Pith. sign in

REVIEW 3 major objections 5 minor 73 references

Supermaps on generalised theories

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read The paper proves that categorical supermaps on any generalised theory with channel-state duality are exactly the concrete CJ supermaps, fixing supermap definitions for Boxworld and real quantum theory.

desk verdict A solid, well-written paper with a plausible central theorem, but the proof as written has a real gap in extending supermaps to non-deterministic instrument components that needs to be pinned down before the Yoneda claim is airtight. read the letter →

arxiv 2602.23865 v3 pith:QYFOXLUY submitted 2026-02-27 quant-ph cs.LOmath.CT

Matt Wilson , James Hefford , Timothée Hoffreumon This is my paper · ORCID
classification quant-phcs.LOmath.CT MSC 18D1018D1581P45
keywords categoricalsupermapsYonedalemmachannel-statedualityChoi-JamiolkowskiisomorphismgeneralisedtheoriesBoxworldNSWSErealquantumtheory
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 sets out to show that there is essentially one way to define higher-order operations—maps that take processes to processes—in any physical theory that has a channel-state duality, a way to identify maps with states like the Choi–Jamiołkowski isomorphism in finite-dimensional quantum theory. Its main theorem proves that the abstract, minimal notion of supermap, called categorical supermaps and defined as families of locally applicable transformations, coincides exactly with the concrete CJ-supermap notion given by ordinary processes in the theory. This eliminates guesswork: wherever a theory has a Choi isomorphism, there is no ambiguity about what its supermaps are. The paper uses this to show that the recently proposed NSWSE process tensors on Boxworld are exactly the categorical supermaps there, and to put forward a definition of supermaps for real quantum theory. A sympathetic reader cares because this stability suggests higher-order causal structures can be studied systematically across classical, quantum, and post-quantum theories.

What carries the argument

The load-bearing tool is the Yoneda lemma adapted to supermaps. In the compact-closed setting, a categorical supermap S: (A⇒A')→(B⇒B') is shown to be representable by a morphism S: B⊗A'→B'⊗A, obtained by evaluating S on the swap process. For generalised theories, the proof uses the cap and cup of the channel-state duality, which are guaranteed to be instrument elements up to scalars, to mimic the Yoneda evaluation inside the supermap: it moves them through S using commutativity with deterministic combs, then uses Lemma 1—equality of instrument components is preserved—to remove the post-selections. The key identity is that S applied to a process φ equals the CJ-supermap connector S placed aro

What would settle it

In a generalised theory with channel-state duality, take a categorical supermap S and compute its purported representative S := S_{B',B}(swap). Then scan all deterministic processes φ; if any S-contracted φ fails to be deterministic, the theorem is false for that theory. A sharper test: pick a theory where the cap is an instrument element only up to a scalar that cannot be scaled away, such as a field not containing all square roots, and check whether the Yoneda construction still lands in deterministic processes.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: if C is a generalised theory with channel-state duality, then categorical supermaps on C are exactly the CJ supermaps. A CJ supermap is just a process S: B⊗A' → B'⊗A in the non-deterministic category that sends every deterministic process φ: A⊗X → A'⊗X' to a deterministic process—a physically motivated, concrete definition. A categorical supermap is a family of functions S_{X,X'} on deterministic process spaces that commutes with arbitrary deterministic processes, so-called combs, on the environment wires; this is a minimal and abstract definition. The theorem shows these two apparently different notions coincide whenever the theory's non-deterministic pro

Load-bearing premise

The theorem's proof evaluates a categorical supermap on the cap and cup instruments and moves them inside S, yet supermaps are formally defined only on deterministic processes; the argument needs a convex-linear extension of S to non-deterministic instrument components, and if that extension fails for some theory with channel-state duality, the concrete representation may not follow.

Editorial extensions

If this is right

  • Categorical supermaps on classical theory equal classical supermaps; on quantum theory they equal quantum supermaps; on Boxworld they equal the NSWSE process tensors.
  • The Yoneda lemma supplies a concrete, unambiguous supermap definition for real quantum theory, and more generally for quantum theory over any field extension of the constructible numbers.
  • For any generalised theory with channel-state duality, defining supermaps concretely through a Choi isomorphism is not an extra choice but the inevitable consequence of the categorical definition.
  • The proof extends to concrete profunctors, so higher-order processes with non-factorised environments, such as channels with memory or shared past and future, also acquire concrete representatives.
  • Boxworld's NSWSE principle, previously introduced as a physical constraint, is recovered from purely categorical reasoning and connected to the BV-logic structure of higher-order theories.

Reading between the lines

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

  • The theorem suggests a practical test for whether a proposed physical higher-order axiom is forced: if the theory has channel-state duality, any supermap definition satisfying locality is uniquely the CJ one, so alternative axioms are either equivalent or not locally applicable.
  • If channel-state duality fails, as in infinite-dimensional quantum theory, the equivalence breaks, so categorical supermaps are strictly more general; the proof identifies exactly which piece of structure is missing.
  • The extension to concrete profunctors hints that the Yoneda representation may survive for causally structured environments, potentially giving a concrete handle on process matrices in theories where the environment wires are not merely parallel.
  • A concrete operational test would be to look for a theory with a Choi isomorphism where a physical supermap axiom disagrees with the categorical one; the theorem predicts none exist, so any candidate axiom must match constraints like NSWSE once locality is imposed.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 introduces a framework of 'generalised theories' (symmetric monoidal categories with deterministic/non-deterministic subcategories and classical interfaces) and defines two notions of supermap: categorical supermaps (locally-applicable transformations on the deterministic subcategory) and CJ-supermaps (concrete processes in the non-deterministic category using channel-state duality). The central claim, Theorem 1, states that on any generalised theory with channel-state duality, the two notions coincide. From this the authors derive Theorem 2, recovering classical, quantum, and Boxworld supermap theories, and Corollary 1, giving a definition of supermaps in real quantum theory. The paper also extends the statement to concrete profunctors and provides appendices on the Boxworld NSWSE characterization, a lemma on equality of instrument components, and the profunctor/Yoneda perspective.

Significance. If Theorem 1 is correct, the paper provides a valuable unification: it turns the abstract, type-theoretic definition of categorical supermaps into a concrete process-level representation for any theory with a Choi isomorphism, eliminating ambiguity in defining higher-order maps in GPTs. The applications to Boxworld and real quantum theory are timely and demonstrate genuine scope. The paper is also refreshingly explicit about its working assumptions and includes substantial appendix material, examples of channel-state duality, and connections to profunctor optics. However, the central proof currently has a load-bearing gap concerning the extension of categorical supermaps to non-deterministic instrument elements, and the proof of the key lemma is partly deferred to a PhD thesis. These issues must be fixed before the main theorem can be regarded as established.

major comments (3)
  1. [§4, Theorem 1; Definition 6; Appendix B] The proof of Theorem 1 evaluates the categorical supermap S on the cap and cup, which are instrument elements in the non-deterministic category C (Definition 3), while Definition 6 defines S only on deterministic processes in C_d. No convex-linear extension of S to C is constructed or proved to exist; Lemma 1, whose proof is in Appendix B, already presupposes such an extension by writing S M^x_y for non-deterministic components. The proof of Lemma 1 also uses a non-convex linear combination and defers justification to [67, Lemma 9] (footnote 9). Since the diagrammatic step 'move the cap and cup inside S' depends on this extension, the central equality is not established as written. Please supply the extension argument or state explicit, verifiable conditions under which it exists.
  2. [§4.1, Theorem 5] Theorem 5 claims the Yoneda lemma extends to all concrete profunctors, but its proof is a single sentence saying the proofs are 'syntactically identical'. This is not adequate: Definition 8 restricts P(X,X') to subsets, while the proof of Theorem 1 uses the full classical-control and instrument-extension structure of a generalised theory (for example, the channels Sigma, Sigma' in Appendix B). The paper should show that every concrete profunctor is closed under the relevant combs, and that the constructed CJ-supermap is independent of the choice of swap. Since Theorem 5 is advertised as a broad generalisation, this is load-bearing.
  3. [Appendix A, Theorem 7] The proof that NSWSE Boxworld tensors are precisely CJ-supermaps on Boxworld is compressed. The forward direction relies on a type-theoretic embedding using BV-logic, and the converse relies on a 'concrete characterisation' of marginals of NS instruments and on the no-signalling result of [68] without stating it. Given that the Boxworld recovery is one of the headline applications of Theorem 1, please expand these steps, quote the exact statements used, and verify that the hypotheses of the cited results are satisfied by the Boxworld embedding.
minor comments (5)
  1. [Abstract / Introduction] There is a typo 'weaking' (should be 'weakening') and the phrase 'forms asymmetric monoidal category' should be 'forms a symmetric monoidal category'.
  2. [Example 11 / Appendix B] Typos: 'Interrpeted' in Example 11 and 'ocntrol' in Appendix B.
  3. [References] Reference [36] has a placeholder-looking DOI (10.1103/kmmy-3dy3); please update to the published data.
  4. [§4, Theorem 1 statement] The equality 'Categorical supermaps = CJ supermaps' is not formalised: is it a bijection, a natural isomorphism, or a set equality? Please state the precise correspondence and its domain/codomain.
  5. [Appendix C, proof of Theorem 8] In the displayed Yoneda proof, the profunctor is written as C(A⊗−, A′⊗=), with '=' instead of '−'; please correct the notation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central equivalence is not a relabeling of its own inputs. The main caveat, an unstated extension of categorical supermaps to non-deterministic cap/cup elements in the proof of Theorem 1, is a proof gap rather than a circular reduction.

full rationale

Theorem 1 is a substantive mathematical equivalence: Definition 6 (categorical supermaps as strong natural transformations on the deterministic category C_d) and Definition 4 (CJ-supermaps as processes in the non-deterministic category C) are not identified by construction. The proof derives the equivalence using Yoneda-style reasoning, and the result is tested against external benchmarks: classical, quantum, and Boxworld/NSWSE are recovered in Theorems 2 and 3, with a detailed appendix for Boxworld. The self-citations to [34,35,69,71] and [67] supply the categorical-supermap framework and some technical lemmas (e.g., non-convex linearity in Appendix B), but the central equality has independent content and is argued in the paper rather than being merely an imported conclusion. The genuine weakness is that the proof of Theorem 1 evaluates S on cap and cup instrument elements, which are non-deterministic processes in C, while Definition 6 defines S only on C_d. Lemma 1 is stated for instrument components and its proof presupposes that such evaluations are meaningful; the paper does not explicitly construct the convex-linear extension of S to all of C. This is an omitted-proof/correctness caveat that could affect whether the Yoneda representation exists in every claimed theory, but it is not a circular step: the theorem is not assumed among its premises, no parameter is fitted to a subset of data and then relabeled as a prediction, and no load-bearing argument reduces to an unverified self-citation. The circularity score therefore remains low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented physical entities appear. The paper introduces a new mathematical abstraction (concrete profunctors) but no new physical degrees of freedom; the central theorem rests on the listed domain axioms and prior results.

assumptions (5)
  • domain assumption A generalised theory is a convex-enriched symmetric monoidal category with a deterministic wide subcategory, embedded classical theories, and classical control (Definition 1).
    Defines the domain of the theorem; if a physical theory lacks classical control, the proof of convex linearity (Lemma 2) fails.
  • domain assumption Channel-state duality: the non-deterministic category C is compact closed and the cup/cap are instrument elements up to scalar (Definition 3).
    Central premise of Theorem 1; theories such as infinite-dimensional quantum mechanics are explicitly excluded.
  • ad hoc to paper The scalars alpha, beta in Definition 3 are nonzero/invertible in the probability semiring so that 1/alpha and 1/beta are available.
    The proof of Theorem 1 divides by alpha and beta; the definition only says 'up to a scalar' and does not explicitly state nonzero/invertible.
  • standard math Representability of strong profunctors (Pastro-Street theorem [47]) and the Yoneda lemma are accepted.
    Used in Appendix C to reduce Theorem 4 to a representability statement.
  • domain assumption For Boxworld, the causal-decomposition property and no-signalling-via-post-selection results of higher-order classical theory / BV logic [60,61,68] are assumed.
    Appendix A's proof that NSWSE tensors equal CJ-supermaps relies on these type-theoretic facts.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Supermaps on generalised theories." pith.science (2026). https://pith.science/paper/QYFOXLUY

@misc{pith2026260223865,
  author       = {Pith},
  title        = {Pith review of: Supermaps on generalised theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QYFOXLUY}},
  note         = {Machine review of arXiv:2602.23865}
}
read the original abstract

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the categorical supermaps on Boxworld are in general characterised by a physically well-motivated weaking of the recently proposed NSWSE principle for higher-order processes on boxworld. Furthermore, via the same Yoneda lemma we put forward a stable definition for supermaps in real quantum theory.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 16 linked inside Pith

  1. [67]

    Wilson.Compositional Frameworks for Supermaps and Causality

    M. Wilson.Compositional Frameworks for Supermaps and Causality. Phd thesis, University of Oxford, 2023

    2023

  2. [68]

    Causality in higher order process theories

    Matt Wilson and Giulio Chiribella. Causality in higher order process theories. In Proceedings QPL 2021, volume 343, pages 265–300. Open Publishing Association,

    2021

  3. [1]

    A categorical semantics of quantum protocols

    Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. InProceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004, pages 415–425, 2004. DOI: 10.1109/LICS.2004.1319636

    arXiv 2004

  4. [2]

    No-signalling constrains quan- tum computation with indefinite causal structure.Quantum, 8:1241, February 2024

    Luca Apadula, Alessandro Bisio, and Paolo Perinotti. No-signalling constrains quan- tum computation with indefinite causal structure.Quantum, 8:1241, February 2024. ISSN 2521-327X. DOI: 10.22331/q-2024-02-05-1241. URLhttps://doi.org/10. 22331/q-2024-02-05-1241

    doi:10.22331/q-2024-02-05-1241 2024

  5. [3]

    Information processing in generalized probabilistic theories.Phys

    Jonathan Barrett. Information processing in generalized probabilistic theories.Phys. Rev. A, 75:032304, Mar 2007. DOI: 10.1103/PhysRevA.75.032304

    doi:10.1103/physreva.75.032304 2007

  6. [4]

    The space of logically consistent classical processes withoutcausalorder.New Journal of Physics, 18(1):013036, 2016

    Ämin Baumeler and Stefan Wolf. The space of logically consistent classical processes withoutcausalorder.New Journal of Physics, 18(1):013036, 2016. DOI:10.1088/1367- 2630/18/1/013036

    doi:10.1088/1367- 2016

  7. [5]

    Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios.Phys

    Ämin Baumeler, Adrien Feix, and Stefan Wolf. Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios.Phys. Rev. A, 90: 042106, 2014. DOI: 10.1103/PhysRevA.90.042106

    doi:10.1103/physreva.90.042106 2014

  8. [6]

    In- definite causal order in boxworld theories, 2024

    Jessica Bavaresco, Ämin Baumeler, Yelena Guryanova, and Costantino Budroni. In- definite causal order in boxworld theories, 2024. URLhttps://arxiv.org/abs/ 2411.00951. 28

    arXiv 2024

Show all 73 references
  1. [7]

    Theoreticalframeworkforhigher-orderquantum theory.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2225):20180706, 2019

    AlessandroBisioandPaoloPerinotti. Theoreticalframeworkforhigher-orderquantum theory.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2225):20180706, 2019. ISSN 1471-2946. DOI: 10.1098/rspa.2018.0706. URLhttp://dx.doi.org/10.1098/rspa.2018.0706

    2019

  2. [8]

    Cornering optics, 2022

    Guillaume Boisseau, Chad Nester, and Mario Román. Cornering optics, 2022. URL https://doi.org/10.48550/ARXIV.2205.00842

    2022 doi arXiv

  3. [9]

    Quantum operations with indefinite time direction

    Giulio Chiribella and Zixuan Liu. Quantum operations with indefinite time direction. Communications Physics, 5(1), 7 2022. ISSN 2399-3650. DOI: 10.1038/s42005-022- 00967-3. URLhttp://dx.doi.org/10.1038/s42005-022-00967-3

    2022 doi

  4. [11]

    Transforming quan- tum operations: Quantum supermaps.EPL (Europhysics Letters), 83(3):30004, 2008

    Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Transforming quan- tum operations: Quantum supermaps.EPL (Europhysics Letters), 83(3):30004, 2008. DOI: 10.1209/0295-5075/83/30004

    2008 doi

  5. [15]

    Quantum computations without definite causal structure.Phys

    Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoît Valiron. Quantum computations without definite causal structure.Phys. Rev. A, 88:022318,

  6. [16]

    Normal completely positive maps on the space of quantum operations.Open Systems & Informa- tion Dynamics, 20(01):1350003, 2013

    Giulio Chiribella, Alessandro Toigo, and Veronica Umanità. Normal completely positive maps on the space of quantum operations.Open Systems & Informa- tion Dynamics, 20(01):1350003, 2013. DOI: 10.1142/S1230161213500030. URL https://doi.org/10.1142/S1230161213500030

    2013 doi

  7. [17]

    Positive linear maps on complex matrices.Linear Algebra and its Applications, 10(3):285–290, 1975

    Man-Duen Choi. Positive linear maps on complex matrices.Linear Algebra and its Applications, 10(3):285–290, 1975. ISSN00243795. DOI:10.1016/0024-3795(75)90075-

    1975 doi

  8. [18]

    Profunctor Optics, a Categorical Update.Composition- ality, 6:1, February 2024

    BryceClarke, DerekElkins, JeremyGibbons, FoscoLoregian, BartoszMilewski, Emily Pillmore, and Mario Román. Profunctor Optics, a Categorical Update.Composition- ality, 6:1, February 2024. ISSN 2631-4444. DOI: 10.32408/compositionality-6-1

    2024 doi

  9. [19]

    URLhttp://dx.doi.org/10.1016/0024-3795(75)90075-0

    doi

  10. [20]

    Causal categories: Relativistically interacting pro- cesses.Found Phys, 43:458–501, 2013

    Bob Coecke and Raymond Lal. Causal categories: Relativistically interacting pro- cesses.Found Phys, 43:458–501, 2013. DOI: 10.1007/s10701-012-9646-8

    2013 doi

  11. [21]

    Quantum picturalism.Contemporary Physics, 51(1):59–83, 2010

    Bob Coecke. Quantum picturalism.Contemporary Physics, 51(1):59–83, 2010. DOI: 10.1080/00107510903257624

    2010 doi

  12. [22]

    Quantum computational networks.Proceedings of the Royal Society of London

    David Deutsch. Quantum computational networks.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 425(1868):73–90, 1989. DOI: 10.1098/rspa.1989.0099

    1989

  13. [23]

    Two roads to classicality.EPTCS, 266: 104–118, 2018

    Bob Coecke, John Selby, and Sean Tull. Two roads to classicality.EPTCS, 266: 104–118, 2018. DOI: 10.4204/eptcs.266.7

    2018 doi

  14. [24]

    Indefinite causal structures for continuous-variable systems.New Journal of Physics, 18(11):113026,

    Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. Indefinite causal structures for continuous-variable systems.New Journal of Physics, 18(11):113026,

  15. [25]

    The produoidal algebra of process decomposition, 2023

    Matt Earnshaw, James Hefford, and Mario Román. The produoidal algebra of process decomposition, 2023. URLhttps://doi.org/10.48550/ARXIV.2301.11867. 29

    2023 doi arXiv

  16. [26]

    Higher-order cpm constructions.EPTCS, 287:145–162, 2019

    Stefano Gogioso. Higher-order cpm constructions.EPTCS, 287:145–162, 2019. DOI: 10.4204/EPTCS.287.8

    2019 doi

  17. [27]

    Categorical probabilistic theories

    Stefano Gogioso and Carlo Maria Scandolo. Categorical probabilistic theories. EPTCS, 266:367–385, 2018. DOI: 10.4204/EPTCS.266.23

    2018 doi

  18. [28]

    Fantastic quantum theories and where to find them.arXiv Preprint, arXiv:1703.10576, 2017

    Stefano Gogioso. Fantastic quantum theories and where to find them.arXiv Preprint, arXiv:1703.10576, 2017

    2017 arXiv

  19. [29]

    A system of interaction and structure.ACM Transactions on Computational Logic, 8(1), 2007

    Alessio Guglielmi. A system of interaction and structure.ACM Transactions on Computational Logic, 8(1), 2007. DOI: 10.1145/1182613.1182614

    2007

  20. [30]

    Toward a general theory of quantum games

    Gus Gutoski and John Watrous. Toward a general theory of quantum games. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC ’07, pages 565–574, New York, NY, USA, 2007. Association for Computing Machinery. ISBN 9781595936318. DOI: 10.1145/12507...

    2007

  21. [31]

    Density hypercubes, higher order inter- ference and hyper-decoherence: A categorical approach

    Stefano Gogioso and Carlo Maria Scandolo. Density hypercubes, higher order inter- ference and hyper-decoherence: A categorical approach. InQuantum Interaction, pages 141–160. Springer International Publishing, 2019. DOI: 10.1007/978-3-030- 35895-2_10

    2019 doi

  22. [32]

    Hyper-decoherence in density hypercubes

    James Hefford and Stefano Gogioso. Hyper-decoherence in density hypercubes. In Proceedings QPL 2020, volume 340, pages 141–159. Open Publishing Association,

    2020

  23. [33]

    Optics for premonoidal categories

    James Hefford and Mario Román. Optics for premonoidal categories. InProceedings ACT 2023, volume 397, pages 152–171. Open Publishing Association, 2023. DOI: 10.4204/eptcs.397.10. URLhttp://dx.doi.org/10.4204/EPTCS.397.10

    2023 doi

  24. [34]

    CPM categories for galois extensions

    James Hefford and Stefano Gogioso. CPM categories for galois extensions. InPro- ceedings QPL 2021, volume 343, pages 165–192. Open Publishing Association, 2021. DOI: 10.4204/eptcs.343.9

    2021 doi

  25. [35]

    A bv-category of spacetime interventions, 2025

    James Hefford and Matt Wilson. A bv-category of spacetime interventions, 2025. URL https://arxiv.org/abs/2502.19022

    2025 arXiv

  26. [36]

    Decoherence to quantum theory from a causally indefinite post-quantum theory.Phys

    James Hefford and Matt Wilson. Decoherence to quantum theory from a causally indefinite post-quantum theory.Phys. Rev. A, 113:042433, Apr 2026. DOI: 10.1103/kmmy-3dy3. URLhttps://link.aps.org/doi/10.1103/kmmy-3dy3

    2026 doi

  27. [37]

    Projective characterization of higher- order quantum transformations, 2022

    Timothée Hoffreumon and Ognyan Oreshkov. Projective characterization of higher- order quantum transformations, 2022. URLhttps://doi.org/10.48550/arXiv. 2206.06206

    2022 doi

  28. [38]

    A Profunctorial Semantics for Quantum Supermaps

    James Hefford and Matt Wilson. A Profunctorial Semantics for Quantum Supermaps. InProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’24, New York, NY, USA, 2024. Association for Computing Machinery. DOI: 10.1145/3661814.3662123

    2024

  29. [39]

    On the structure of higher order quantum maps

    Anna Jenčová. On the structure of higher order quantum maps. 2024. URLhttps: //arxiv.org/abs/2411.09256. 30

    2024 arXiv

  30. [40]

    Traced monoidal categories.Math- ematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, 1996

    André Joyal, Ross Street, and Dominic Verity. Traced monoidal categories.Math- ematical Proceedings of the Cambridge Philosophical Society, 119(3):447–468, 1996. DOI: 10.1017/S0305004100074338

    1996 doi

  31. [41]

    Acategoricalsemanticsforcausalstructure.Logical Methods in Computer Science, 15, 2019

    AleksKissingerandSanderUijlen. Acategoricalsemanticsforcausalstructure.Logical Methods in Computer Science, 15, 2019. ISSN 18605974. DOI: 10.23638/LMCS- 15(3:15)2019

    2019 doi

  32. [42]

    Linear transformations which preserve trace and positive semidefinitenessofoperators.Reports on Mathematical Physics, 3(4):275–278, 121972

    Andrej Jamiołkowski. Linear transformations which preserve trace and positive semidefinitenessofoperators.Reports on Mathematical Physics, 3(4):275–278, 121972. ISSN 00344877. DOI: 10.1016/0034-4877(72)90011-0. URLhttp://linkinghub. elsevier.com/retrieve/pii/0034487772900110

  33. [43]

    Parity erasure: a foundational principle for indef- inite causal order, 2025

    Zixuan Liu and Ognyan Oreshkov. Parity erasure: a foundational principle for indef- inite causal order, 2025. URLhttps://arxiv.org/abs/2512.08635

    2025

  34. [44]

    London Mathematical Society Lecture Note Series

    Fosco Loregian.(Co)end Calculus. London Mathematical Society Lecture Note Series. Cambridge University Press, 2021. DOI: 10.1017/9781108778657

    2021 doi

  35. [45]

    Indefinite causal structure and causal inequalities with time-symmetry, 2024

    Luke Mrini and Lucien Hardy. Indefinite causal structure and causal inequalities with time-symmetry, 2024. URLhttps://arxiv.org/abs/2406.18489

    2024 arXiv

  36. [46]

    Graduate Texts in Mathematics

    Saunders Mac Lane.Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer Science+Business Media, New York, NY, 2 edition, Septem- ber 1998. ISBN 978-0-387-98403-2, 978-1-4419-3123-8, 978-1-4757-4721-8. DOI: 10.1007/978-1-4757-4721-8. URLhttps://doi.o...

    1998 doi

  37. [47]

    Doubles for monoidal categories.Theory and Applica- tions of Categories, 21(4):61–75, 2008

    Craig Pastro and Ross Street. Doubles for monoidal categories.Theory and Applica- tions of Categories, 21(4):61–75, 2008

    2008

  38. [48]

    Challenges for extensions of the process matrix formalism to quantum field theory, 2023

    Nikola Paunkovic and Marko Vojinovic. Challenges for extensions of the process matrix formalism to quantum field theory, 2023. URLhttps://arxiv.org/abs/ 2310.04597

    2023 arXiv

  39. [49]

    Categorical semantics for time travel, 2019

    Nicola Pinzani, Stefano Gogioso, and Bob Coecke. Categorical semantics for time travel, 2019. URLhttps://arxiv.org/abs/1902.00032

    2019 arXiv

  40. [50]

    Quantum correlations with no causal order.Nature Communications, 3(1092), 2012

    Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. Quantum correlations with no causal order.Nature Communications, 3(1092), 2012. DOI: 10.1038/ncomms2076

    2012 doi

  41. [51]

    Quantum nonlocality as an axiom.Foundations of Physics, 24(3):379–385, 1994

    Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom.Foundations of Physics, 24(3):379–385, 1994. DOI: 10.1007/BF02058098. URLhttps://doi.org/ 10.1007/BF02058098

    1994 doi

  42. [52]

    Categories of optics

    Mitchell Riley. Categories of optics. 2018. DOI: 10.48550/arXiv.1809.00738. URL https://doi.org/10.48550/arXiv.1809.00738

    2018 doi arXiv

  43. [53]

    Comb diagrams for discrete-time feedback, 2020

    Mario Román. Comb diagrams for discrete-time feedback, 2020. URLhttps://doi. org/10.48550/arXiv.2003.06214

    2020 doi arXiv

  44. [54]

    Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternos- tro, and Kavan Modi

    Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternos- tro, and Kavan Modi. Non-markovian quantum processes: Complete framework and efficient characterization.Phys. Rev. A, 97:012127, Jan 2018. DOI: 10.1103/Phys- RevA.97.012127. URLhttps://link.aps.org/d...

    2018 doi

  45. [55]

    Open diagrams via coend calculus

    Mario Román. Open diagrams via coend calculus. InProceedings ACT 2020, volume 333, pages 65–78. Open Publishing Association, Feb 2021. DOI: 10.4204/eptcs.333.5

    2020 doi

  46. [56]

    PhD thesis, Tallinn University of Technol- ogy, 2023

    Mario Román.Monoidal Context Theory. PhD thesis, Tallinn University of Technol- ogy, 2023. URLhttps://doi.org/10.23658/taltech.54/2023

    2023 doi

  47. [57]

    Selby, Carlo Maria Scandolo, and Bob Coecke

    John H. Selby, Carlo Maria Scandolo, and Bob Coecke. Reconstructing quan- tum theory from diagrammatic postulates.Quantum, 5:445, April 2021. ISSN 2521-327X. DOI: 10.22331/q-2021-04-28-445. URLhttps://doi.org/10.22331/ q-2021-04-28-445. 31

    2021 doi

  48. [58]

    Profunctor optics and traversals, 2020

    Mario Román. Profunctor optics and traversals, 2020. URLhttps://doi.org/10. 48550/arXiv.2001.08045

    2020 doi arXiv

  49. [59]

    Achieving maximal causal indefiniteness in a maximally nonlocal theory, 2024

    Kuntal Sengupta. Achieving maximal causal indefiniteness in a maximally nonlocal theory, 2024. URLhttps://arxiv.org/abs/2411.04201

    2024 arXiv

  50. [60]

    Higher-Order Causal Theories Are Models of BV-Logic

    Will Simmons and Aleks Kissinger. Higher-Order Causal Theories Are Models of BV-Logic. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors,47th In- ternational Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 ofLeibniz International ...

    2022 doi

  51. [61]

    A complete logic for causal consistency, 2024

    Will Simmons and Aleks Kissinger. A complete logic for causal consistency, 2024

    2024

  52. [62]

    Idempotents in dagger categories: (extended abstract).Elec- tronic Notes in Theoretical Computer Science, 210:107–122, 2008

    Peter Selinger. Idempotents in dagger categories: (extended abstract).Elec- tronic Notes in Theoretical Computer Science, 210:107–122, 2008. DOI: 10.1016/j.entcs.2008.04.021

    2008 doi

  53. [63]

    Higher-order quantum operations, 2025

    Philip Taranto, Simon Milz, Mio Murao, Marco Túlio Quintino, and Kavan Modi. Higher-order quantum operations, 2025. URLhttps://arxiv.org/abs/2503.09693

    2025 arXiv

  54. [64]

    Device-independent certification of indefinite causal order in the quantum switch.Nature Communications, 14(1):5811, 2023

    Tein van der Lugt, Jonathan Barrett, and Giulio Chiribella. Device-independent certification of indefinite causal order in the quantum switch.Nature Communications, 14(1):5811, 2023. DOI: 10.1038/s41467-023-40162-8. URLhttps://doi.org/10. 1038/s41467-023-40162-8

    2023 doi

  55. [65]

    Abbott, and Cyril Branciard

    Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard. Quan- tum circuits with classical versus quantum control of causal order.PRX Quan- tum, 2:030335, Aug 2021. DOI: 10.1103/PRXQuantum.2.030335. URLhttps: //link.aps.org/doi/10.1103/PRXQuantum.2.030335

    2021 doi

  56. [66]

    Distributors on a tensor category.Hokkaido Mathematical Journal, 35(2):379 – 425, 2006

    Daisuke Tambara. Distributors on a tensor category.Hokkaido Mathematical Journal, 35(2):379 – 425, 2006. DOI: 10.14492/hokmj/1285766362

    2006

  57. [69]

    Free polycategories for unitary supermaps of arbitrary dimension, 2022

    Matt Wilson and Giulio Chiribella. Free polycategories for unitary supermaps of arbitrary dimension, 2022. URLhttps://doi.org/10.48550/ARXIV.2207.09180

    2022 doi arXiv

  58. [70]

    Wilde.Quantum Information Theory

    Mark M. Wilde.Quantum Information Theory. Cambridge University Press, 2013

    2013

  59. [71]

    Quantum supermaps are char- acterized by locality, 2022

    Matt Wilson, Giulio Chiribella, and Aleks Kissinger. Quantum supermaps are char- acterized by locality, 2022. URLhttps://doi.org/10.48550/ARXIV.2205.09844

    2022 doi

  60. [72]

    tensoring

    Mário Ziman. Process positive-operator-valued measure: A mathematical frame- work for the description of process tomography experiments.Physical Review A, 77(6), 6 2008. DOI: 10.1103/physreva.77.062112. URLhttps://doi.org/10.1103% 2Fphysreva.77.062112. A Categorical Supermaps ...

    2008 doi

  61. [73]

    URLhttp://dx.doi.org/10.4204/EPTCS.343

    DOI: 10.4204/eptcs.343.12. URLhttp://dx.doi.org/10.4204/EPTCS.343. 12

    doi

  62. [75]

    On the origin of linearity and unitarity in quantum theory, 2023

    Matt Wilson and Nick Ormrod. On the origin of linearity and unitarity in quantum theory, 2023. URLhttps://arxiv.org/abs/2305.20063

    2023

  63. [2013]

    DOI: 10.1103/PhysRevA.88.022318

    doi

  64. [2016]

    URLhttps://dx.doi.org/10.1088/ 1367-2630/18/11/113026

    DOI: 10.1088/1367-2630/18/11/113026. URLhttps://dx.doi.org/10.1088/ 1367-2630/18/11/113026

    doi

  65. [2021]

    DOI: 10.4204/eptcs.340.7

    doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.