A type-based framework for higher-order quantum theory

The study of quantum information protocols that process quantum gates has led to an interest in ‘higher-order’ quantum transformations, which are transformations that act on other quantum transformations as inputs and/or return them as outputs. Mathematically, such transformations form an inductive hierarchy of infinitely many levels, or ‘orders’: quantum states are at order zero; channels, which transform quantum states, are at order one; superchannels, which transform channels, are at order two; maps that transform superchannels are at order three, and so on. To proceed toward a general understanding of higher-order quantum theory, we need methods to study the full inductive hierarchy in a uniform manner. In the context of the Hilbert space formalism of quantum theory, this means studying operator spaces (used to represent states), linear operators between operator spaces (used to represent channels), etc.; in other words, the inductive hierarchy of higher-order linear maps. We introduce a methodological framework for higher-order quantum theory, partly modeled on that of [Bisio \& Perinotti, 2019]. The framework is based on a set of types, denoted TypesA. Types may be thought of as labels that we use to identify and differentiate all the different kinds of objects of higher-order quantum theory. The central objects of the framework are typed (higher-order) linear maps, as given by the construction of a set L(x) of linear maps of type x for each x ∈ TypesA. We equip the framework with a parallel product operation for typed linear maps, which modifies and generalizes the tensor product so as to be physically meaningful when applied to maps of distinct and arbitrary orders. With the aim of characterizing the higher-order linear maps that represent valid quantum objects, we give a construction, consisting of a convex cone K(x) for each x ∈ TypesA, which generalizes the notion of complete positivity to all orders. We construct a version of the Choi isomorphism for our framework, and we use it to show that K(x) is self-dual for all x ∈ TypesA.

Name thesis:degree_name

Master of Science (MSc)

Discipline thesis:degree_discipline

Mathematics & Statistics

Grantor dc:publisher.institution

Graduate Studies

Year dc:date.issued

2025

Author dc:creator

• Steakley, Samuel Boudreaux

Advisors dc:contributor.advisor

• Scandolo, Carlo Maria
• Gour, Gilad

Committee members dc:contributor.committeemember

• Hamilton, Ryan
• Cunningham, Clifton

• Higher-order quantum theory

Statement dc:rights

• University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.

Language dc:language.iso

en

DOI

https://dx.doi.org/10.11575/PRISM/50889

OAI identifier oai:identifier

oai:ucalgary.scholaris.ca:1880/123655