Categories of quantum cpos

Andre Kornell; Bert Lindenhovius; Michael Mislove

arxiv: 2406.01816 · v2 · submitted 2024-06-03 · 🧮 math-ph · cs.LO· math.CT· math.MP· math.OA· quant-ph

Categories of quantum cpos

Andre Kornell , Bert Lindenhovius , Michael Mislove This is my paper

Pith reviewed 2026-05-24 00:27 UTC · model grok-4.3

classification 🧮 math-ph cs.LOmath.CTmath.MPmath.OAquant-ph

keywords quantum cposdiscrete quantizationcategorical modelsquantum programming languagesdomain theoryvon Neumann algebrasω-complete partial ordersquantum relations

0 comments

The pith

Quantum cpos obtained by discrete quantization retain the categorical properties of classical cpos.

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

The paper defines quantum cpos as a noncommutative generalization of ω-complete partial orders by internalizing the classical structure inside the category of von Neumann algebras and quantum relations. It establishes that these quantum cpos satisfy the same key categorical properties as classical cpos, including those central to domain theory. This preservation allows the same kinds of categorical constructions used for programming language models to apply in the quantum setting, and the paper gives concrete examples of such models. The outcome indicates that quantum cpos can serve as the basis for a quantum domain theory.

Core claim

Discrete quantization produces quantum cpos that behave categorically like classical cpos, so the standard constructions from domain theory carry over directly and support the building of categorical models for quantum programming languages.

What carries the argument

Quantum cpos, formed by internalizing cpo structure in von Neumann algebras equipped with quantum relations, which transfers ω-completeness and the required order properties.

Load-bearing premise

The discrete quantization procedure preserves ω-completeness and the other order-theoretic properties when the structures are internalized in von Neumann algebras and quantum relations.

What would settle it

A concrete quantum cpo that fails to have directed suprema or another required categorical limit that classical cpos possess would show the properties do not carry over.

read the original abstract

This paper unites two research lines. The first involves finding categorical models of quantum programming languages and their type systems. The second line concerns the program of quantization of mathematical structures, which amounts to finding noncommutative generalizations (also called quantum generalizations) of these structures. Using a quantization method called discrete quantization, which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of $\omega$-complete partial orders (cpos), called quantum cpos. Cpos are central in domain theory, and are widely used to construct categorical models of programming languages. We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages, which is illustrated with some examples. For this reason, quantum cpos may form the backbone of a future quantum domain theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines quantum cpos via discrete quantization and shows they carry the expected categorical properties, giving a concrete starting point for quantum domain theory.

read the letter

The central point is that discrete quantization applied to cpos produces quantum cpos that keep enough of the classical structure to support the same kinds of categorical constructions used in domain theory. The authors lay out the internalization into von Neumann algebras and quantum relations, then verify the properties hold with examples that tie back to quantum programming models. This directly links the quantization program to the needs of categorical quantum computation without obvious circularity or extra parameters. The construction itself is the new piece, and the examples make the claim usable rather than purely formal. The paper does a clean job of scoping the result as a foundation rather than a finished theory. The main vulnerability is whether the quantization step preserves ω-completeness and the other order properties in full detail. The abstract and claim structure suggest it follows from the definition, and no internal contradiction appears, but the transfer needs to be spelled out explicitly enough for readers to check. If the full text has the derivations or checks, the result stands; if it leans heavily on the general method without case-by-case verification, that section would benefit from more expansion. This is aimed at people already working on categorical models for quantum languages or on quantizing order structures. A reader in either subfield gets a usable new object and a clear path to test it further. It is coherent and engages the right prior lines without overreach. Send it to peer review. The idea is specific enough and the execution careful enough that referees can usefully tighten the proofs and examples.

Referee Report

0 major / 2 minor

Summary. The paper unites categorical models for quantum programming languages with the program of quantizing mathematical structures. Using discrete quantization (internalization of structures in the category of von Neumann algebras and quantum relations), it defines quantum cpos as a noncommutative generalization of ω-complete partial orders. The central claim is that quantum cpos possess analogous categorical properties to classical cpos, rendering them suitable for constructing categorical models of quantum programming languages; this is illustrated with examples and positioned as a potential foundation for quantum domain theory.

Significance. If the preservation of ω-completeness and related order-theoretic properties under discrete quantization holds, the construction supplies a concrete bridge between domain theory and quantum information, enabling semantic models for quantum languages. The approach leverages an established quantization method and focuses on categorical properties rather than ad-hoc definitions, which strengthens its potential utility if the examples are shown to be representative.

minor comments (2)

[Abstract] The abstract and introduction would benefit from an explicit statement of which specific cpo properties (e.g., existence of suprema for directed sets, fixed-point theorems) are shown to carry over, with pointers to the relevant theorems or propositions.
Notation for quantum relations and the internalization functor should be introduced with a short reminder of the ambient category before the definition of quantum cpos, to aid readers unfamiliar with the quantization framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on quantum cpos and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines quantum cpos by applying the established discrete quantization procedure (internalization into von Neumann algebras and quantum relations) to classical cpos, then proves that the resulting structures inherit ω-completeness and categorical properties. This is a standard mathematical construction followed by verification, not a reduction of any claimed result to fitted parameters, self-definitional equations, or load-bearing self-citations. The abstract and claim structure position the work as an extension suitable for models, with properties demonstrated rather than smuggled in by ansatz or renaming. No load-bearing step equates a prediction to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of a new structure via an established quantization technique inside standard categories of von Neumann algebras; no free parameters are mentioned and the axioms invoked are the background axioms of category theory and operator algebras.

axioms (1)

standard math Standard axioms of category theory, von Neumann algebras, and quantum relations
The quantization method and property verifications presuppose the usual definitions and theorems of these mathematical frameworks.

invented entities (1)

quantum cpos no independent evidence
purpose: Noncommutative generalization of ω-cpos for use in categorical models of quantum programming languages
New mathematical structure introduced by the paper; the abstract provides no independent falsifiable evidence outside the construction itself.

pith-pipeline@v0.9.0 · 5687 in / 1265 out tokens · 29344 ms · 2026-05-24T00:27:08.809148+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean; IndisputableMonolith/Foundation/AlexanderDuality.lean reality_from_one_distinction; alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using a quantization method called discrete quantization, which essentially amounts to the internalization of structures in a category of von Neumann algebras and quantum relations, we find a noncommutative generalization of ω-complete partial orders (cpos), called quantum cpos.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; Jcost uniqueness unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that quantum cpos have similar categorical properties to cpos and are therefore suitable for the construction of categorical models for quantum programming languages

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 1 internal anchor

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M. Pagani, P. Selinger, and B. Valiron, Applying quantitative semantics to higher-order quantum c om- puting, POPL 2014, 2014, pp. 647–658

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R. P´ echoux, S. Perdrix, M. Rennela, and V. Zamdzhiev,Quantum programming with inductive datatypes: Causality and aﬃne type theory (2020), 562–581

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Abramsky and B

S. Abramsky and B. Coecke, Categorical quantum mechanics , 2008

work page 2008

[2] [2]

Abramsky and A

S. Abramsky and A. Jung, Handbook of logic in computer science (vol. 3) , 1994, pp. 1–168

work page 1994

[3] [3]

Ad´ amek, H

J. Ad´ amek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories: The Joy of Cats , John Wiley and Sons, Inc, 1990

work page 1990

[4] [4]

Betti, A

R. Betti, A. Carboni, R. Street, and R. Walters, Variation through enrichment , Journal of Pure and Applied Algebra 29 (1983), 109–127

work page 1983

[5] [5]

Borceux, Handbook of categorical algebra 1: Basic category theory , Cambridge University Press, 1994

F. Borceux, Handbook of categorical algebra 1: Basic category theory , Cambridge University Press, 1994

work page 1994

[6] [6]

Cho, Semantics for a quantum programming language by operator al gebras, New Generation Comput

K. Cho, Semantics for a quantum programming language by operator al gebras, New Generation Comput. 34 (2016), no. 1-2, 25–68

work page 2016

[7] [7]

Cho and A

K. Cho and A. Westerbaan, Von Neumann Algebras form a Model for the Quantum Lambda Calc ulus (2016), available at 1603:02113

work page 2016

[8] [8]

Coecke and A

B. Coecke and A. Kissinger, Picturing quantum processes: A ﬁrst course in quantum theor y and dia- grammatic reasoning, Cambridge University Press, 2017

work page 2017

[9] [9]

Connes, Noncommutative geometry, Academic Press, 1994

A. Connes, Noncommutative geometry, Academic Press, 1994

work page 1994

[10] [10]

R. Duan, S. Severini, and A. Winter, Zero-error communication via quantum channels, noncommut ative graphs, and a quantum Lov´ asz number, IEEE Trans. Inform. Theory 59 (2013), 1164–1174

work page 2013

[11] [11]

Eklund, J

P. Eklund, J. Guti´ errez Garc ´ ıa, U. H¨ ohle, and J. Kortelainen, Semigroups in complete lattices: Quantales, modules and related topics , Developments in Mathematics, vol. 54, Springer, Cham, 2018

work page 2018

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Fiore, Axiomatic domain theory in categories of partial maps , Ph.D

M.P. Fiore, Axiomatic domain theory in categories of partial maps , Ph.D. Thesis, 1994

work page 1994

[13] [13]

P. Fu, K. Kishida, N.J. Ross, and S. Selinger, A biset-enriched categorical model for Proto-Quipper with dynamic lifting , Proceedings 19th International Conference on Quantum Physic s and Logic (2022), 302–342

work page 2022

[14] [14]

ACM Program

, Proto-quipper with dynamic lifting , Proc. ACM Program. Lang. 7 (2023jan), no. POPL

work page

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Girard, Linear logic, Theoretical Computer Science 50 (1987), 1 –101

J.-Y. Girard, Linear logic, Theoretical Computer Science 50 (1987), 1 –101

work page 1987

[16] [16]

, Proofs and types , Cambridge University Press, 1989

work page 1989

[17] [17]

Grover, A fast quantum mechanical algorithm for database search , STOC (symposium on the theory of computing), 1996

L.K. Grover, A fast quantum mechanical algorithm for database search , STOC (symposium on the theory of computing), 1996

work page 1996

[18] [18]

Heunen and J

C. Heunen and J. Vicary, Categories for quantum theory, an introduction , Oxford University Press, 2019

work page 2019

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Heymans and I

H. Heymans and I. Stubbe, Grothendieck quantaloids for allegories of enriched categ ories, Bulletin of the Belgian Mathematical Society - Simon Stevin 19 (2012), no. 5, 859 –888

work page 2012

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Jenˇ ca and B

G. Jenˇ ca and B. Lindenhovius, Quantum suplattices , QPL 2023: Proceedings of the twentieth interna- tional conference on quantum physics and logic, 2023

work page 2023

[21] [21]

X. Jia, A. Kornell, B. Lindenhovius, M. Mislove, and V. Zamdzhiev, Semantics for variational quantum programming, Proc. ACM Program. Lang. 6 (2022jan), no. POPL

work page

[22] [22]

X. Jia, B. Lindenhovius, M. Mislove, and V. Zamdzhiev, Commutative monads for probabilistic pro- gramming languages, 36th annual acm/ieee symposium on logic in computer science (lics), 2021

work page 2021

[23] [23]

Jia and M

X. Jia and M. Mislove, Completing simple valuations in K-categories, Topology and its Applications 318 (2022)

work page 2022

[24] [24]

Jones, Probabilistic Nondeterminism, Ph.D

C. Jones, Probabilistic Nondeterminism, Ph.D. dissertation, University of Edinburgh (1990)

work page 1990

[25] [25]

V*-algebras

A. Kornell, Operator algebras in Solovay’s model , Ph.D. dissertation, University of California, Berkeley (2015), available at arXiv:1502.01516

work page internal anchor Pith review Pith/arXiv arXiv 2015

[26] [26]

, Quantum sets , J. Math. Phys. 61 (2020), 102202

work page 2020

[27] [27]

Noncommut

, Discrete quantum structures I: Quantum predicate logic , J. Noncommut. Geom. 18 (2024), no. 1, 337–382

work page 2024

[28] [28]

Noncommut

, Discrete quantum structures II: Examples , J. Noncommut. Geom. 18 (2024), no. 2, 411–450

work page 2024

[29] [29]

Kornell, B

A. Kornell, B. Lindenhovius, and M. Mislove, A category of quantum posets , Indagationes Mathematicae 33 (2022), 1137–1171

work page 2022

[30] [30]

Kuperberg and N

G. Kuperberg and N. Weaver, A Von Neumann Algebra Approach to Quantum Metrics: Quantum Relations, Memoirs of the American Mathematical Society, American Mathema tical Society, 2012

work page 2012

[31] [31]

Mac Lane, Categories for the working mathematician (2nd ed.) , Springer, 1998

S. Mac Lane, Categories for the working mathematician (2nd ed.) , Springer, 1998. 79

work page 1998

[32] [32]

Paul Blain Levy, Call-by-push-value: A functional/imperative synthesis , Springer, 2004

work page 2004

[33] [33]

Lindenhovius, M

B. Lindenhovius, M. Mislove, and V. Zamdzhiev, LNL-FPC: The Linear/Non-linear Fixpoint Calculus , Logical Methods in Computer Science 17 (2021), 9:1––9:61

work page 2021

[34] [34]

, Semantics for a lambda calculus for string diagrams , Samson Abramsky on Logic and Structure in Computer Science and Beyond, 2023

work page 2023

[35] [35]

Moggi, Computational lambda-calculus and monads , Proceedings of the Fourth Annual Symposium on Logic in Computer Science (1989), 14–23

E. Moggi, Computational lambda-calculus and monads , Proceedings of the Fourth Annual Symposium on Logic in Computer Science (1989), 14–23

work page 1989

[36] [36]

1, 55–92

Eugenio Moggi, Notions of computation and monads , Information and Computation 93 (1991), no. 1, 55–92. Selections from 1989 IEEE Symposium on Logic in Computer Sc ience

work page 1991

[37] [37]

Nakagawa, Chapter 14 categorical topology , Topics in general topology, 1989, pp

R. Nakagawa, Chapter 14 categorical topology , Topics in general topology, 1989, pp. 563 –623

work page 1989

[38] [38]

M. P. Olson, The self-adjoint operators of a von neumann algebra form a co nditionally complete lattice , Proc. Amer. Math. Soc. 28 (1971)

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Pagani, P

M. Pagani, P. Selinger, and B. Valiron, Applying quantitative semantics to higher-order quantum c om- puting, POPL 2014, 2014, pp. 647–658

work page 2014

[40] [40]

P´ echoux, S

R. P´ echoux, S. Perdrix, M. Rennela, and V. Zamdzhiev,Quantum programming with inductive datatypes: Causality and aﬃne type theory (2020), 562–581

work page 2020

[41] [41]

Pt´ ak and S

P. Pt´ ak and S. Pulmannov´ a, Orthomodular structures as quantum logics , Fundamental Theories of Physics, Kluwer Academic Publishers, 1991

work page 1991

[42] [42]

Rennela, Towards a quantum domain theory: order-enrichment and ﬁxpo ints in W*-algebras , Mfps 30, 2014, pp

M. Rennela, Towards a quantum domain theory: order-enrichment and ﬁxpo ints in W*-algebras , Mfps 30, 2014, pp. 289–307

work page 2014

[43] [43]

Seal, Tensors, monads and actions , Theory and Applications of Categories 28 (2013), no

G.J. Seal, Tensors, monads and actions , Theory and Applications of Categories 28 (2013), no. 15, 403– 434

work page 2013

[44] [44]

Shor, Polynomial-time algorithms for prime factorization and di screte logarithms on a quantum computer, SIAM Review 41 (1999), no

P.W. Shor, Polynomial-time algorithms for prime factorization and di screte logarithms on a quantum computer, SIAM Review 41 (1999), no. 2, 303–332

work page 1999

[45] [45]

Smyth and G.D

M.B. Smyth and G.D. Plotkin, The category-theoretic solution of recursive domain equat ions, Siam J. Comput. (1982)

work page 1982

[46] [46]

Szigeti, On limits and colimits in the Kleisli category , Cahiers de topologie et g’eom’etrie diﬀ’erentielle cat’egoriques 24 (1983), no

J. Szigeti, On limits and colimits in the Kleisli category , Cahiers de topologie et g’eom’etrie diﬀ’erentielle cat’egoriques 24 (1983), no. 4, 381–391

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