arXiv subjects

Priyaa Varshinee Srinivasan

Publications and source records attributed to Priyaa Varshinee Srinivasan.

14 recordsLinked to original sources

arXiv2026

Categorical Message Passing Language (CaMPL): Syntax and Semantics

We introduce a novel functional-style concurrent programming language called Categorical Message Passing Language (CaMPL) which is designed using the mathematics of linear actegories. This mathematical underpinning gives CaMPL programs useful properties such as deadlock freedom, and additionally, livelock freedom for programs without general recursive processes. We explore CaMPL's type system through a series of code examples. The current proto-alpha version of the compiler and the abstract machine is implemented in Haskell. A reader is encouraged to experiment with writing CaMPL programs either using the online compiler https://campl-app.vercel.app/ or by installing CaMPL from https://campl-ucalgary.github.io -- our website has detailed instructions on how to run CaMPL code.

cs.PL↗

arXiv2026

Categorical Message Passing Language (CaMPL) for programmers

Categorical Message Passing Language (CaMPL) is a functional-style concurrent programming language whose semantics is in category theory, more specifically, linear actegories. Its core programming feature is message passing along typed communication channels between concurrent processes. CaMPL also supports controlled non-determinism via 'races' which allow processes to adapt dynamically while they are running, higher-order processes which pass other processes as messages, and custom channel datatypes called protocols and coprotocols which allow one to define infinite channel types or implement session types. The type system of CaMPL arises from a Curry-Howard-Lambek-like correspondence for concurrent programming, established by Cockett and Pastro in their paper titled "The logic of message passing". This type system ensures that a formal CaMPL program, i.e., one which does not allow general recursion, will never become deadlocked or livelocked. In this article, we explore the type system of CaMPL, custom channel types, and controlled non-determinism using code examples after briefly introducing its mathematical underpinnings.

cs.PL↗

arXiv2026

Unitary, Inner product, and Dagger categories

This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories. By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.

math.CT↗

arXiv2026

Generalized Inverses of Quantum Channels: a categorical perspective

A quantum channel is defined as being completely positive (CP) and trace preserving (TP). While not every quantum channel is invertible or reversible, every quantum channel admits various kinds of generalized inverses such as the Moore-Penrose inverse and the Drazin inverse. A generalized inverse of a quantum channel may not itself be a quantum channel: it often fails to be CP. However, generalized inverses still play an important role in quantum error mitigation. Here, because it is often desirable for the generalized inverse of a quantum channel to be at least TP, the Drazin inverse, which is TP, is favoured over the Moore-Penrose inverse, which is not in general TP. In this paper, we take a categorical perspective on generalized inverses of quantum channels. This allows us to give a simple proof of the fact that the Drazin inverse of a quantum channel is always TP. It also allows us to show that for unital quantum channels, the Drazin inverse is also unital. We then generalize this result to dagger Drazin inverses, which allows us to show that for unital quantum channels, the Moore-Penrose inverse is both TP and unital as well. This opens the door to new applications of both the Drazin inverse and Moore-Penrose inverse in quantum information theory and, in particular, in quantum error mitigation.

quant-ph↗

arXiv2025

A Compositional Account of Generalized Reversible Computing

We develop a compositional framework for generalized reversible computing using copy-discard categories and resource theories. We introduce partitioned matrices between partitioned sets as subdistribution matrices which preserve the equivalence relation of its domain. We model computational and physical transformations as subdistribution matrices over the category of sets and partitioned matrices on partitioned sets, respectively. We show that the interactions between the physical and computational transformations are governed by an aggregation functor whose functoriality and monoidality we deduce from general principles of the formal theory of monads. We study the associated copy-discard structures, in particular, general conditions for determinism and partial invertibility. We then define several notions of entropies that we use to state and prove the fundamental theorem of generalized reversible computing.

math.CT↗

arXiv2025

Dagger-Drazin Inverses

Drazin inverses are a special kind of generalized inverses that can be defined for endomorphisms in any category. A natural question to ask is whether one can somehow extend the notion of Drazin inverse to arbitrary maps - not simply endomorphisms. It turns out that this is possible and, indeed, natural to do so for dagger categories. This paper, thus, introduces the notion of a dagger-Drazin inverse, which is a new kind of generalized inverse appropriate for arbitrary maps in a dagger category. This inverse is closely related to the Drazin inverse, for having dagger-Drazin inverses is equivalent to asking that positive maps have Drazin inverses. Moreover, dagger-Drazin inverses are also closely related to Moore-Penrose inverses as we observe that a map has a Moore-Penrose inverse if and only if it is a Drazin inverse. Furthermore, we explain how Drazin inverses of opposing pairs correspond precisely to dagger-Drazin inverses in cofree dagger categories. We also give examples of dagger-Drazin inverses for matrices over (involutive) fields, bounded linear operators, and partial injections.

math.CT↗

arXiv2025

Drazin Inverses in Categories

Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory, ring theory, and matrix theory. Drazin inverses can also be defined for endomorphisms in any category. However, beyond a paper by Puystjens and Robinson from 1987, there has been almost no further development of Drazin inverses in category theory. Here we provide a survey of the theory of Drazin inverses from a categorical perspective. We introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, the category of finite length modules over a ring, and finite set enriched categories. We also introduce the notion of expressive rank and prove that a category with expressive rank is Drazin. Moreover, we not only study Drazin inverses in mere categories, but also in additive categories and dagger categories. In an arbitrary category, we show how a Drazin inverse corresponds to an isomorphism in the idempotent splitting, as well as explain how Drazin inverses relate to Leinster's notion of eventual image duality. In additive categories, we consider core-nilpotent decompositions, image-kernel decompositions, and Fitting decompositions. We also develop the notion of Drazin inverses for pairs of opposing maps, generalizing the usual notion of Drazin inverse for endomorphisms. As an application of this new kind of Drazin inverse, for dagger categories, we provide a novel characterization of the Moore-Penrose inverse in terms of being a Drazin inverse of the pair of a map and its adjoint.

math.CT↗

arXiv2024

What kind of linearly distributive category do polynomial functors form?

This paper has two purposes. The first is to extend the theory of linearly distributive categories by considering the structures that emerge in a special case: the normal duoidal category $(\mathsf{Poly} ,\mathcal{y}, \otimes, \triangleleft )$ of polynomial functors under Dirichlet and substitution product. This is an isomix LDC which is neither $*$-autonomous nor fully symmetric. The additional structures of interest here are a closure for $\otimes$ and a co-closure for $\triangleleft$, making $\mathsf{Poly}$ a bi-closed LDC, which is a notion we introduce in this paper. The second purpose is to use $\mathsf{Poly}$ as a source of examples and intuition about various structures that can occur in the setting of LDCs, including duals, cores, linear monoids, and others, as well as how these generalize to the non-symmetric setting. To that end, we characterize the linearly dual objects in $\mathsf{Poly}$: every linear polynomial has a right dual which is a representable. It turns out that the linear and representable polynomials also form the left and right cores of $\mathsf{Poly}$. Finally, we provide examples of linear monoids, linear comonoids, and linear bialgebras in $\mathsf{Poly}$.

math.CT↗

arXiv2023

Normalizing Resistor Networks

Star to mesh transformations are well-known in electrical engineering, and are reminiscent of local complementation for graph states in qudit stabilizer quantum mechanics. This paper describes a rewriting system for resistor circuits over any positive division rig using general star to mesh transformations. We show how these transformations can be organized into a confluent and terminating rewriting system on the category of resistor circuits. Furthermore, based on the recently established connections between quantum and electrical circuits, this paper pushes forward the quest for approachable normal forms for stabilizer quantum circuits.

math.CT↗

arXiv2023

Extending Resource Monotones using Kan Extensions

In this paper we generalize the framework proposed by Gour and Tomamichel regarding extensions of monotones for resource theories. A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monotone extensions using Kan extensions, we introduce partitioned categories (pCat)as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into the preorder of non-negative real numbers, and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.

quant-ph↗

arXiv2023

Complete Positivity for Mixed Unitary Categories

Coecke and Heunen described completely positive maps in dagger monoidal categories and the {\sf CP}-infinity construction on these categories in order to construct a category of arbitrary dimensional quantum processes. This article generalizes the ${\sf CP}$-infinity construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories, on the one hand, generalize the (compact) dagger monoidal categories, and on the other hand, accommodate arbitrary dimensional quantum processes, both without sacrificing the notion of dual objects. This means that the ${\sf CP}$-infinity construction for mixed unitary categories provides a suitable semantics for higher-order quantum programming languages which employ arbitrary dimensional structures. The existing results for the ${\sf CP}$-infinity construction are shown to generalize to the new setting. In particular, the notion of environment structures generalizes to mixed unitary categories and it is shown that the ${\sf CP}$-infinity construction for mixed unitary categories is characterized by this generalized environment structure.

math.CT↗

arXiv2023

Dagger linear logic and categorical quantum mechanics

This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and $*$-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger-linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger-LDC. From every (isomix) dagger-LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger-LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.

math.CT↗

arXiv2022

Exponential Modalities and Complementarity (extended abstract)

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.

math.CT↗

arXiv2015

On the robustness of bucket brigade quantum RAM

We study the robustness of the bucket brigade quantum random access memory model introduced by Giovannetti, Lloyd, and Maccone [Phys. Rev. Lett. 100, 160501 (2008)]. Due to a result of Regev and Schiff [ICALP '08 pp. 773], we show that for a class of error models the error rate per gate in the bucket brigade quantum memory has to be of order $o(2^{-n/2})$ (where $N=2^n$ is the size of the memory) whenever the memory is used as an oracle for the quantum searching problem. We conjecture that this is the case for any realistic error model that will be encountered in practice, and that for algorithms with super-polynomially many oracle queries the error rate must be super-polynomially small, which further motivates the need for quantum error correction. By contrast, for algorithms such as matrix inversion [Phys. Rev. Lett. 103, 150502 (2009)] or quantum machine learning [Phys. Rev. Lett. 113, 130503 (2014)] that only require a polynomial number of queries, the error rate only needs to be polynomially small and quantum error correction may not be required. We introduce a circuit model for the quantum bucket brigade architecture and argue that quantum error correction for the circuit causes the quantum bucket brigade architecture to lose its primary advantage of a small number of "active" gates, since all components have to be actively error corrected.

quant-ph↗