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Separability of Quantum States via Algebraic Geometry : 대수기하학을 통한 양자 상태의 분리가능성 연구

DC Field Value Language
dc.contributor.advisor김영훈-
dc.contributor.author나주한-
dc.date.accessioned2017-07-14T00:41:07Z-
dc.date.available2017-07-14T00:41:07Z-
dc.date.issued2014-08-
dc.identifier.other000000022165-
dc.identifier.urihttps://hdl.handle.net/10371/121284-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2014. 8. 김영훈.-
dc.description.abstractIn this thesis, we study the quantum separability problem by taking advantage of various methods in algebraic geometry.

In order to explore the separability of quantum states, we begin with the range criterion for separability. It leads us to examine the condition that $		-
dc.description.abstract\psi_1 \rangle \otimes-
dc.description.abstract\psi_2 \rangle \in D$ and $-
dc.description.abstract\overline{\psi_1} \rangle \otimes-
dc.description.abstract\psi_2 \rangle \in E$ for subspaces $D$ and $E$ of a finite-dimensional composite quantum system $\mathcal{H}_A \otimes \mathcal{H}_B$. More explicitly, the following two questions naturally arise : (1) For which conditions there is a nonzero product vector $-
dc.description.abstract\psi_2 \rangle$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ such that $-
dc.description.abstract\psi_2 \rangle \in E$? (2) if it exists, how many such nonzero product vectors in $\mathcal{H}_A \otimes \mathcal{H}_B$ exist up to constant?

We investigate the question (1) and generalize it for the multipartite cases. Moreover, we answer the question (2) so that the upper bound for the number of vectors $		-
dc.description.abstract\psi_2 \rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ satisfying the condition that $-
dc.description.abstract\psi_2 \rangle \in E$ is expected to be sharp for the qubit-qunit case.-
dc.description.tableofcontents1 Introduction 1
1.1 Basics on quantum entanglement 2
1.2 Quantum separability problem 5
1.3 Classification of entangled states 7
1.4 Content of this thesis 9
2 Classical Algebraic Geometry 11
2.1 Affine varieties 12
2.2 Projective varieties 18
2.3 Dimension and degree 24
2.4 Smooth varieties 31
2.5 Grassmann varieties 36
2.6 Segre varieties 38
2.7 Join varieties, secant varieties and tangential varieties 42
2.8 Dual varieties and hyperdeterminants 47
2.9 Newton polytopes and Bernsteins theorem 52
2.10 Classical resultants 57
2.11 Permanents 60
3 Quantum Separability Problem 62
3.1 Separability for pure states 62
3.2 PPT criterion and positive linear maps 65
3.3 Range criterion 68
4 Algebraic Criterion for Separability 72
4.1 Algebraic criterion for bipartite cases 73
4.2 Algebraic criterion for multipartite cases 76
4.3 Proof of Theorem 4.2.1 77
4.3.1 Over-determined case 77
4.3.2 Critical case 81
4.3.3 Under-determined case 83
4.4 Multi-qubit cases and permanents of matrices 88
5 Upper Bounds for the Number of Product Vectors 97
5.1 Transformed into a system of equations 98
5.2 Qubit-qunit case 102
5.3 Examples 108
6 Classification of Entangled States 118
6.1 Bipartite cases 119
6.2 Three qubit case 120
6.3 Other cases 125
Abstract (in Korean) 135		-
dc.formatapplication/pdf-
dc.format.extent3643342 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject양자얽힘-
dc.subject분리가능 상태-
dc.subject얽힌 상태-
dc.subject곱벡터-
dc.subject치역판별법-
dc.subject대수기하학-
dc.subject.ddc510-
dc.titleSeparability of Quantum States via Algebraic Geometry-
dc.title.alternative대수기하학을 통한 양자 상태의 분리가능성 연구-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pages134-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2014-08-
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