低维量子系统中两体与多体量子关联指示相变的研究

Quantum correlation is an important concept in many-body quantum physics. Quantum entanglement(QE)-a kind of quantum correlation-has been investigated extensively, as its non-local connotation provides novel perspectives for phase transitions in low-dimensional quantum systems. However, QE is not the only nature of quantum correlation. Recently, Olliver et al. have introduced a quantity-the quantum discord(QD)- to quantify all the quantum correlation present in a quantum state. Both QD and QE describe the nature of quantum correlation in quantum states, however, QD can even be present in un-entangled (or separable) states. In this project, by classifying general states into classical states, quantum separable states and quantum entangled states, we will show that QD has a broader scope of application than QE in detecting phase transition, that is, QD can capture the signal of phase transition in a separable state, which QE cannot. We will study QE and QD in several one-dimensional systems, including a four-site interaction spin ladder model, in which it's already known that two-point QE fails to detect a phase transition point of the system. We will investigate the ability of QD in detecting phase transitions in quantum separate states..Furthermore, quite recently, people have tried to generalize two-point quantum discord to multipartite states. With the help of relative entropy, which allows for a unified view of different correlations including QE and QD, people have proposed a global measure of quantum discord for multipartite quantum states, named as global quantum discord (GQD). The calculation of GQD is generally difficult, because it depends on the multipartite reduced density matrix and requires an extremization procedure. Then a witness for non-classical multipartite states is proposed. The witness operator can provide a sufficient condition for non-vanishing global quantum discord, but it does not require any extremization procedure. By classifying phase transitions into two-point-quantum-correlation-driven(TD) phase transitions and multipartite-quantum-correlation-driven(MD) phase transitions, we see that GQD would have an important application in detecting phase transitions: in an MD phase transition, two-point quantum correlation is not in a key position, thus cannot detect this kind of phase transition, while GQD and its witness operator, which describe the multipartite correlation in the system, still have the ability to detect the phase transition. In this project, with the help of matrix product theory and density matrix renormalization group(DMRG) method, we will study the GQD and its witness operator in one-dimensional quantum systems, and try to relate GQD to the phase diagram of the models. Special attention would be paid to MD phase transitions, for which other quantum correlations fail to detect the quantum critical points.

随着信息科学与凝聚态物理的交融，量子纠缠（Quantum Entanglement(QE)）指示相变的能力得到了广泛的研究。近年来，另一种量子关联-Quantum Discord(QD)-也引起了很多关注。QD与QE具有一定的相似性，但作为相变指示器，QD自身的特色并未得到充分发掘：QD可捕捉非纠缠态中的相变信号。我们将从"纠缠态相变、可分离量子态相变、经典态相变"这一新的角度分析相变，凸显QD指示可分离量子态相变的能力。最近，QD的概念被推广到多体系统的全局量子关联（Global Quantum Discord(GQD)）。GQD也有指示相变的能力，更重要的是，在多体关联驱动的相变中，两体关联将失去指示器的作用，GQD的价值则被凸显出来。我们将从"两体关联驱动的相变、多体关联驱动的相变"这一角度分析相变，阐述GQD指示相变的能力。这些信息科学中的概念将为加深我们对相变的认识提供新的视角。

低维量子系统中的量子关联在量子信息理论和凝聚态物理中均引起了广泛的兴趣。一方面，量子关联可以作为信息的载体，为量子信息处理和量子计算服务；另一方面，量子关联可以用来刻画凝聚态物理所关心的量子相变。目前，大部分研究局限于两分关联，并多用奇异性来刻画量子相变；由于多分关联计算困难，相关研究甚少，人们还未构建出相变中多体关联的直观图像。. 本课题研究了低维量子系统发生量子相变时两分和多分关联的行为。. 在两分关联的研究中，第一，研究了非纠缠态中量子相变的探测。在梯子模型中，纠缠和QD在临界点均无奇异性；我们发现QD在临界点出现奇-偶效应，并用对称破缺进行了解释。第二，研究了量子非定域性(QN, quantum nonlocality)与对称性的关系，并通过计入自旋-晶格耦合，破坏平移不变性，观测到了非定域性；可见，低维自旋链系统可用做潜在的材料来存储QN并进行量子计算。. 在多分关联的研究中，我们重点研究了全局关联(GQD, Global quantum discord)和多分非定域性(MQN, multipartite quantum nonlocality)。多分关联最大的难题是数值计算，已有研究多只考虑短链，极大约束了人们的研究深度和广度。本项目中，第一，我们解决了多分关联的数值计算难题。(1)借助虚时演化算法(iTEBD)，我们把一维自旋链的基态近似表达为矩阵积态(MPS)的形式；(2)在MPS态的框架下，我们把多分关联的数学表达式，无缝地嵌入到密度矩阵重整化群算法(DMRG)中，从而用局部变分的方法完成了数值优化，极大地提高了计算效率。以上思路成功用于计算GQD和MQN，亦易移植于其他多体关联。第二，我们探索了GQD和MQN在大N极限下的标度行为，并发现：(1) 在短程量子关联的一维量子链中，GQD将呈现线性增长；因此，GQD具有宏观量的特征(类似于质量)，可定义关联密度(类似于质量密度)。(2)多分贝尔关联函数(多分贝尔不等式的左值)将呈现指数增长；可用对应的指数来更好地刻画多体关联。第三，我们描绘了量子相变中多体关联的直观图像：当系统趋近于临界点时，各格点逐渐凝聚成高度多体关联着的小团簇，且小团簇彼此间逐渐凝聚、团簇尺寸增大。该图像与“磁畴”的图象相通，加深了人们对多体关联的直观理解。