{"id":{"repo_id":"calgary","oai_identifier":"oai:ucalgary.scholaris.ca:1880/123655"},"canonical_url":"https://search.dev.ndltd.org/etd/calgary/oai:ucalgary.scholaris.ca:1880/123655","repository":{"repo_id":"calgary","name":"University of Calgary","base_url":"https://ucalgary.scholaris.ca/server/oai/request"},"display":{"title":"A type-based framework for higher-order quantum theory","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Steakley, Samuel Boudreaux"],"institution":"Graduate Studies","degree_name":"Master of Science (MSc)","degree_level":null,"degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":[],"advisors":["Scandolo, Carlo Maria","Gour, Gilad"],"committee_chairs":[],"committee_members":["Hamilton, Ryan","Cunningham, Clifton"],"year":2025,"date_issued":"2025-12-18","date_published":"2025-12-18","updated_at":"2026-07-24T01:30:40Z","subjects":["Higher-order quantum theory"],"languages":["en"],"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."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://dx.doi.org/10.11575/PRISM/50889"],"render_values":[{"text":"https://dx.doi.org/10.11575/PRISM/50889","href":"https://dx.doi.org/10.11575/PRISM/50889","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1880/123655","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Scandolo, Carlo Maria","Gour, Gilad"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Hamilton, Ryan","Cunningham, Clifton"]},{"key":"dc:creator","label":"Author","values":["Steakley, Samuel Boudreaux"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2026-02"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-12-22T23:12:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-12-18"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Calgary"]},{"key":"dc:type","label":"Dc Type","values":["master thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics & Statistics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MSc)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Calgary"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Higher-order quantum theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://dx.doi.org/10.11575/PRISM/50889"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1880/123655"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["A type-based framework for higher-order quantum theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Scandolo, Carlo Maria","Gour, Gilad"],"dc:contributor.committeemember":["Hamilton, Ryan","Cunningham, Clifton"],"dc:creator":["Steakley, Samuel Boudreaux"],"dc:date":["2026-02"],"dc:date.accessioned":["2025-12-22T23:12:56Z"],"dc:date.issued":["2025-12-18"],"dc:description.abstract":["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."],"dc:identifier.doi":["https://dx.doi.org/10.11575/PRISM/50889"],"dc:identifier.uri":["https://hdl.handle.net/1880/123655"],"dc:language.iso":["en"],"dc:publisher.institution":["University of Calgary"],"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."],"dc:subject":["Higher-order quantum theory"],"dc:title":["A type-based framework for higher-order quantum theory"],"dc:type":["master thesis"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_name":["Master of Science (MSc)"],"thesis:institution_name":["University of Calgary"]},"updated_at":"2026-07-24T01:30:40Z"}