonal and off-diagonal symmetric states) while Option II gives one. Since the paper does not select a preferred option or provide a criterion, the predicted set of particle types is not uniquely determined by the GPT alone. Please either justify a canonical choice or present the framework as a family of definitions parameterised by this choice.
[3.4 and 3.6] The claim that the diagrammatic construction recovers bosons and fermions for all finite-dimensional quantum systems is only illustrated for two qubits in §3.6 (Sym^2_B(C^2) = B(C^3) ⊕ B(C^1)); the general-dimensional statement in §3.4 is asserted without proof. Once the uniqueness theorem from §3.6 is supplied, please give the general argument and state the sense in which the two summands are the bosonic and fermionic sectors.

minor comments (5)

[2.1] In the paragraph after Eq. (5), "pure quantum states satisfying Eq. (5)" is imprecise because Eq. (5) defines the action of the swap on basis vectors; the intended condition is Pψ = ±ψ, which follows from P^2 = 1.
[2.3.2] In Option I, the states 1/2([j]⊗[k]+[k]⊗[j]) for j ≠ k are extremal in the symmetric state space but are mixtures in the full state space; calling them "pure states" could confuse readers, so adding a clarifying remark would help.
[3.4] In Eq. (35), the rightmost diagram is labelled "Sym^n_A" but the surrounding text and the left side concern Sym^3_A; this appears to be a typo.
[3.6] The notation (S^n_A)_i is used before it is defined; please introduce it explicitly.
[Throughout] There are several typos, e.g., "indistinguishablility" (§2.1), "simpliticy" (§3.4), and "Symn_A" versus "Sym3_A" in Eq. (35); a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the orbit-based and diagrammatic particle-type definitions are explicit constructions, and the quantum boson/fermion result follows from Schur–Weyl duality rather than from assuming the conclusion.

full rationale

The paper's two frameworks are constructional rather than circular. In the GPT approach, particle types are defined as orbits of symmetric pure states under symmetry-preserving transformations (Eq. 13), and the quantum classification into two orbits is derived from Schur–Weyl duality, not from assuming the boson/fermion dichotomy. The diagrammatic approach defines operational symmetrisation as an equal mixture of identity and swap (Eq. 23), forms the Karoubi envelope, and then splits the symmetrisation idempotent into orthogonal idempotents whose summands are declared particle types (Eqs. 42-43). The quantum result Sym^2 = B(C3) ⊕ B(C1) is a standard computation, and no fitted parameter is renamed as a prediction. The paper does cite Ref. [28]—which shares an author—for the relation between biproduct completion and the Karoubi envelope, but that categorical result is independent of the particle-type conclusion and the relevant constructions are also recapitalized in Sections 3.3 and 3.5, so the citation is not load-bearing in a circular way. The genuine weaknesses are non-circular incompleteness: the uniqueness of the 'most refined' decomposition is deferred ('This fact will be demonstrated in a forthcoming second version of this work'), and the classical direct-sum case is left unformalized ('We, however, leave formalising this to a second version of this work'). Likewise, Sec. 2.2 leaves open which of two pure-state conventions to adopt, honestly reporting that quantum theory coincides but classical GPT gives different counts. These are proof gaps and definitional ambiguities, not reductions of outputs to inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 4 invented entities

The central framework rests on standard GPT and category-theoretic assumptions, plus one asserted-but-unproved uniqueness statement. The only modeling freedom that changes the output is the choice between Option I and Option II for pure states. No numerical free parameters are fitted to data, and no physically independent new entities are introduced; the new systems are internal mathematical constructions.

free parameters (1)

Choice of pure-state set (Option I vs Option II)
Sec. 2.2 offers two inequivalent definitions of the pure states of a particle type. In the classical GPT (Sec. 2.3.2) Option I yields two particle types while Option II yields one. The paper does not fix a unique choice, so the particle-type classification depends on an unresolved modeling choice.

assumptions (6)

domain assumption The GPT is tomographically local, so a pair of systems has state and effect spaces embedded in V⊗V rather than V⊗V⊕W.
Stated in Sec. 2 before Eq. (3): 'under the assumption of local tomography'; the paper omits holistic degrees of freedom.
domain assumption The GPT satisfies the no-restriction hypothesis for effects and transformations.
Sec. 2: 'For simplicity, in this work we will assume our GPTs satisfy the no-restriction hypothesis.' The Spekkens toy model does not satisfy it, yet the method is applied there.
domain assumption Indistinguishability is equivalent to exchange symmetry of the (operationally symmetric) state, not to trajectory distinguishability.
Sec. 1: 'This invariance of the state under exchange of the labels will form the notion of indistinguishability we will study'; Eq. (4) formalizes it.
standard math Process theories are symmetric monoidal categories with sums of processes, and operational symmetrisation is the equal mixture of the identity and swap (Eq. 23).
Sec. 3.1 defines process theories and sums; Eq. (23) defines the specific symmetrisation process used throughout.
standard math For quantum pairs, Schur-Weyl duality gives H⊗H = Hs⊕Ha, and unitaries commuting with the swap are block-diagonal Us⊕Ua.
Used in Sec. 2.3.1 to compute the two orbits and in Sec. 3.6 for the decomposition Sym²_{B(C2)} = B(C3)⊕B(C1).
ad hoc to paper There exists a unique most refined decomposition of the symmetrisation idempotent into orthogonal idempotents, and it can be obtained by coarse graining from any other decomposition.
Sec. 3.6: 'There is a canonical choice for this as it can be shown that there is a unique most refined such decomposition... This fact will be demonstrated in a forthcoming second version of this work.' The particle-type definition depends on this unproved assertion.

invented entities (4)

Operationally symmetrised system Symⁿ_A
purpose: Represents a collection of n indistinguishable particles of type A, defined by splitting the operational symmetrisation idempotent in the Karoubi envelope.
Introduced in Sec. 3.4; it is a new system type in the extended process theory KSym[P], with no observable predicted outside the framework.
Particle-type sector (Sⁿ_A)_i
purpose: Labels the different particle types as biproduct summands of Symⁿ_A.
Defined in Sec. 3.6 via the decomposition of the symmetrisation idempotent. In quantum theory it matches bosonic/fermionic sectors, but as a GPT-level object it has no independent falsifiable handle in this paper.
Part_i idempotents (particle-type measurement outcomes)
purpose: Mathematical components whose sum is the symmetrisation idempotent; interpreted as outcomes of a non-disturbing measurement of particle type.
Sec. 3.6 treats Part_i as a measurement on the symmetrised system; the existence and uniqueness of the maximal refinement is asserted without proof.
Orbit-based particle type π
purpose: A particle type in the traditional GPT framework, defined as an orbit of symmetric pure states under symmetry-preserving transformations.
Defined in Sec. 2.2, Eq. (13); a mathematical equivalence class rather than a physically evidenced new entity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Indistinguishability in general probabilistic theories." pith.science (2026). https://pith.science/paper/JX5XION2

@misc{pith2026241220963,
  author       = {Pith},
  title        = {Pith review of: Indistinguishability in general probabilistic theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JX5XION2}},
  note         = {Machine review of arXiv:2412.20963}
}

read the original abstract

The existence of indistinguishable quantum particles provides an explanation for various physical phenomena we observe in nature. We lay out a path for the study of indistinguishable particles in general probabilistic theories (GPTs) via two frameworks: the traditional GPT framework and the diagrammatic framework of process theories. In the first approach we define different types of indistinguishable particle by the orbits of symmetric states under transformations. In the diagrammatic approach, we find a decomposition of the symmetrised state space using two key constructions from category theory: the biproduct completion and the Karoubi envelope. In both cases for pairs of indistinguishable particles in quantum theory we recover bosons and fermions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 24 canonical work pages

[1]

Indistinguishable particles in quantum mechanics: An introduction

Yasser Omar. Indistinguishable particles in quantum mechanics: An introduction. Contemporary Physics, 46(6):437–448, 2005

work page 2005
[2]

Indistinguishability

Nick Huggett and Tom Imbo. Indistinguishability. In Compendium of quantum physics , pages 311–317. Springer, 2009

work page 2009
[3]

W. Pauli. ¨Uber den Einfluß der Geschwindigkeitsabh¨ angigkeit der Elek- tronenmasse auf den Zeemaneffekt. Zeitschrift f¨ ur Physik, 31(1):373– 385, 1925. 23

work page 1925
[4]

L. M. Falicov. Niels Bohr Collected Works Vol 4: The Periodic System (1920–1923). 1977

work page 1920
[5]

W. Pauli. ¨Uber den Zusammenhang des Abschlusses der Elektronen- gruppen im Atom mit der Komplexstruktur der Spektren. 1925

work page 1925
[6]

A. M. L. Messiah and O. W. Greenberg. Symmetrization postulate and its experimental foundation. Physical Review, 136(1B):B248–B267

work page
[7]

University Press, 1952

Erwin Schr¨ odinger.Science and humanism . University Press, 1952

work page 1952
[8]

Clarendon Press, 2006

D´ ecio Krause Steven French.Identity in Physics: A Historical, Philo- sophical, and Formal Analysis . Clarendon Press, 2006

work page 2006

Show all 28 references

[9]

Identity, individuality and indistinguishability in physics and mathematics

Gabriel Catren. Identity, individuality and indistinguishability in physics and mathematics. Philosophical Transactions of the Royal So- ciety A: Mathematical, Physical and Engineering Sciences , 381(2255), 2023

2023
[10]

Gigena, M

N. Gigena, M. Di Tullio, and R. Rossignoli. One-body entanglement as a quantum resource in fermionic systems. Physical Review A , 102(4)
[11]

Indistinguishability of elementary systems as a resource for quantum information processing

Rosario Lo Franco and Giuseppe Compagno. Indistinguishability of elementary systems as a resource for quantum information processing. Phys. Rev. Lett. , 120(24)
[12]

Information processing in generalized probabilistic theories

Jonathan Barrett. Information processing in generalized probabilistic theories. Physical Review A , 75(3)
[13]

General probabilistic theories: An introduction

Martin Pl´ avala. General probabilistic theories: An introduction. Physics Reports, 1033:1–64, 2023

2023
[14]

Quantum theory from five reasonable axioms

Lucien Hardy. Quantum theory from five reasonable axioms. arXiv preprint quant-ph/0101012, 2001

2001 arXiv
[15]

M¨ uller

Llu ´ ıs Masanes and Markus P. M¨ uller. A derivation of quantum theory from physical requirements. New Journal of Physics , 13(6):063001
[16]

Limits on nonlocal correlations from the structure of the local state space

Peter Janotta, Christian Gogolin, Jonathan Barrett, and Nicolas Brun- ner. Limits on nonlocal correlations from the structure of the local state space. New Journal of Physics , 13(6):063024
[17]

Selby, Elie Wolfe, Ravi Kunjwal, and Robert W

David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens. Characterization of noncontextuality in the framework of generalized probabilistic theories. PRX Quantum , 2(1), 2021

2021
[18]

Prob- abilistic theories with purification

Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Prob- abilistic theories with purification. Physical Review A , 81(6)
[19]

Selby, Carlo Maria Scandolo, and Bob Coecke

John H. Selby, Carlo Maria Scandolo, and Bob Coecke. Reconstructing quantum theory from diagrammatic postulates. Quantum, 5:445
[20]

Particle exchange in post-quantum theories

Oscar CO Dahlsten, Andrew JP Garner, Jayne Thompson, Mile Gu, and Vlatko Vedral. Particle exchange in post-quantum theories. arXiv preprint arXiv:1307.2529, 2013

2013 arXiv
[21]

Generalised phase kick-back: the structure of computational algorithms from physical principles

Ciar´ an M Lee and John H Selby. Generalised phase kick-back: the structure of computational algorithms from physical principles. New Journal of Physics , 18(3):033023, 2016

2016
[22]

Strongly symmetric spec- tral convex bodies are jordan algebra state spaces

Howard Barnum and Joachim Hilgert. Strongly symmetric spec- tral convex bodies are jordan algebra state spaces. arXiv preprint 24 arXiv:1904.03753, 2019

1904 arXiv
[23]

Invertible quantum operations and perfect encryption of quantum states

Ashwin Nayak and Pranab Sen. Invertible quantum operations and perfect encryption of quantum states. Quantum Information and Com- putation, 7(1):103–110, 2007

2007
[24]

All reversible dynamics in maximally nonlocal theories are trivial.Phys- ical review letters , 104(8):080402, 2010

David Gross, Markus M¨ uller, Roger Colbeck, and Oscar CO Dahlsten. All reversible dynamics in maximally nonlocal theories are trivial.Phys- ical review letters , 104(8):080402, 2010

2010
[25]

Spekkens

Robert W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Physical Review A , 75(3)
[26]

Generalized probabilistic theories without the no-restriction hypothesis

Peter Janotta and Raymond Lal. Generalized probabilistic theories without the no-restriction hypothesis. Physical Review A , 87(5)
[27]

Picturing quantum processes: A first course on quantum theory and diagrammatic reasoning

Bob Coecke and Aleks Kissinger. Picturing quantum processes: A first course on quantum theory and diagrammatic reasoning. In Di- agrammatic Representation and Inference: 10th International Confer- ence, Diagrams 2018, Edinburgh, UK, June 18-22, 2018, Proceedings 10, pages 28–3...

2018
[28]

Two roads to classicality

Bob Coecke, John Selby, and Sean Tull. Two roads to classicality. Electronic Proceedings in Theoretical Computer Science, 266:104–118. 25

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Indistinguishable particles in quantum mechanics: An introduction

Yasser Omar. Indistinguishable particles in quantum mechanics: An introduction. Contemporary Physics, 46(6):437–448, 2005

work page 2005

[2] [2]

Indistinguishability

Nick Huggett and Tom Imbo. Indistinguishability. In Compendium of quantum physics , pages 311–317. Springer, 2009

work page 2009

[3] [3]

W. Pauli. ¨Uber den Einfluß der Geschwindigkeitsabh¨ angigkeit der Elek- tronenmasse auf den Zeemaneffekt. Zeitschrift f¨ ur Physik, 31(1):373– 385, 1925. 23

work page 1925

[4] [4]

L. M. Falicov. Niels Bohr Collected Works Vol 4: The Periodic System (1920–1923). 1977

work page 1920

[5] [5]

W. Pauli. ¨Uber den Zusammenhang des Abschlusses der Elektronen- gruppen im Atom mit der Komplexstruktur der Spektren. 1925

work page 1925

[6] [6]

A. M. L. Messiah and O. W. Greenberg. Symmetrization postulate and its experimental foundation. Physical Review, 136(1B):B248–B267

work page

[7] [7]

University Press, 1952

Erwin Schr¨ odinger.Science and humanism . University Press, 1952

work page 1952

[8] [8]

Clarendon Press, 2006

D´ ecio Krause Steven French.Identity in Physics: A Historical, Philo- sophical, and Formal Analysis . Clarendon Press, 2006

work page 2006