進階搜尋


 
系統識別號 U0026-1701202001423800
論文名稱(中文) 對稱高斯態糾纏度量之上界,及糾纏強度
論文名稱(英文) Supremum of Entanglement Measure for Symmetric Gaussian states, and Entangling Capacity
校院名稱 成功大學
系所名稱(中) 物理學系
系所名稱(英) Department of Physics
學年度 108
學期 1
出版年 109
研究生(中文) 高至遠
研究生(英文) Jhih-Yuan Kao
學號 L28031017
學位類別 博士
語文別 英文
論文頁數 124頁
口試委員 指導教授-周忠憲
口試委員-李哲明
口試委員-陳岳男
口試委員-管希聖
口試委員-蘇正耀
口試委員-林豐利
中文關鍵字 糾纏  量子操作  糾纏強度  高斯態  正部分轉置 
英文關鍵字 entanglement  quantum operation  entangling capacity  Gaussian state  PPT 
學科別分類
中文摘要 這篇論文有兩大主題: 一個是關於量子操作的糾纏強度,另一個是對稱高斯態的糾纏度量的最低上界。
正部分轉置態在量子資訊中是一類很重要的量子態。在這裡我們展示了一個可以針對負度,得到糾纏強度上下界的方法。這些糾纏強度的上界被發現和一個量子操作的非正部分轉置性有關。一個可以度量量子操作和量子態,而且和正部分轉置相關的長度可以被定義,因而建立了一個和正部分轉置性有關的幾何。從此衍伸出來的距離可以限制相對的糾纏能力,對這個幾何附加了更多的物理意涵。
對於一個對稱高斯態,藉由識別出好的量子態的邊界並搭配適當的變數變換,我們得以證明對稱量子態中兩個團塊間的負度存在最小上界之存在及明確值。只包含了團塊以及總體的大小,這是結果可以輕易地套用到所有此類的量子態。
英文摘要 In this thesis there are two topics: On the entangling capacity, in terms of negativity, of quantum operations, and on the supremum of negativity for symmetric Gaussian states.
Positive partial transposition (PPT) states are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies both entangling capacity/entanglement and PPT-ness of an operation or state can be defined, establishing a geometry characterized by PPT-ness. The distance derived from the length bounds the relative entangling capability, endowing the geometry with more physical significance.
For a system composed of permutationally symmetric Gaussian modes, by identifying the boundary of valid states and making necessary change of variables, the existence and exact value of the supremum of logarithmic negativity (and negativity likewise) between any two blocks can be shown analytically. Involving only the total number of interchangeable modes and the sizes of respective blocks, this result is general and easy to be applied for such a class of states.
論文目次 1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Quantum Operations and Entangling Capability . . . . . . . . . . . . . . . . . 1
1.1.2 Symmetric Gaussian States and Suprema of Entanglement Measure . . . . . . 2
1.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Entanglement Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 How This Thesis Is Organized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Mathematical Premier 5
2.1 Sets and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Index, Countability, Sequences and Series . . . . . . . . . . . . . . . . . . . . . 6
2.1.4 Ordered Sets, Intervals and Bounds . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Vector Spaces and Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Vector Spaces and Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Linear Mappings, Operators and the Dual Space . . . . . . . . . . . . . . . . . 11
2.3 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Inner Product, Inner Product Spaces and Hilbert Spaces . . . . . . . . . . . . . 13
2.3.2 Fourier Expansion and Orthonormal Bases . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Adjoint, Self-adjoint Operators and Unitary Mappings . . . . . . . . . . . . . . 15
2.3.4 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.5 Orthogonal Direct Sum, Projections and the Identity Operator . . . . . . . . . 17
2.3.6 Isomorphism with Cn and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.7 Spectrum and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 L2-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Lp-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 L2-space and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.3 Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Bilinear mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.3 Linear Mappings and Tensor Product . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.4 Multilinear Mappings and Tensor Product . . . . . . . . . . . . . . . . . . . . . 28
2.5.5 Hilbert Space Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.6 Tensor Product of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Quantum States and Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.2 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.3 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Group Actions and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7.1 Group Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7.2 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7.3 Unitary Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 Quantum Operators on L2(Rn) and Representations . . . . . . . . . . . . . . . . . . . 35
2.8.1 Position and Momentum Operators . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8.2 Unitary Representations on L2(Rn) . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8.3 Representations and Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8.4 Spectral Theorem and Realization of a State . . . . . . . . . . . . . . . . . . . 37
2.9 Phase Spaces and Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 37
2.9.1 Classical Mechanics and the Phase Space . . . . . . . . . . . . . . . . . . . . . 38
2.9.2 Symplectic Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Several Topics of Operators 40
3.1 Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Hilbert-Schmidt and Trace-class Operators . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Trace and Hilbert-Schmidt Inner Product . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Hilbert-Schmidt and Trace-class Operators . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Another Look at Density Operators . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.4 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Weyl Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Weyl Quantization and Dequantization . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Wigner Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Wigner Quasi-probability Distribution . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.4 Metaplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Hermitian and Positive Operators on a Finite-Dimensional Hilbert Space . . . . . . . 45
3.4.1 Norms on Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.2 Inner Product between Hermitian and Positive Operators . . . . . . . . . . . . 47
3.4.3 Decomposition of a Hermitian Operator . . . . . . . . . . . . . . . . . . . . . . 47
3.4.4 Ensembles of Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Linear Mappings from Operators to Operators 51
4.1 Linear Mappings from Operators to Operators and Quantum Operations . . . . . . . . 51
4.1.1 Quantum Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Adjoint of a Linear Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Choi Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Choi Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Choi Isomorphism is Indeed an Isomorphism . . . . . . . . . . . . . . . . . . . 54
4.4 Tensor Product Space, Transposition and Partial Transposition . . . . . . . . . . . . . 55
4.4.1 Choi Isomorphism on a Bipartite System . . . . . . . . . . . . . . . . . . . . . 55
4.4.2 Transposition and Partial Transposition . . . . . . . . . . . . . . . . . . . . . . 55
4.4.3 PPT, Peres Criterion and Negativity . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Adjoint, Choi Isomorphism and Complex Conjugation . . . . . . . . . . . . . . . . . . 57
4.5.1 Adjoint and Choi Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5.2 Complex Conjugation and Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5.3 Adding an Ancilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 HP and TP Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6.1 Decomposition of an HP Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6.2 Operator-sum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6.3 Trace Norm of a Hermitian Operator after an HP or HPTP Mapping . . . . . 60
5 Entangling Capacity of a Quantum Operation 63
5.1 Entangling Capacity and Perfect Entangler . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Entangling Capacity and the Ancilla . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.2 Perfect Entangler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Bounds on Entangling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.1 The Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.3 Proof of the Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Geometrical Interpretation of Bounds on Entangling Capacity and Entanglement . . . 70
5.3.1 Norms and Metrics Induced by Partial Transposition . . . . . . . . . . . . . . . 70
5.3.2 Distance and Difference in Negativity after Operations . . . . . . . . . . . . . . 72
5.4 Upper Bounds Using Schatten p-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Comparison between Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.1 Proof for the Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.2 Proof for the Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6 PPT-ness and Separability of Unitary Operations and of Pure States . . . . . . . . . . 76
5.6.1 Proof for Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.6.2 Proof for Unitary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.7 Exact Entangling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.7.1 Basic Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.7.2 Reaching the Entangling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Density Operators on L2(RN) and Gaussian States 84
6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.1 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.2 Symplectic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.3 Gaussian State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.4 Valid Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.5 Entanglement of a Density Operator on L2(Rn) . . . . . . . . . . . . . . . . . . 86
6.2 Multisymmetric Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.1 Symmetric Gaussian States, and the Standard Form . . . . . . . . . . . . . . . 87
6.2.2 Multisymmetric Gaussian state . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Metaplectic Transform on States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.1 Motive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.2 Symmetric State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3.3 Multisymmetric State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.4 Blocks of Modes from a Symmetric Gaussian State . . . . . . . . . . . . . . . . 90
7 Supremum of Block Entanglement for Symmetric Gaussian States 92
7.1 Block Entanglement of a Symmetric Gaussian State . . . . . . . . . . . . . . . . . . . 92
7.1.1 Parameters for the Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.1.2 Block Entanglement for a Symmetric Gaussian State . . . . . . . . . . . . . . . 93
7.2 Search for the Supremum/infimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2.1 Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2.2 Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2.3 Value at the Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3 The Suprema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4 Decrease of Entanglement with Increasing nd . . . . . . . . . . . . . . . . . . . . . . . 97
7.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.5.1 Boundedness and Block Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.5.2 Monogamy of Entanglement and Multipartite Entanglement . . . . . . . . . . . 98
7.5.3 Available Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.5.4 Purities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.5.5 Approximation and Two Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8 Summary 102
8.1 Entangling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2 Supremum of Block Entanglement for Symmetric Gaussian States . . . . . . . . . . . 103
8.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A Equivalence of Norms 111
B Isometry with Respect to Hilbert-Schmidt Inner Product 112
B.1 Choi Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2 Transposition and Partial Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.2.2 Linear Mappings from Operators to Operators . . . . . . . . . . . . . . . . . . 113
C Si- does not Necessarily Have the Smallest Operator Norm 114
D Schmidt Decomposition and Spectral decomposition 116
E Spectrum of a Symmetric Matrix with the Same Diagonal Entries and the Same Off-diagonal Ones 118
F Covariances of a Multisymmetric System 120
F.1 First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
F.2 Variances and Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
F.2.1 The Same Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
F.2.2 Different Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
G Supremum for Symmetric Pure Gaussian States 122
G.1 One Family of Symmetric Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
G.1.1 b < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
G.1.2 b > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
G.2 Two Families of Symmetric Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
G.2.1 κ > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
G.2.2 κ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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