经典隐变量模型与量子关联的研究
摘要
纠缠是量子力学现象中的突出特征之一,它在量子信息处理,如量子计算,量子离物传态,量子密钥的建立,量子纠错中起着非常重要的作用.本文主要通过量子非定域性和纠缠的研究做出了一个理论的梳理,一方面从最开始的EPR佯谬引发的对量子力学的质疑,到隐变量理论,直至Bell提出的检验非定域性的不等式以及后来拓展的无不等式的Bell定理简单回顾了量子非定域性的研究发展；另一方面,我们从量子力学基本理论的层面讨论了量子态,可分离和纠缠的判据,度量,量子非定域性不等式对纠缠的成功验证等.一般来说,多离子量子态可以在定域性角度分为定域态和非定域态,或者分为可分离态(非纠缠态)和纠缠态,然而,这两种分类是不等价的.在定性方面,Werner对此做出了详细讨论,并给出了例证——Werner态,再定量研究上,本文还描述了纠缠度和Bell不等式二者之间定量的关系.最后本文还关注了局域观测量的定域性和Bell算子最大期望之间的定量关系.我们开始沿着这样两条条发展路线阐述量子非定域性和纠缠的研究发展,然后对他们的交汇部分做了比较详细的叙述.其中,在用不等式方法验证纠缠方面,我们提出了一种定域联合观测量不等式,而在对易性和Bell算子期望值方面,我们讨论了三粒子直积态的情形,发现了这样的
一种奇异的现象,平庸的直积态情形同样拥有与两粒子情形不同的非平庸现象.
本论文主要内容和结构如下：
第一章介绍定域隐变量模型,包括EPR佯谬,Bohm模型,和隐变量模型的提出、基本理论等.
第二章介绍了Bell不等式和CHSH不等式的构造过程,和较为一般情况(2,2,d)情形的不等式模型.
第三章介绍了无不等式Bell定理,包括GHZ定理,Hardy定理和Cabello定理.
第四章首先介绍了量子一些系统的基本概念,包括量子态,密度矩阵,可分离态和纠缠态,及若干判据.随后介绍了一些量子体系纠缠的度量,和两体系统对CHSH不等式的违反,以及例外——Werner态及其相关讨论,再者,定量的讨论了两体系统中纠缠度和CHSH不等式的最大违反之间的关联,最后,我们给出了一种定域联合观测量不等式.
第五章介绍了量子熵和经典关联的相关内容.
第六章回顾了两粒子情形下局域观测量的对易性和Bell算子最大期望值之间的关系——Trade-off关联,既然后讨论了平庸的三粒子直积态情形,证明了三粒子直积态情形Trade-off关联有时不成立.
Entanglement is one of the outstanding characteristic phenomena of quantum mechanics. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, has potential for many quantum processes, including canonical ones:quantum cryptog-raphy, quantum teleportation, and dense coding. The main content of this thesis is research on entanglement and quantum non-locality. On one hand, from the EPR paradox leading to oppugn for quantum mechanics, to the institution of local hidden variable theory, until Bell inequality and Bell theorem without inequalities, we track such path to describe the devel-opment of quantum non-locality; on the other hand, we show aspects of entanglement including its characterization, detection, quantification, and quantum non-locality inequality as a test of quantum entanglement. Mul-tipartite quantum states can be classified into local and non-local states or into separable (non-entangled) and entangled states. These two kinds of classification based upon Bell-type inequalities are not equivalent.To the qualitative aspects, Werner gave the detailed analysis and the typical examples-Werner states, one the quantitative point of view, the relations between entanglement quantity and the violation of CHSH inequality is also discussed in this paper. On the last of this thesis, we show relation-ship between the local commutativity and CHSH inequality violation. We track such path to describe the development of quantum non-locality and quantum entanglement,and discuss the intersection carefully.
As to the manifestations of entanglement, we present a succinct in-equality for two-qubit quantum states reveal the non-locality and entan-glement. Furthermore, we extend relation between the local commutativity and CHSH inequality violation to3-qubit system,and shows that the con-verse trade-off relations do not always hold for three-qubit states, and that there exists some correlation even though the state is the simple product state.
The thesis structured as follows:
Chapter No.l We introduce the local hidden variable (LHV) theorem precisely, including the EPR disputation, Bohm model, the origin and the basic fundamentals of the LHV theory.
Chapter No.2We first review the construction of Bell inequality and CHSH inequality, then describe Bell inequality on the general (2,2,d) case.
Chapter No.3We introduce another kind of proof about non-locality, that is, all-versus-nothing violation of local realism, such as GHZ theorem, Hardy theorem and the Cabello's contribution.
Chapter No.4We first show fundamentals of quantum mechanics, in-cluding quantum state, density matrix, entanglement, criteria and quan-tification of entanglement, especially various manifestations of entangle-ment via Bell inequalities and the Wernrer discussion of classic correla-tions vs violation of Bell inequality. Then we discussion relation between the quantification of entanglement and the violation of Bell inequality.
Chapter No.5We discuss the quantum entropic and the classical corre-lations.
Chapter No.6We show the local commutativity and CHSH inequality violation on2-qubit case-conversed trade-off relations. Further more, we extended it to3-qubit case, and show the converse trade-off relations do not always hold for three-qubit states, and that there exists some correlation even though the state is the simple product state.
一种奇异的现象,平庸的直积态情形同样拥有与两粒子情形不同的非平庸现象.
本论文主要内容和结构如下：
第一章介绍定域隐变量模型,包括EPR佯谬,Bohm模型,和隐变量模型的提出、基本理论等.
第二章介绍了Bell不等式和CHSH不等式的构造过程,和较为一般情况(2,2,d)情形的不等式模型.
第三章介绍了无不等式Bell定理,包括GHZ定理,Hardy定理和Cabello定理.
第四章首先介绍了量子一些系统的基本概念,包括量子态,密度矩阵,可分离态和纠缠态,及若干判据.随后介绍了一些量子体系纠缠的度量,和两体系统对CHSH不等式的违反,以及例外——Werner态及其相关讨论,再者,定量的讨论了两体系统中纠缠度和CHSH不等式的最大违反之间的关联,最后,我们给出了一种定域联合观测量不等式.
第五章介绍了量子熵和经典关联的相关内容.
第六章回顾了两粒子情形下局域观测量的对易性和Bell算子最大期望值之间的关系——Trade-off关联,既然后讨论了平庸的三粒子直积态情形,证明了三粒子直积态情形Trade-off关联有时不成立.
Entanglement is one of the outstanding characteristic phenomena of quantum mechanics. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, has potential for many quantum processes, including canonical ones:quantum cryptog-raphy, quantum teleportation, and dense coding. The main content of this thesis is research on entanglement and quantum non-locality. On one hand, from the EPR paradox leading to oppugn for quantum mechanics, to the institution of local hidden variable theory, until Bell inequality and Bell theorem without inequalities, we track such path to describe the devel-opment of quantum non-locality; on the other hand, we show aspects of entanglement including its characterization, detection, quantification, and quantum non-locality inequality as a test of quantum entanglement. Mul-tipartite quantum states can be classified into local and non-local states or into separable (non-entangled) and entangled states. These two kinds of classification based upon Bell-type inequalities are not equivalent.To the qualitative aspects, Werner gave the detailed analysis and the typical examples-Werner states, one the quantitative point of view, the relations between entanglement quantity and the violation of CHSH inequality is also discussed in this paper. On the last of this thesis, we show relation-ship between the local commutativity and CHSH inequality violation. We track such path to describe the development of quantum non-locality and quantum entanglement,and discuss the intersection carefully.
As to the manifestations of entanglement, we present a succinct in-equality for two-qubit quantum states reveal the non-locality and entan-glement. Furthermore, we extend relation between the local commutativity and CHSH inequality violation to3-qubit system,and shows that the con-verse trade-off relations do not always hold for three-qubit states, and that there exists some correlation even though the state is the simple product state.
The thesis structured as follows:
Chapter No.l We introduce the local hidden variable (LHV) theorem precisely, including the EPR disputation, Bohm model, the origin and the basic fundamentals of the LHV theory.
Chapter No.2We first review the construction of Bell inequality and CHSH inequality, then describe Bell inequality on the general (2,2,d) case.
Chapter No.3We introduce another kind of proof about non-locality, that is, all-versus-nothing violation of local realism, such as GHZ theorem, Hardy theorem and the Cabello's contribution.
Chapter No.4We first show fundamentals of quantum mechanics, in-cluding quantum state, density matrix, entanglement, criteria and quan-tification of entanglement, especially various manifestations of entangle-ment via Bell inequalities and the Wernrer discussion of classic correla-tions vs violation of Bell inequality. Then we discussion relation between the quantification of entanglement and the violation of Bell inequality.
Chapter No.5We discuss the quantum entropic and the classical corre-lations.
Chapter No.6We show the local commutativity and CHSH inequality violation on2-qubit case-conversed trade-off relations. Further more, we extended it to3-qubit case, and show the converse trade-off relations do not always hold for three-qubit states, and that there exists some correlation even though the state is the simple product state.
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