Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Generalised Process Theories

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper proposes that every process theory, traditional or generalised, is an algebra of a wiring operad, and shows the definition recovers all known variants.

desk verdict A genuine unification proposal for process theories, but the enrichment theorem is underdefined and needs a real proof before the central claim holds. read the letter →

arxiv 2502.10368 v2 pith:DFJYCRTL submitted 2025-02-14 math.CT math-phmath.MPquant-ph

John H. Selby , Maria E. Stasinou , Matt Wilson , Bob Coecke This is my paper · ORCID
classification math.CTmath-phmath.MPquant-ph
keywords processtheoriesoperadalgebraswiringoperadssymmetricmonoidalcategoriescausaltime-neutralhigher-orderprocessesenriched
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

The paper argues that the usual home of process theories, symmetric monoidal categories, is too rigid to cover variants that have appeared in quantum foundations, and that a slightly more general formalism does the job: a process theory is simply an algebra of an operad of wiring diagrams. Different choices of the operad reproduce traditional, time-neutral, causal, and higher-order process theories, and different choices of the codomain reproduce enrichment. The paper proposes this as the definition of a generalised process theory and shows by explicit constructions that each previously known variant is an instance. A reader should care because it offers a single composition-aware framework, with string-diagram notation throughout, in which maps between theories are natural transformations preserving wirings.

What carries the argument

The machinery is the wiring operad and its algebras. A wiring operad has boxes or dots as objects and wirings as operations, with composition by diagram substitution; an algebra sends each object to a set of processes and each wiring to the corresponding composition function. Varying the operad changes the allowed notion of composition, with acyclic wirings for traditional theories, arbitrary wirings on dots for time-neutral theories, wirings with discarding for causal theories, and undirected acyclic wirings for polycategorical higher-order theories, while varying the codomain from $\mathbf{Set}$ to a general operad $\mathbb{K}$ changes the enrichment. The paper uses string diagrams for both the operads and their algebras to make the constructions visually checkable.

What would settle it

Exhibit an acyclic wiring diagram that cannot be assembled from sequential, parallel, identity, and swap wirings, or an operad algebra for which two different decompositions of the same wiring produce different composite processes; either would break the claimed equivalence between symmetric monoidal categories and acyclic-wiring operad algebras, and with it the recovery of traditional process theories.

Watch

Extended reading notes

Core claim

The paper's central claim is Definition 6.2: a generalised process theory is an operad algebra $F : \mathcal{O} \to \mathbb{K}$, where $\mathcal{O}$ is an operad whose operations are wirings, $\mathbb{K}$ is an operad whose operations supply enrichment, and $F$ sends each type in $\mathcal{O}$ to a set or enriched hom-object of processes. The paper claims this single definition recovers traditional process theories as algebras of the acyclic wiring operad $\mathbb{W}_A$, time-neutral theories as algebras of the dot wiring operad $\mathbb{W}_D$, causal theories as algebras of the causal wiring operad $\mathbb{W}$, higher-order polycategorical structures as algebras of the undirected acyclic wiring operad $\mathbb{W}_{UA}$, and enriched theories as algebras with codomain other than $\mathbf{Set}$. The constructions are carried out explicitly, including a proof that algebras of the causal wiring operad are causal symmetric monoidal categories and a theorem that algebras of $\mathbb{W}_{UA}$ are polycategories with space.

Load-bearing premise

The framework rests on the previously proven fact that every allowable wiring of boxes can be built from sequential, parallel, identity, and swap wirings, no matter how the diagram is drawn; if that decomposition failed, the operad algebra could not be guaranteed to define a well-behaved composition law.

Editorial extensions

If this is right

  • Traditional process theories need no fictitious trivial system, no promoted identity and swap morphisms, and no preferred foliation into sequential and parallel composition; the operad algebra carries the full composition structure directly.
  • Time-neutral theories can be studied natively via algebras of the dot wiring operad $\mathbb{W}_D$, and these are equivalent to compact-closed categories with cups and caps, making the two presentations interchangeable.
  • Causal process theories arise exactly as algebras of the causal wiring operad $\mathbb{W}$, with the discard equation imposed freely; the resulting symmetric monoidal category has a terminal monoidal unit.
  • Enrichment is achieved by replacing $\mathbf{Set}$ with an operad $\mathbb{K}$: for instance, taking $\mathbb{K}$ to be convex spaces yields convexly enriched process theories such as quantum channels.
  • Higher-order supermaps, whose plugging rules are restricted to avoid causal loops, are captured by algebras of the undirected acyclic wiring operad, whose categorical counterpart is a polycategory with space.

Reading between the lines

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

  • If the operadic definition is taken as primary, compositionality itself can be turned into a resource: one can compare process theories by which wiring operad they are algebras of, so the question of which wirings are free becomes a resource-theoretic choice.
  • The same framework suggests a direct translation between tensor-network contractions and operadic composition in $\mathbb{W}_D$, giving tensor networks a presentation that does not privilege inputs over outputs; the paper gestures at this but does not develop it into a full calculus.
  • The sketch of partially time-neutral theories in Section 3.1 could be made precise as an operad whose objects are three-dimensional plates with acyclic vertical wiring and unrestricted horizontal wiring, modelling mixed space-like and time-like connectivity in tensor networks.
  • The resemblance noted in Appendix D between the enriched string diagrams and the causal-inferential diagrammatic language suggests that that language may itself be an enriched operad algebra, a connection the paper leaves for future work.
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.

Referee Report

3 major / 5 minor

Summary. The paper proposes replacing symmetric monoidal categories as the formal foundation of process theories by operad algebras over wiring operads. It defines an acyclic wiring operad WA whose Set-algebras are identified with traditional process theories (Definition 2.2); a wiring operad of dots WD whose Set-algebras are time-neutral process theories (Definition 3.5), with an operad equivalence between WD and the cyclic wiring operad WC of cups and caps established in Appendix A; an undirected acyclic wiring operad WUA whose algebras are claimed to be 'polycategories with space' (Section 4, Theorem 4.1); and a causal wiring operad W, obtained from WA by adding discarding and quotienting by the discard equation, whose Set-algebras are claimed to be causal symmetric monoidal categories (Section 5, Definition 5.1). Section 6 then allows the target of the algebra to be an arbitrary operad K, claiming in Theorem 6.1 that algebras F: WA to K encode 'K-enriched symmetric monoidal categories', and Definition 6.2 defines a generalised process theory as any operad algebra F: O to K with O capturing composition, K capturing enrichment, and F the specific theory. The final section sketches applications to tensor networks, ZX calculus, and compositional quantum field theory.

Significance. If the recovery theorems were complete, this would be a genuinely unifying and valuable framework: ordinary, time-neutral, higher-order, causal, and enriched process theories would all be algebras of suitably chosen wiring operads, and the paper's string-diagrammatic language would extend diagrammatic reasoning uniformly across all of them. The paper does contain original mathematical content with complete proofs in places: Appendix A establishes the operad equivalence WD isomorphic to WC; Appendix B gives a formal affine-completion and quotient machinery (Theorems B.8-B.13) for imposing the discard equation; and the WD and W constructions are plausible and well motivated. The exposition is unusually accessible and the examples (tensor networks, ZX calculus, time-neutral quantum theory, convex enrichment) are well chosen. However, two of the paper's headline claims are currently ahead of their proofs: Theorem 6.1 is a four-sentence gloss resting on an undefined notion of K-enrichment, and Theorem 4.1 is a sketch whose appendix proof contains an improperly typeset diagram and an asserted rather than demonstrated round-trip.

major comments (3)
  1. [§6, Theorem 6.1; Definition 6.2] The central clause of Definition 6.2, that 'K captures a theory of enrichment', is unsupported. Theorem 6.1 asserts that an operad algebra F: WA to K 'encodes the information that constitutes a K-enriched symmetric monoidal category', but the paper never defines a K-enriched symmetric monoidal category for a general operad K. Standard enrichment is over a monoidal category V: hom-objects are objects of V and composition is a V-morphism V(Hom(B,C), Hom(A,B)) to Hom(A,C). In contrast, an operad operation is multi-input/single-output data, and K carries no tensor product, no unit object, and no notion of identity element. The proof's sentence that the construction is 'given in exactly the same way as it was presented for set' cannot transfer the set-based construction, which uses cartesian products of hom-sets and the unique map to the singleton at every step; in particular, the identities 1_A in Eq. (40) and the symmetry S_AB in Eq. (41) are defined as maps from the singleton set, and no analogue exists in a general operad. The paper neither lists the enrichment data (identity, composition, parallel composition) nor verifies the enrichment axioms, and it does not give the converse direction, from a K-enriched SMC to an algebra F: WA to K. The example K = ConvSpc is also underspecified: the paper should state the operad operations on convex spaces and how hom-objects and composition maps are obtained. Since Definition 6.2 is the paper's central definition, this should be either a real theorem with a precisely delimited class of target operads (for example, operads arising from cartesian monoidal categories, for which the set-based transfer genuinely works) or an explicitly downgraded conjecture.
  2. [§4, Theorem 4.1; Appendix C] Theorem 4.1, identifying algebras of the undirected acyclic wiring operad WUA with 'polycategories with space', is the sole support for the higher-order/polycategorical claims of Section 4, yet it is not established as stated. The proof is deferred to Appendix C, where it is labelled '(Sketch)'; the key interchange-law diagram is accompanied by the leftover editorial note '[JHS: This diagram doesn't fit very well, but messing around with the scale option here doesn't seem to do anything!]', so the central diagram is not properly typeset; and the round-trip between WUA-algebras and polycategories with space is asserted ('in this case the roundtrip holds on the nose') rather than demonstrated, even though the polycategory P-hat of the polyfunctor tensor: P-hat to P is defined only inside the appendix. The theorem should either be proved completely, with the polycategory P-hat, the parallel-composition rule, and the round-trip made explicit, or stated at its true strength as a partial characterization.
  3. [§5, Definition 5.1; Appendix B, Definition B.12] Definition 5.1 defines the causal wiring operad W as WA extended with a discarding diagram and an operadic discarding operation 'such that conditions (64) and (65) are satisfied'. The formal construction in Appendix B (Definition B.12) imposes only equation (64)/(95) on the affine completion of the discard wiring operad; condition (65), that discarding the output box of a multi-box wiring is equivalent to discarding all the input boxes, is nowhere imposed in the appendix and is not shown to follow from (64). Since (65) is exactly what is needed for discarding to interact correctly with parallel composition in the category constructed from a W-algebra, the two definitions must be reconciled: either derive (65) from (64) together with functoriality, or include it in the formal quotient, or the forward claim that every algebra F: W to Set gives a causal SMC (Section 5, Eqs. (68)-(72)) is incomplete.
minor comments (5)
  1. [§5, after Eq. (67)] The sentence 'We therefore have that CF(A,B) = CF(A,B)' is vacuous as printed because the two displayed expressions are identical; the intended comparison (presumably between the hom-set of the W-algebra and that of the underlying WA-algebra, or between the two sides of the imposed equation) should be written out explicitly.
  2. [§5, Eq. (64)] The middle term of the chain in Eq. (64) does not render the intended three diagrams, which makes it harder than necessary to verify Definition 5.1 and the causality proof in Eqs. (68)-(72).
  3. [§2, paragraph after Eq. (50)] Definition 2.2 inherits its adequacy entirely from the decomposition theorem of Ref. [34], that every wiring in WA decomposes into sequential, parallel, identity, and swap wirings, but the theorem is neither stated nor summarized. A short statement of it, together with an explanation of how the additional wirings of W fit the same decomposition, would make the paper substantially more self-contained; this is a legitimate reliance on prior work, but readers of the generalized sections currently have no precise statement of what is being assumed.
  4. [Throughout] Typos and infelicities: 'anther process' (Definition 1.2); 'straightforwaqrd' (Appendix A); 'the porevious results' (Section 5); 'Moeover' and 'with it's motivations' (Section 7); 'bares a striking resemblance' (Section 6); 'physcal evolution map' (Section 7). References [26] and [27] are the same Kissinger-Uijlen paper in two versions and should be cross-referenced.
  5. [Appendix B, Lemma B.4] The proof that the extended map L2[F] is well defined is asserted ('by the affine structure of the codomain') rather than shown in detail; since the affine completion is the technical core justifying the causal construction, this step should be spelled out, including why the equivalence relation on A to B tensor X is preserved by the images under F.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main operad-algebra results are explicit constructions based on external theorems, and the causal, time-neutral, and higher-order sections are definitional designs rather than predictions forced by fitted inputs. Theorem 6.1 is a rigor gap but not a circular reduction.

full rationale

The paper's derivation chain is largely a set of explicit constructions: it defines new wiring operads (WA, WD, WUA, W) and then shows that their Set-algebras carry the corresponding categorical structure. The traditional-process-theory/SMC correspondence is imported from the external theorem of Patterson–Spivak–Vagner (Ref. [34]), and the polycategory statement from Yau (Ref. [54]); these are independent results, not author self-citations. The causal construction is deliberately definitional: the causal wiring operad W is defined by imposing the discard equations (64) and (65), and the proof that an algebra F: W -> Set gives a causal SMC applies exactly those imposed equations to verify the discard axiom. This is a consistency argument for a designed structure, not a claim that causality is predicted from causality-free inputs, and not a fitted parameter renamed as a result. The time-neutral and higher-order sections similarly build new operads and prove or sketch correspondences with compact closed categories and polycategories; no step reduces to a quantity defined by the target claim. The genuine weakness is Theorem 6.1: 'K-enriched symmetric monoidal category' is never rigorously defined for a general operad K, and the proof only says the construction is 'given in exactly the same way as it was presented for set.' That is an under-specified claim and a rigor gap, but it is not circular in the sense required here: there is no explicit equation or fitted input that forces the conclusion by construction, nor is the claim supported by a load-bearing chain of author self-citations. Self-citations to Refs. [39,40] appear only as motivation or analogy and do not carry the derivations. Therefore the appropriate circularity finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numbers are fitted to data and no physical entities are postulated. The paper's new constructs (wiring operads of dots, causal wiring operads, operadic discarding, polycategories with space) are mathematical definitions, not empirical postulates, so they do not count as invented physical entities.

assumptions (6)
  • standard math Equivalence between symmetric monoidal categories and algebras of the acyclic wiring operad (Ref. [34]).
    Used throughout Section 2 to identify traditional process theories with WA-algebras and to import SMC structure.
  • standard math Every acyclic wiring diagram decomposes into sequential, parallel, identity, and swap wirings (Ref. [34]).
    Invoked in Section 2 to define the algebra F_C on arbitrary wirings and to guarantee well-definedness.
  • domain assumption A process theory is a collection of processes closed under acyclic wirings with type-matching, with equality by connectivity (Definition 1.1).
    This is the paper's starting notion of traditional process theory, which the operadic framework is designed to capture.
  • domain assumption Time-neutral theories have no input/output distinction and equality of diagrams is by connectivity (Definition 1.2).
    Motivates the wiring operad of dots WD in Section 3.
  • domain assumption Causality is characterized by the discard equation: discarding the output of a box equals discarding its inputs (Eq. (64)).
    Imposed in Section 5 to define the causal wiring operad W.
  • standard math The affine completion L2 and the monoidal congruence quotienting machinery preserve the correspondence between operad algebras and prop algebras (Appendix B).
    Used to construct operadic discarding and to impose Eq. (64) while retaining monoidal structure.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalised Process Theories." pith.science (2026). https://pith.science/paper/DFJYCRTL

@misc{pith2026250210368,
  author       = {Pith},
  title        = {Pith review of: Generalised Process Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFJYCRTL}},
  note         = {Machine review of arXiv:2502.10368}
}
read the original abstract

Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories (SMCs). However, various generalizations, including time-neutral, higher-order, and enriched process theories, do not naturally conform to this structure. In this work, we propose an alternative formalization using operad algebras, motivated by recent results connecting SMCs to operadic structures, which captures a broader class of process theories. By leveraging the string-diagrammatic language, we provide an accessible yet rigorous formulation that unifies and extends traditional process-theoretic approaches. Our operadic framework not only recovers standard process theories as a special case but also enables new insights into quantum foundations and compositional structures. This work paves the way for further investigations into the algebraic and operational properties of generalised process theories within an operadic setting.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Towards a double operadic theory of systems

    math.CT 2025-05 conditional novelty 6.0 of 10

    The paper proposes symmetric monoidal loose right modules over double categories as a unifying double operadic framework for categorical systems theory.

Reference graph

Works this paper leans on

55 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [34]

    Wiring diagrams as normal forms for computing in symmetric monoidal categories

    Evan Patterson, David I Spivak, and Dmitry Vagner. Wiring diagrams as normal forms for computing in symmetric monoidal categories. arXiv preprint arXiv:2101.12046 , 2021

    work page Pith review arXiv 2021

  2. [1]

    No-signalling constrains quantum com- putation with indefinite causal structure

    Luca Apadula, Alessandro Bisio, and Paolo Perinotti. No-signalling constrains quantum com- putation with indefinite causal structure. arXiv, 2022. DOI: 10.48550/ARXIV.2202.10214. URL https://arxiv.org/abs/2202.10214

    arXiv doi:10.48550/arxiv.2202.10214 2022

  3. [2]

    J. Barrett. Information processing in generalized probabilistic theories. Physical Review A , 75:032304, 2007

    work page 2007

  4. [3]

    Theoretical framework for higher-order quantum theory

    Alessandro Bisio and Paolo Perinotti. Theoretical framework for higher-order quantum theory. Proceedings of the Royal Society A , 475(2225):20180706, 2019

    work page 2019

  5. [4]

    Cornering optics

    Guillaume Boisseau, Chad Nester, and Mario Roman. Cornering optics. 2022

    work page 2022

  6. [5]

    Chiribella, G

    G. Chiribella, G. M. D 'Ariano, and P. Perinotti. Transforming quantum operations: Quan- tum supermaps. EPL (Europhysics Letters) , 83(3):30004, jul 2008. DOI: 10.1209/0295- 5075/83/30004. URL https://doi.org/10.12092F0295-50752F832F30004. 27

    doi:10.1209/0295- 2008

  7. [6]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P. Perinotti. Quantum circuit architecture. Phys. Rev. Lett., 101:060401, Aug 2008. DOI: 10.1103/PhysRevLett.101.060401. URL https://link. aps.org/doi/10.1103/PhysRevLett.101.060401

    doi:10.1103/physrevlett.101.060401 2008

  8. [7]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P. Perinotti. Probabilistic theories with purification. Physical Review A, 81(6):062348, 2010

    work page 2010

Show all 55 references
  1. [8]

    Theoretical framework for quantum networks

    Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Theoretical framework for quantum networks. Phys. Rev. A , 80:022339, Aug 2009. DOI: 10.1103/PhysRevA.80.022339. URL https://link.aps.org/doi/10.1103/PhysRevA.80.022339

    2009 doi

  2. [9]

    Quan- tum computations without definite causal structure

    Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. Quan- tum computations without definite causal structure. Phys. Rev. A , 88:022318, Aug 2013. DOI: 10.1103/PhysRevA.88.022318. URL https://link.aps.org/doi/10.1103/PhysRevA. 88.022318

    2013 doi

  3. [10]

    B. Coecke. Kindergarten quantum mechanics. In A. Khrennikov, editor,Quantum Theory: Re- considerations of the Foundations III, pages 81–98. AIP Press, 2005. arXiv:quant-ph/0510032

    2005 arXiv

  4. [11]

    Coecke and A

    B. Coecke and A. Kissinger. Picturing Quantum Processes. A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2016

    2016

  5. [12]

    Coecke and R

    B. Coecke and R. Lal. Causal categories: relativistically interacting processes. Foundations of Physics, 43:458–501, 2013. arXiv:1107.6019

    2013 arXiv

  6. [13]

    Coecke, T

    B. Coecke, T. Fritz, and R. W. Spekkens. A mathematical theory of resources. Information and Computation, to appear , 2014. arXiv:1409.5531

    2014 arXiv

  7. [14]

    Categories for the practising physicist

    Bob Coecke and Eric Oliver Paquette. Categories for the practising physicist. In New struc- tures for physics , pages 173–286. Springer, 2010

    2010

  8. [15]

    Kindergarden quantum mechanics graduates

    Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang. Kindergarden quantum mechanics graduates... or how i learned to stop gluing lego together and love the zx-calculus. Theoretical Computer Science, 897:1–22, 2022

    2022

  9. [16]

    Quantum theory from first principles: an informational approach

    Giacomo Mauro D’Ariano, Giulio Chiribella, and Paolo Perinotti. Quantum theory from first principles: an informational approach . Cambridge University Press, 2017

    2017

  10. [17]

    The produoidal algebra of process decom- position

    Matt Earnshaw, James Hefford, and Mario Rom´ an. The produoidal algebra of process decom- position. arXiv, 2023. DOI: 10.48550/ARXIV.2301.11867. URL https://arxiv.org/abs/ 2301.11867

    2023 doi arXiv

  11. [18]

    Seven sketches in compositionality: An invitation to applied category theory

    Brendan Fong and David I Spivak. Seven sketches in compositionality: An invitation to applied category theory. arXiv preprint arXiv:1803.05316 , 2018

    2018 arXiv

  12. [19]

    Convex spaces i: Definition and examples

    Tobias Fritz. Convex spaces i: Definition and examples. 2015

    2015

  13. [20]

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

    2001 arXiv

  14. [21]

    Harrigan and R

    N. Harrigan and R. W. Spekkens. Einstein, incompleteness, and the epistemic view of quantum states. Foundations of Physics, 40:125–157, 2010

    2010

  15. [22]

    Coend optics for quantum combs

    James Hefford and Cole Comfort. Coend optics for quantum combs. arXiv, 2022. DOI: 10.48550/ARXIV.2205.09027. URL https://arxiv.org/abs/2205.09027

    2022 doi arXiv

  16. [23]

    Monoidal indeterminates and categories of possible worlds

    Claudio Hermida and Robert D Tennent. Monoidal indeterminates and categories of possible worlds. Theoretical Computer Science, 430:3–22, 2012

    2012

  17. [24]

    Projective characterization of higher-order quantum transformations

    Timoth´ ee Hoffreumon and Ognyan Oreshkov. Projective characterization of higher-order quantum transformations. arXiv, 2022. DOI: 10.48550/ARXIV.2206.06206. URL https: //arxiv.org/abs/2206.06206

    2022 doi

  18. [25]

    Universal properties in quantum theory.Electronic Proceedings in Theoretical Computer Science, 287:213–223, jan 2019

    Mathieu Huot and Sam Staton. Universal properties in quantum theory.Electronic Proceedings in Theoretical Computer Science, 287:213–223, jan 2019. DOI: 10.4204/eptcs.287.12

    2019 doi

  19. [26]

    A categorical semantics for causal structure

    Aleks Kissinger and Sander Uijlen. A categorical semantics for causal structure. In Logic in Computer Science (LICS), 2017 32nd Annual ACM/IEEE Symposium on , pages 1–12. IEEE, 2017

    2017

  20. [27]

    A categorical semantics for causal structure

    Aleks Kissinger and Sander Uijlen. A categorical semantics for causal structure. Logical Methods in Computer Science , Volume 15, Issue 3, August 2019. DOI: 10.23638/LMCS- 15(3:15)2019. URL https://lmcs.episciences.org/5681

    2019 doi

  21. [28]

    Mac Lane

    S. Mac Lane. Natural associativity and commutativity. The Rice University Studies , 49(4): 28–46, 1963

    1963

  22. [29]

    Mac Lane

    S. Mac Lane. Categories for the working mathematician . Springer-verlag, 1998

    1998

  23. [30]

    Compositional quantum field theory: An axiomatic presentation

    Robert Oeckl and Juan Orendain Almada. Compositional quantum field theory: An axiomatic presentation. 2024. URL https://arxiv.org/pdf/2208.10385. 28

    2024 arXiv

  24. [31]

    Operational formulation of time reversal in quantum theory

    Ognyan Oreshkov and Nicolas J Cerf. Operational formulation of time reversal in quantum theory. Nature Physics, 11(10):853–858, 2015

    2015

  25. [32]

    Operational quantum theory without predefined time

    Ognyan Oreshkov and Nicolas J Cerf. Operational quantum theory without predefined time. New Journal of Physics , 18(7):073037, 2016

    2016

  26. [33]

    A practical introduction to tensor networks: Matrix product states and pro- jected entangled pair states

    Rom´ an Or´ us. A practical introduction to tensor networks: Matrix product states and pro- jected entangled pair states. Annals of physics , 349:117–158, 2014

    2014

  27. [35]

    Causal boxes: Quantum information-processing systems closed under composition

    Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Bjorn Tackmann. Causal boxes: Quantum information-processing systems closed under composition. IEEE Transactions on Information Theory , pages 1–1, 2017. DOI: 10.1109/tit.2017.2676805

    2017

  28. [36]

    Compo- sition rules for quantum processes: a no-go theorem

    Philippe Allard Gue rin, Marius Krumm, Costantino Budroni, and Caslav Brukner. Compo- sition rules for quantum processes: a no-go theorem. New Journal of Physics , 21(1):012001, jan 2019. DOI: 10.1088/1367-2630/aafef7

    2019 doi

  29. [37]

    Mapping indefinite causal order processes to composable quantum protocols in a spacetime

    Matthias Salzger and V Vilasini. Mapping indefinite causal order processes to composable quantum protocols in a spacetime. arXiv preprint arXiv:2404.05319 , 2024

    2024 arXiv

  30. [38]

    A structure theorem for generalized-noncontextual ontological models

    David Schmid, John H Selby, Matthew F Pusey, and Robert W Spekkens. A structure theorem for generalized-noncontextual ontological models. arXiv preprint arXiv:2005.07161 , 2020

    2005 arXiv

  31. [39]

    Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

    David Schmid, John H Selby, and Robert W Spekkens. Unscrambling the omelette of causation and inference: The framework of causal-inferential theories. arXiv preprint arXiv:2009.03297, 2020

    2009 arXiv

  32. [40]

    Time symmetry in quantum theories and beyond

    John H Selby, Maria E Stasinou, Stefano Gogioso, and Bob Coecke. Time symmetry in quantum theories and beyond. arXiv preprint arXiv:2209.07867 , 2022

    2022 arXiv

  33. [41]

    Higher-order causal theories are models of bv-logic

    Will Simmons and Aleks Kissinger. Higher-order causal theories are models of bv-logic. arXiv,

  34. [42]

    R. W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Physical Review A, 75(3):032110, 2007

    2007

  35. [43]

    Quasi-quantization: classical statistical theories with an epistemic re- striction

    Robert W Spekkens. Quasi-quantization: classical statistical theories with an epistemic re- striction. Quantum Theory: Informational Foundations and Foils , pages 83–135, 2016

    2016

  36. [44]

    The operad of wiring diagrams: Formalizing a graphical language for databases, recursion, and plug-and-play circuits

    David I Spivak. The operad of wiring diagrams: Formalizing a graphical language for databases, recursion, and plug-and-play circuits. arXiv preprint arXiv:1305.0297 , 2013

    2013 arXiv

  37. [45]

    String diagrams for traced and compact categories are oriented 1-cobordisms

    David I Spivak, Patrick Schultz, and Dylan Rupel. String diagrams for traced and compact categories are oriented 1-cobordisms. Journal of Pure and Applied Algebra, 221(8):2064–2110, 2017

    2017

  38. [46]

    M.E. Szabo. Polycategories. Communications in Algebra , 3(8):663–689, 1975. DOI: 10.1080/00927877508822067. URL https://doi.org/10.1080/00927877508822067

    1975 doi

  39. [47]

    Zx-calculus for the working quantum computer scientist.arXiv preprint arXiv:2012.13966, 2020

    John van de Wetering. Zx-calculus for the working quantum computer scientist.arXiv preprint arXiv:2012.13966, 2020

    2012 arXiv

  40. [48]

    Consistent circuits for indefinite causal order

    Augustin Vanrietvelde, Nick Ormrod, Hl´ er Kristj´ ansson, and Jonathan Barrett. Consistent circuits for indefinite causal order. arXiv, 2022. DOI: 10.48550/ARXIV.2206.10042. URL https://arxiv.org/abs/2206.10042

    2022 doi

  41. [49]

    Vilasini and Renato Renner

    V. Vilasini and Renato Renner. Embedding cyclic causal structures in acyclic spacetimes: no-go results for process matrices. arXiv, 2022. DOI: 10.48550/ARXIV.2203.11245. URL https://arxiv.org/abs/2203.11245

    2022 doi arXiv

  42. [50]

    Distilling text into circuits

    Vincent Wang-Mascianica, Jonathon Liu, and Bob Coecke. Distilling text into circuits. arXiv preprint arXiv:2301.10595, 2023

    2023 arXiv

  43. [51]

    Free polycategories for unitary supermaps of arbitrary dimension

    Matt Wilson and Giulio Chiribella. Free polycategories for unitary supermaps of arbitrary dimension. arXiv, 2022. DOI: 10.48550/ARXIV.2207.09180. URL https://arxiv.org/abs/ 2207.09180

    2022 doi arXiv

  44. [52]

    Quantum supermaps are characterized by locality

    Matt Wilson, Giulio Chiribella, and Aleks Kissinger. Quantum supermaps are characterized by locality. arXiv, 2022. DOI: 10.48550/ARXIV.2205.09844. URL https://arxiv.org/abs/ 2205.09844

    2022 doi

  45. [53]

    Higher dimensional algebras via colored props

    Donald Yau. Higher dimensional algebras via colored props. arXiv preprint arXiv:0809.2161, 2008

    2008 arXiv

  46. [54]

    round-trips

    Donald Yau. Operads of wiring diagrams . Springer, 2018. 29 Appendix A Time neutrality from cups and caps In this appendix we extend the acyclic wiring operad, WA, by adding in new wirings in which inputs and outputs can be freely connected to one another. For example, unlike ...

    2018

  47. [2022]

    URL https://arxiv.org/abs/2205.11219

    DOI: 10.48550/ARXIV.2205.11219. URL https://arxiv.org/abs/2205.11219

    doi arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.