arXiv subjects

Carmen Constantin

Publications and source records attributed to Carmen Constantin.

6 recordsLinked to original sources

arXiv2026

Algebraic paradoxes in adaptive quantum computation

Measurement-based quantum computation (MBQC) is a universal model of quantum computation whose full power requires adaptivity. Contextuality is known to power quantum advantage in MBQC, yet it has resisted algebraic analysis in the adaptive setting. We show that if an adaptive $\mathbb{Z}_2$-linear measurement-based quantum computing protocol deterministically computes a non-affine Boolean function, then the underlying quantum resource satisfies an inconsistent set of linear equations. This witnesses an algebraic form of strong contextuality generalising Mermin's All-versus-Nothing arguments. Such algebraic contextuality can be detected cohomologically, resolving an open question posed by Raussendorf, who had established cohomological witnesses of contextuality for non-adaptive protocols, but left the adaptive case open. We prove this result constructively: we model adaptive measurement protocols as ordinary measurements on a larger scenario of tree-like measurements, and explicitly build the inconsistent equations inductively.

quant-ph↗

arXiv2023

Weak cartesian properties of simplicial sets

Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálvez, Kock, and Tonks, are characterized by the property of sending certain commuting squares in the simplex category $Δ$ to pullback squares of sets. We introduce weaker analogues of these properties called completeness conditions, which require squares in $Δ$ to be sent to weak pullbacks of sets, defined similarly to pullback squares but without the uniqueness property of induced maps. We show that some of these completeness conditions provide a simplicial set with lifts against certain subsets of simplices first introduced in the theory of database design. We also provide reduced criteria for checking these properties using factorization results for pushouts squares in $Δ$, which we characterize completely, along with several other classes of squares in $Δ$. Examples of simplicial sets with completeness conditions include quasicategories, many of the compositories and gleaves of Flori and Fritz, and bar constructions for algebras of certain classes of monads. The latter is our motivating example.

math.CT↗

arXiv2023

Partial Evaluations and the Compositional Structure of the Bar Construction

The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $\mathsf{Set}$ (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on $\mathsf{Set}$ for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in $Δ^n$, but generally not all inner horns.

math.CT↗

arXiv2021

Localisable Monads

Monads govern computational side-effects in programming semantics. They can be combined in a ''bottom-up'' way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we equip monads with fine-grained structure in a ''top-down'' way, using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes.

cs.LO↗

arXiv2014

A classification of multipartite states by degree of non-locality

We propose a novel form of classification of multipartite states, in terms of the maximum degree of non-locality they can exhibit under any choice of local observables. This uses the hierarchy of notions previously introduced by Abramsky and Brandenburger: strong contextuality, logical contextuality, and probabilistic contextuality. We study n-qubit pure states. We conjecture that for more than 2 parties, all entangled states are logically contextual. We prove a number of results in support of this conjecture: (1) We show that all permutation-symmetric states are logically non-local. (2) We study the class of balanced states with functional dependencies. These states are described by Boolean functions and have a rich structure, allowing a detailed analysis, which again confirms the conjecture in this case.

quant-ph↗

arXiv2013

Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.

quant-ph↗