{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/311595"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/311595","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels","abstract":"This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting.","abstract_html":"This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting.","abstract_has_math":false,"creators":["Hanson, Eric"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Datta, Nilanjana"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-04-03","date_published":"2020-04-03","updated_at":"2026-07-22T22:24:06Z","subjects":["continuity","entropy","repeated interaction systems","quantum"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/06f2f6a6-4201-45b4-b447-a3d4146b9cf0/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000246754348"],"render_values":[{"text":"0000-0002-4675-4348","href":"https://orcid.org/0000-0002-4675-4348","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.58687","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Datta, Nilanjana"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Cantab Capital Institute for the Mathematics of Information"]},{"key":"dc:creator","label":"Author","values":["Hanson, Eric"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000246754348"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2020-04-03"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/311595"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["continuity","entropy","repeated interaction systems","quantum"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/06f2f6a6-4201-45b4-b447-a3d4146b9cf0/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.58687"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6d3e209e-6bbc-4585-ae58-9d3e1bffdf24/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. 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In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["d39e3c7201bb620754a3e7cf81d8d0d9","353adac0d1ebdfd65ab16480263c3c87"]},{"key":"dc:title","label":"Title","values":["Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels"]}]}],"canonical_facts":{"dc:contributor.advisor":["Datta, Nilanjana"],"dc:contributor.sponsor":["Cantab Capital Institute for the Mathematics of Information"],"dc:creator":["Hanson, Eric"],"dc:creator.authoridentifier":["0000000246754348"],"dc:date.issued":["2020-04-03"],"dc:description.abstract":["This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting."],"dc:format.checksum.md5":["d39e3c7201bb620754a3e7cf81d8d0d9","353adac0d1ebdfd65ab16480263c3c87"],"dc:identifier.doi":["10.17863/CAM.58687"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6d3e209e-6bbc-4585-ae58-9d3e1bffdf24/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/311595"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/06f2f6a6-4201-45b4-b447-a3d4146b9cf0/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"dc:subject":["continuity","entropy","repeated interaction systems","quantum"],"dc:title":["Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:06Z"}