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系統識別號 U0026-1806201920095900
論文名稱(中文) 量子關聯性測度: 量子操縱性之幾何量化以及時間量子關聯性之階層關係
論文名稱(英文) Measuring quantum correlations: geometrically quantifying quantum steering and hierarchy in temporal quantum correlations.
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 107
學期 2
出版年 108
研究生(中文) 古煥宇
研究生(英文) Huan-Yu Ku
電子信箱 huan_yu@phys.ncku.edu.tw
學號 L28041012
學位類別 博士
語文別 英文
論文頁數 88頁
口試委員 口試委員-管希聖
口試委員-李哲明
召集委員-周忠憲
口試委員-陳光胤
指導教授-陳岳男
口試委員-尼爾 藍伯特
中文關鍵字 量子非局域性  量子操控性  時間量子關聯性  量子糾纏 
英文關鍵字 quantum steering  quantum entanglement  quantum nonlocality  temporal quantum correlations 
學科別分類
中文摘要 為了回應愛因斯坦(Einstein)-波多爾斯基(Podolsky)-羅森(Rosen)悖論,薛丁格(Schro ̈dinger)宣稱一個子系統的量測可以讓另一個空間分離的子系統瞬間被投影到不同的狀態。愛因斯坦-波多爾斯基-羅森操縱性的嚴格定義,是在2007年由懷斯曼(Wiseman),瓊斯(Jones)和多爾蒂(Doherty)所提出的。在他們的工作中,量子操縱性是介於量子非分離性(quantum nonseparability)和貝爾非局域性(Bell nonlocality)之間的量子關聯性,這三種量子關聯性之間具有階層關係。具體而言,違反貝爾不等式的量子態是量子可操縱而且有量子糾纏。可操縱的量子態必定具有量子糾纏,但是反之則否。因此,貝爾非局域性和量子糾纏性分別是最強和最弱的空間量子關聯性。量子操縱性在與單側與設備無關的量子信息處理中提供了許多實際應用,例如與單側設備無關的量子密鑰分配及與單側設備無關的量子隨機數發生器。

類似於貝爾不等式,宏觀系統上的宏觀實在性和非侵入性測量的假設可用於構造Leggett-Garg(LG)不等式。 LG不等式可用於對觀察或實驗結果進行分類,以滿足宏觀實在性。一個宏觀的理論(或等效地,對於給定的觀察,一個宏觀的對象)是指一個狀態總是“真實的”並且可以以非侵入的方式觀察之理論。通常,當系統的尺寸或尺寸增加時,可以應用宏觀理論,使得系統傾向於宏觀地表現並且可以非侵入地觀察。

類似於LG和貝爾不等式之間的關係,我們最近也提出了空間操縱性的時間類比,稱為時間量子操縱性。時間量子操控性可以應用於量子信息,開放量子系統和量子網路。時間量子操縱性顯示早期量測的結果如何影響時間演化後的量子系統。

在此篇論文中,我們首先證明在受限制的單向局部操作和古典通信之後量子操控性的幾何量測是單調方程。我們透過半正定規劃來計算量子操控性之幾何量測的上限和下限。對於維爾納態(Werner states),我們還給出量子操控性之幾何量測的下界。其次,我們證明了三種時間量子關聯性之間也存在階層關係:時間量子不可分離性,時間量子操縱和宏觀實在論。由於時間量子不可分離性可用於度量量子因果關係,所以,時間量子操縱性的量化可被視為直接關聯性的較弱度量,並且可用於區分量子網路中的直接關聯性和共同關聯性。
英文摘要 In response to Einstein-Podolsky-Rosen (EPR) paradox, Schro ̈dinger claimed that one subsystem can be instantaneously steered into different states by the other spatially separated subsystem. It was not clear until 2007 that the rigorous definition of Einstein-Podolsky-Rosen steering was proposed by Wiseman, Jones, and Doherty. They have shown that EPR steering is the intermediated class of quantum correlation between nonseparability and Bell nonlocality, which forms a hierarchy relation.

Specifically, a state, which violates a Bell inequality, is not only steerable but also entangled. And a steerable state must be entangled, but not vice versa. Therefore, Bell nonlocality and entanglement are respectively the strongest and weakest spatial quantum correlations. The EPR steering provides many practical applications in the one-sided device-independent quantum information processing, such as one-sided device-independent quantum key distribution and one-sided device-independent quantum random number generator.

Similar to the Bell inequality, the assumptions of macrorealism and noninvasive measurement on a macroscopic system can be used to construct an inequality, commonly called the Leggett-Garg (LG) inequality. The LG inequality can be used to classify observations or experimental outcomes that satisfy macrorealism or not. Here, a macrorealistic theory (or equivalently, for a given observation, a macrorealistic object) is the one whose state is always `realistic' and can be observed in a noninvasive manner. Generally, it is thought that macrorealistic theories may apply when the dimension or the size of the system is increased, such that the system tends to behave macroscopically and can be observed noninvasively.

Similar to the relation between Leggett-Garg and Bell inequality, a temporal analogy of spatial steering, referred to temporal steering, was also proposed recently. Temporal steering has recently been applied to quantum information, open quantum systems, and quantum network. Temporal steering shows how the earlier measurements influence the later ones on the very same party.

In this thesis, first, we define a geometrical measure of EPR steering, by showing that it is a monotone after performing a restricted one-way local operation and classical communication. We provide practical methods to estimate the upper and lower bounds of the geometrical EPR steering measure by a semidefinite program. We also analytically show that for the Werner states the bound is tight. Second, we show that there exists a hierarchy relation among the three temporal quantum correlations: temporal inseparability, temporal steering, and macrorealism. Given that the temporal inseparability can be used to define a measure of quantum causality, similarly, the quantification of temporal steering can be viewed as a weaker measure of direct cause and can be used to distinguish between direct cause and common cause in a quantum network.
論文目次 1 Introduction 1
1.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bell inequality and device-independent quantum information processing 2
1.3 Einstein-Podolsky-Rosen steering . . . . . . . . . . . . . . . . . . . . 3
1.4 Leggett-Garg and temporal steering inequalities . . . . . . . . . . . . 4
1.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Einstein-Podolsky-Rosen Steering 9
2.1 The Einstein-Podolsky-Rosen paradox . . . . . . . . . . . . . . . . . 9
2.2 An operational de nition of EPR steering . . . . . . . . . . . . . . . 10
2.3 Hierarchy relation among EPR unsteerability, Bell locality, and separability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Resource theory of steering . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Quanti cations of quantum steerability . . . . . . . . . . . . . . . . . 19
2.5.1 Steering witness . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.2 Steering robustness . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Geometrical quanti ers of quantum steerability 25
3.1 Trace-distance steerability measure . . . . . . . . . . . . . . . . . . . 25
3.1.1 Upper bound based on the restricted-noise consistent steering
robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Upper bound based on the consistent steering robustness . . . 29
3.1.3 Lower bound based on operator norm . . . . . . . . . . . . . . 29
3.2 Geometrical witness of steerability . . . . . . . . . . . . . . . . . . . . 30
I
CONTENTS
3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Steerability of the Werner states . . . . . . . . . . . . . . . . . 32
3.3.2 Steerability of the Horodecki states . . . . . . . . . . . . . . . 35
3.3.3 Steerability of the rank-2 Bell-diagonal states . . . . . . . . . 35
4 Hierarchy in temporal quantum correlations 39
4.1 Macrorealism and Leggett-Garg inequality . . . . . . . . . . . . . . . 40
4.2 Temporal steerability . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Pseudo density matrix and temporal inseparability . . . . . . . . . . 44
4.4 A hierarchy relation of temporal quantum correlations . . . . . . . . . 45
4.5 Classical steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Inferring causal structure with temporal steerability . . . . . . . . . . 51
5 Conclusions 53
6 Appendices 57
6.1 Metric properties of trace distance between two assemblages . . . . . 57
6.2 Restricted convex steering monotone . . . . . . . . . . . . . . . . . . 58
6.3 Semide nite programming formulation of SR
CSR . . . . . . . . . . . . 62
6.4 LHS surface of the Werner states . . . . . . . . . . . . . . . . . . . . 63
6.5 Obtaining a set of temporal correlations and a temporal assemblage
from a pseudo density matrix . . . . . . . . . . . . . . . . . . . . . . 65
6.6 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.7 Proof of identical relation between temporal steerable weight and
trace distance when considering a classical steerable temporal assemblage
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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