Duistermaat-Heckman measure and the mixture of quantum states

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum \bsA+\bsB of two independent random Hermitian matrices \bsA and \bsB. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Comment: 40 pages, 10 figures, LaTeX, the final version accepted for publication in J. Phys.

How Do I Love Thee? Adult Attachment and Reinforcement Sensitivity

This thesis aimed to examine the nature of the relations between individual differences in adult attachment patterns and the sensitivity of motivational systems – the Behavioural Approach System (BAS), the Fight-Flight-Freeze System (FFFS), and the Behavioural Inhibition System (BIS) – proposed by the revised reinforcement sensitivity theory (r-RST). In Study 1, psychology undergraduates (N=225) completed self-reported measures of adult attachment and reinforcement sensitivity. Both attachment dimensions were significantly related to BIS sensitivity, which suggests that sensitivity to motivational ambivalence is a central feature of attachment insecurity. In Study 2, psychology undergraduates (N=200) experienced virtual separation and reunion scenarios with a ‘virtual spouse,’ and subsequently completed adult attachment and reinforcement sensitivity questionnaires. Adult attachment, but not reinforcement sensitivity, were predictive of behavioural and emotional responses to separation and reunion. This suggests that adult attachment has unique predictive power to dyadic behaviour. Finally, Study 3 (N=63) examined the links between self-reported adult attachment and reinforcement sensitivity and neurobiological markers of approach and avoidance motivation (8 minutes of resting EEG). Neither adult attachment nor reinforcement sensitivity exhibited robust associations with the resting EEG indices. This may reflect the construct heterogeneity of the attachment dimensions and reinforcement sensitivity, such that they do not neatly map onto neural correlates of approach and avoidance. Together, the studies reported in this thesis suggest modest overlaps between individual differences in adult attachment and reinforcement sensitivities at the self-report level, but the two domains are largely independent in relation to attachment behaviour and neural correlates of approach-avoidance

The Chinese translation of the Francis Scale of Attitude toward Christianity : factor structure, internal consistency reliability, and construct validity among Protestant Christians in Shanghai

A sample of 131 Chinese Christians attending a Protestant church in Shanghai completed the Chinese translation of the Francis Scale of Attitude toward Christianity developed originally by Francis et al. (North American Journal of Psychology 4, 431–440, 2002) in Hong Kong. The data support the factor structure, internal the internal consistency reliability, and construct validity of this instrument and commend it for further use in studies conducted among Christians in China

Unveiling Correlated Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices, harboring extraordinary boundary modes protected by anomalous symmetry actions. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wavefunctions for numerical simulations is highly desirable. In this paper, we tackle this challenge by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connections between these tensor equations and topological insulators by construing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for corresponding topological insulator phases. Our methodology provides a crucial first step towards simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.Comment: 32+20 pages, 11 figure

Revisiting Nyquist-Like Impedance-Based Criteria for Converter-Based AC Systems

Multiple types of Nyquist-like impedance-based criteria are utilized for the small-signal stability analysis of converter-based AC systems. It is usually considered that the determinant-based criterion can determine the overall stability of a system while the eigenvalue-based criterion can give more insights into the mechanism of the instability. This paper specifies such understandings starting with the zero-pole calculation of impedance matrices obtained by state-spaces with the Smith-McMillan form, then clarifying the absolute reliability of determinant-based criterion with the common assumption for impedance-based analysis that each subsystem can stably operate before the interconnection. However, ambiguities do exist for the eigenvalue-based criterion when an anticlockwise encirclement around the origin is observed in the Nyquist plot. To this end, a logarithmic derivative-based criterion to directly identify the system modes using the frequency responses of loop impedances is proposed, which owns a solid theoretical basis of the Schur complement of transfer function matrices. The theoretical analysis is validated using a PSCAD simulation of a grid-connected two-level voltage source converter.Comment: Accepted by CSEE JPE

Design flexibility in complex engineering systems under multiple uncertainties

Master'sMASTER OF ENGINEERIN

Variational Tensor Wavefunctions for the Interacting Quantum Spin Hall Phase

The quantum spin hall (QSH) phase, also known as the 2D topological insulator, is characterized by protected helical edge modes arising from time reversal symmetry. While initially proposed for band insulators, this phase can also manifest in strongly-correlated systems where conventional band theory fails. To overcome the challenge of simulating this phase in realistic correlated models, we propose a novel framework utilizing fermionic tensor network states. Our approach involves constructing a tensor representation of the fixed-point wavefunction based on an exact solvable model, enabling us to derive a set of tensor equations governing the transformation rules of local tensors under symmetry operations. These tensor equations lead to the anomalous edge theory, which provides a comprehensive description of the QSH phase. By solving these tensor equations, we obtain variational ansatz for the QSH phase, which we subsequently verify through numerical calculations. This method serves as an initial step towards employing tensor algorithms to simulate the QSH phase in strongly-correlated systems, opening new avenues for investigating and understanding topological phenomena in complex materials.Comment: 6+15 pages,12 figures. Numerical calculations are adde