Quantum Latin squares and quantum functions: applications in quantum information
<p>A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent mathematical objects such as Latin squares or functions as string diagrams over the category...
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| Format: | Thesis |
| Language: | English |
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2019
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| _version_ | 1826298962547572736 |
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| author | Musto, BJ |
| author2 | Vicary, J |
| author_facet | Vicary, J Musto, BJ |
| author_sort | Musto, BJ |
| collection | OXFORD |
| description | <p>A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent mathematical objects such as Latin squares or functions as string diagrams over the category of finite-dimensional Hilbert spaces obeying various diagrammatic axioms. This leads to a direct understanding of how these objects arise in quantum mechanics and gives insight into how quantum analogues can be defined.</p>
<p> In Part I of this thesis we introduce quantum Latin squares (QLS), quantum objects which generalize the
classical Latin squares from combinatorics. We present a new method for constructing a unitary error basis
(UEB) from a quantum Latin square equipped with extra data, which we show simultaneously generalizes the
existing shift-and-multiply and Hadamard methods. We introduce two different notions of orthogonality for QLS
which we use to construct families of mutually unbiased bases and perfect tensors respectively. We also introduce a further generalization of Latin squares called quantum Latin isometry squares. We use these to produce quantum error codes and to give a new way of characterizing UEBs.</p>
<p> In Part II we show that maximal families of mutually unbiased bases (MUBs) are characterized in all
dimensions by partitioned UEBs, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned UEBs, and thus maximal families of MUBs, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of MUBs, partitioned UEBs, and finite fields as algebraic structures defined over Hilbert spaces.</p>
<p> In Part III we introduce quantum functions and quantum sets, which quantize the classical notions. We
show that these structures form a 2-category <b>QSet</b>. We extend this framework to introduce quantum graphs
and quantum homomorphisms which form a 2-category <b>QGraph</b>. We show that these 2-categories capture
several different notions of quantum morphism and noncommutative graph from various previous papers in
noncommutative topology, quantum non-local games and quantum information. We later use the correspondence between our quantum graph morphisms and those from quantum non-local games to show that pairs of quantum isomorphic graphs with multiple connected components are built up of quantum isomorphisms on those components.</p> |
| first_indexed | 2024-03-07T04:54:43Z |
| format | Thesis |
| id | oxford-uuid:d632d71c-cb07-45da-bc9b-778efad9da0c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:54:43Z |
| publishDate | 2019 |
| record_format | dspace |
| spelling | oxford-uuid:d632d71c-cb07-45da-bc9b-778efad9da0c2022-03-27T08:31:39ZQuantum Latin squares and quantum functions: applications in quantum informationThesishttp://purl.org/coar/resource_type/c_db06uuid:d632d71c-cb07-45da-bc9b-778efad9da0cCategory theoryQuantum pseudo-telepathyQuantum error correctionQuantum non-local gamesQuantum algebraQuantum channelsQuantum supremacyQuantum teleportationCategorical quantum mechanicsQuantum combinatoricsQuantum physicsQuantum informationEnglishHyrax Deposit2019Musto, BJVicary, JCoecke, B<p>A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent mathematical objects such as Latin squares or functions as string diagrams over the category of finite-dimensional Hilbert spaces obeying various diagrammatic axioms. This leads to a direct understanding of how these objects arise in quantum mechanics and gives insight into how quantum analogues can be defined.</p> <p> In Part I of this thesis we introduce quantum Latin squares (QLS), quantum objects which generalize the classical Latin squares from combinatorics. We present a new method for constructing a unitary error basis (UEB) from a quantum Latin square equipped with extra data, which we show simultaneously generalizes the existing shift-and-multiply and Hadamard methods. We introduce two different notions of orthogonality for QLS which we use to construct families of mutually unbiased bases and perfect tensors respectively. We also introduce a further generalization of Latin squares called quantum Latin isometry squares. We use these to produce quantum error codes and to give a new way of characterizing UEBs.</p> <p> In Part II we show that maximal families of mutually unbiased bases (MUBs) are characterized in all dimensions by partitioned UEBs, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned UEBs, and thus maximal families of MUBs, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of MUBs, partitioned UEBs, and finite fields as algebraic structures defined over Hilbert spaces.</p> <p> In Part III we introduce quantum functions and quantum sets, which quantize the classical notions. We show that these structures form a 2-category <b>QSet</b>. We extend this framework to introduce quantum graphs and quantum homomorphisms which form a 2-category <b>QGraph</b>. We show that these 2-categories capture several different notions of quantum morphism and noncommutative graph from various previous papers in noncommutative topology, quantum non-local games and quantum information. We later use the correspondence between our quantum graph morphisms and those from quantum non-local games to show that pairs of quantum isomorphic graphs with multiple connected components are built up of quantum isomorphisms on those components.</p> |
| spellingShingle | Category theory Quantum pseudo-telepathy Quantum error correction Quantum non-local games Quantum algebra Quantum channels Quantum supremacy Quantum teleportation Categorical quantum mechanics Quantum combinatorics Quantum physics Quantum information Musto, BJ Quantum Latin squares and quantum functions: applications in quantum information |
| title | Quantum Latin squares and quantum functions: applications in quantum information |
| title_full | Quantum Latin squares and quantum functions: applications in quantum information |
| title_fullStr | Quantum Latin squares and quantum functions: applications in quantum information |
| title_full_unstemmed | Quantum Latin squares and quantum functions: applications in quantum information |
| title_short | Quantum Latin squares and quantum functions: applications in quantum information |
| title_sort | quantum latin squares and quantum functions applications in quantum information |
| topic | Category theory Quantum pseudo-telepathy Quantum error correction Quantum non-local games Quantum algebra Quantum channels Quantum supremacy Quantum teleportation Categorical quantum mechanics Quantum combinatorics Quantum physics Quantum information |
| work_keys_str_mv | AT mustobj quantumlatinsquaresandquantumfunctionsapplicationsinquantuminformation |