Taksu Cheon, Atushi Tanaka
We present a unified formulation of quantum holonomy which is capable of describing all of its known varieties. The full role of non-Abelian gauge connection is elucidated. As examples, solvable models of quantum kicked spin are analyzed. They are shown to exhibit spiral holonomy, as well as Berry phase and Wilczek-Zee holonomy.
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A periodic change of slow environmental parameters of a quantum system induces quantum holonomy. The phase holonomy is a well-known example. Another is a more exotic kind that exhibits eigenvalue and eigenspace holonomies. We introduce a theoretical formulation that describes the phase and eigenspace holonomies on an equal footing. The building block of the theory is a gauge connection for an ordered basis, which is conceptually distinct from Mead-Truhlar-Berry's connection and its Wilczek-Zee extension. A gauge invariant treatment of eigenspace holonomy based on Fujikawa's formalism is develo…
A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.
Given a completely integrable system, we associate to any connection on its invariant tori fibred over a parameter manifold the classical and quantum holonomy operator (generalized Berry's phase factor), without any adiabatic approximation.
In the project AQV the issue of quantum constraints, KMS-states and algebras of quantum configuration and momentum variables in Loop Quantum Gravity has been argued. There a physical algebra has been required to contain complete observables and the quantum constraints, or at least the quantum constraints are affilliated with this algebra. In this context a first conjecture for a physical algebra is presented in this article. A new *-algebra for LGQ, which is called the localised holonomy-flux cross-product *-algebra, is studied. A suggestion for a physical *-algebra, which contains the localis…
An adiabatic change of parameters along a closed path may interchange the (quasi-)eigenenergies and eigenspaces of a closed quantum system. Such discrepancies induced by adiabatic cycles are refereed to as the exotic quantum holonomy, which is an extension of the geometric phase. "Small" adiabatic cycles induce no change on eigenspaces, whereas some "large" adiabatic cycles interchange eigenspaces. We explain the topological formulation for the eigenspace anholonomy, where the homotopy equivalence precisely distinguishes the larger cycles from smaller ones. An application to two level systems …
An adiabatic change of a bound state along a closed circuit in the parameter space can induces holonomies not only in the phase of the state, but also in the associated eigenspace and eigenvalue. The former is the well-known Berry phase while the latter, namely the exotic holonomy, is found a decade ago and its origin has not been understood yet. By extending the parameter into the complex number, the correspondence of the exotic holonomies and the degeneracy of the non-Hermitian Hamiltonian, or the exceptional points, is revealed. We show that this explains all the known non-trivial character…
We examine a parametric cycle in the N-body Lieb-Liniger model that starts from the free system and goes through Tonks-Girardeau and super-Tonks-Girardeau regimes and comes back to the free system. We show the existence of exotic quantum holonomy, whose detailed workings are analysed with the specific sample of two- and three body systems. The classification of eigenstates based on clustering structure naturally emerges from the analysis.
A quantum holonomy reflects the curvature of some underlying structure of quantum mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace preserving completely positive maps. By the use of the Jamiołkowski isomorphism, we show that the proposed channel holonomy is related to the Uhlmann holonomy. The general theory is illustrated for specific examples. We put forward a physical realization of the channel holonomy in terms of interferometry. This enables us to identify a gauge invariant physical object …
Uhlmann's concept of quantum holonomy for paths of density operators is generalised to the off-diagonal case providing insight into the geometry of state space when the Uhlmann holonomy is undefined. Comparison with previous off-diagonal geometric phase definitions is carried out and an example comprising the transport of a Bell-state mixture is given.
In this article a new C*-algebra derived from the basic quantum variables: holonomies along paths and group-valued quantum flux operators in the framework of Loop Quantum Gravity is constructed. This development is based on the theory of cross-products and C*-dynamical systems. The author has presented a set of actions of the flux group associated to a surface set on the analytic holonomy C*-algebra, which define C*-dynamical systems. These objects are used to define the holonomy-flux cross-product C*-algebra associated to a surface set. Furthermore surface-preserving path- and graph-diffeomor…
The correspondence between exotic quantum holonomy that occurs in families of Hermitian cycles, and exceptional points (EPs) for non-Hermitian quantum theory is examined in quantum kicked tops. Under a suitable condition, an explicit expressions of the adiabatic parameter dependencies of quasienergies and stationary states, which exhibit anholonomies, are obtained. It is also shown that the quantum kicked tops with the complexified adiabatic parameter have a higher order EP, which is broken into lower order EPs with the application of small perturbations. The stability of exotic holonomy again…
We give an update on a quantum adiabatic algorithm for the Turing noncomputable Hilbert's tenth problem, and briefly go over some relevant issues and misleading objections to the algorithm.
Completely integrable Hamiltonian systems look promising for controllability since their first integrals are stable under an internal evolution, and one may hope to find a perturbation of a Hamiltonian which drives the first integrals at will. Action-angle coordinates are most convenient for this purpose. Written with respect to these coordinates, a Hamiltonian and first integrals depend only on the action ones. We introduce a suitable perturbation of a Hamiltonian by a term containing time-dependent parameters so that an evolution of action variables is nothing else than a displacement along …
An interplay of an exotic quantum holonomy and exceptional points is examined in one-dimensional Bose systems. The eigenenergy anholonomy, in which Hermitian adiabatic cycle induces nontrivial change in eigenenergies, can be interpreted as a manifestation of eigenenergy's Riemann surface structure, where the branch points are identified as the exceptional points which are degeneracy points in the complexified parameter space. It is also shown that the exceptional points are the divergent points of the non-Abelian gauge connection for the gauge theoretical formulation of the eigenspace anholono…
We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogen-like atoms where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian, while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlma…
We study the evolution of quantum eigenstates in the presence of level crossing under adiabatic cyclic change of environmental parameters. We find that exotic holonomies, indicated by exchange of the eigenstates after a single cyclic evolution, can arise from non-Abelian gauge potentials among non-degenerate levels. We illustrate our arguments with solvable two and three level models.
In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sjöqvist et al. We develop a holonomy theory for the latter which we also relate to the already existing theory for the former. This makes it clear what the similarities and differences between the two geometric phases are. We discuss and motivate constraints on the two phases. Furthermore, we discuss some topological properties of the holonomy of `real' quantum systems, and we introduce higher-order geometric phases for not necessari…
We discuss some of the properties of the `collision' of a quantum mechanical wave packet with an infinitely high potential barrier, focusing on novel aspects such as the detailed time-dependence of the momentum-space probability density and the time variation of the uncertainty principle product $Δx_t \cdot Δp_t$. We make explicit use of Gaussian-like wave packets in the analysis, but also comment on other general forms.
The philosophy of the Loop Quantum Gravity approach is to construct the canonical variables by using the duality of infinitesimal connections and holonomies along loops. Based on this fundamental property for example the holonomy-flux *-algebra has been formulated. A generalisation of the one-to-one correspondence between infinitesimal objects: connections and curvature and path based objects: holonomy maps and parallel transports is used to replace the configuration space of the theory. This generalised duality is related to the concept of path connections and holonomy groupoids, which origin…
The Influence of Quantum Correlation on the Holonomy of Spatially-Structured Bi-Photons
本文研究空间结构双光子中量子关联如何影响几何相位，提出投影规范不变度量以区分统计叠加与量子关联的贡献，并通过可调纠缠光子对的光学线路实验加以验证。
Abelian and non-Abelian geometric phases, known as quantum holonomies, have attracted considerable attention in the past. Here, we show that it is possible to associate nonequivalent holonomies to discrete sequences of subspaces in a Hilbert space. We consider two such holonomies that arise naturally in interferometer settings. For sequences approximating smooth paths in the base (Grassmann) manifold, these holonomies both approach the standard holonomy. In the one-dimensional case the two types of holonomies are Abelian and coincide with Pancharatnam's geometric phase factor. The theory is il…
Holonomy and vortex structures in quantum hydrodynamics
本文通过规范联络理论为量子流体动力学引入常曲率与内禀完整性，使流体速度可包含涡旋细丝解，并将涡旋动力学作为几何相来源纳入薛定谔方程和泡利方程。
Anatomy of Non-Hermitian Dynamical Quantum Phase Transitions
本文建立统一框架研究非厄米系统中的动力学量子相变，揭示其几何正交性与涌现拓扑特征，为开放系统量子临界性奠定原理基础。
Quasi-holonomy in non-adiabatic quantum evolution
研究非绝热量子时间演化中沿复格拉斯曼流形回路子空间的准全纯性，通过分解薛定谔演化，获依赖动力学传播子的有效几何生成元，探讨多种分解方式及相关问题。
We present a review on the progress in the understanding and characterization of holonomy and topology of a discrete-time quantum walk architecture, consisting of a unitary step given by a sequence of two non-commuting rotations in parameter space \cite{Puentesarxiv}. Unlike other similar systems recently studied in detail in the literature, this system does not present continous 1D topological boundaries, it only presents a discrete number of Dirac points where the quasi- energy gap closes. At these discrete points the topological winding number is not defined. Therefore, such discrete points…
We investigate the continuous quantum measurement of a superconducting qubit undergoing fluorescence. The fluorescence of the qubit is detected via a phase-preserving heterodyne measurement, giving the fluorescence quadrature signals as two continuous qubit readout results. By using the stochastic path integral approach to the measurement physics, we derive most likely paths between boundary conditions on the state, and compute approximate time correlation functions between all stochastic variables via diagrammatic perturbation theory. We focus on paths that increase in energy during the conti…
Quantum phase transitions reflect singular changes taking place in a many-body ground state, however, computing and analyzing large-scale critical wave functions constitutes a formidable challenge. New physical insights into the sub-Ohmic spin-boson model are provided by the coherent state expansion (CSE), which represents the wave function by a linear combination of classically displaced configurations. We find that the distribution of low-energy displacements displays an emergent symmetry in the absence of spontaneous symmetry breaking, while experiencing strong fluctuations of the order par…
A unifying framework for identifying distance and holonomy for decompositions of density operators is introduced. Parallelity between quantum ensembles is defined by minimizing this distance over allowed decompositions. The minimum is a property of a pair of states and coincides with the Bures distance. The parallelity condition imposes a connection (rule for parallel transport) that results in the Uhlmann holonomy for sequences of density operators. A distance and holonomy for spectral decompositions of density operators is identified as a sub-group restriction of the full decomposition freed…
Anatomy of localisation protected quantum order on Hilbert space
本文用本征态振幅在Hilbert空间图上的非局域及高点关联刻画局域化保护量子序与纠缠结构，并为分析多体局域化相到遍历相的转变提供新标度途径。
Comment on "Evolution Operator Can Always Be Separated into the Product of Holonomy and Dynamic Operators"
该文指出Ref中关于量子时间演化可分解为holonomy和动态算子乘积的结论错误，因该结论基于时间演化算子的循环定义。
Phase space anatomy of dynamical quantum phase transitions
本研究通过离散相空间区分了动力学量子相变中序参量与返回率两种定义，揭示了干涉在返回奇异分支选择中的独立作用。
Deterministic photonic entanglement arising from non-Abelian quantum holonomy
本文利用片上三维非阿贝尔量子和乐系统提出一种协议，实现了确定性、高保真度的光子纠缠，并推广至任意高维量子系统，为光的确定性量子信息处理提供了新机制。
Evolution Operator Can Always be Separated into the Product of Holonomy and Dynamic Operators
本研究证明量子系统演化算符总能分解为和乐与动力学算符的乘积，给出循环演化分解的矩阵表示及纯和乐条件，统一四类几何相位演化表示。
Holonomy and Complementarity in Open Quantum Systems
本文研究开放量子系统中互补性与耗散的关系，通过准静态传输赋予几何解释，揭示了耗散如何影响几何热力学响应。
Anatomy of Dynamical Quantum Phase Transitions
本文利用量子多体疤痕模型研究动力学量子相变，证明周期性DQPT通常源于简并初态流形中主导贡献的变化，其本质超越两能级系统的拉比振荡图像。
Liouvillian Geometry of Multidimensional Spectra: Pathway Transport and Observational Holonomy in Open Quantum Systems
针对开放量子系统中环境引起的Liouville路径间振幅传输，提出基于Liouvillian联络、曲率和观测完整度的几何框架，并利用Duhamel展开重构传输算符，将光谱畸变解释为弯曲Liouville空间流形的几何特征。
Quantum-State Projectors on Grassmannian: Geometry, Holonomy, and Topology
该研究针对量子系统的带隙哈密顿量，以全局投影算子的微分dP为核心，构建格拉斯曼流形上的最短路径，推导多带威尔逊圈行列式的基无关表达式，统一局域量子几何、和乐与拓扑，避免全局规范固定。
Krylov-space anatomy and spread complexity of a disordered quantum spin chain
通过Krylov基分析无序相互作用自旋链在遍历和MBL相中的量子态扩散复杂度，发现长时间复杂度在遍历相随Fock空间维度线性增长，在MBL相呈次线性增长且态分布呈拉伸指数衰减。
Continuous measurement-based holonomic quantum computation
该研究提出利用量子芝诺效应基于连续测量生成完整圈的方案，解析得出相关序列和时间，给出处理测量跃迁的方法，建立保持可纠错性的条件，还提及不满足纠错时用辅助量子比特使协议适用。
Stabilizing the Kerr arbitrary cat states and holonomic universal control
本研究提出参量驱动KNR的新型工程哈密顿量，可稳定任意猫态、操控势阱生成碰撞态，首次探究其量子绝热相位，开发通用绝热量子计算协议，并给出超导电路实验方案，提升了玻色模式控制自由度。
A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, these conditions determine that the holonomy group of the two-qubit interaction model contains $SU(2) \times SU(2)$. In particular, a universal two-qubit logic gate is attainable for this model.
Quantum Kernels and the Cross-Section of Stock Returns: Anatomy of a Vanishing Advantage
研究量子核能否改善股票回报预测，通过在中国A股市场对照实验发现，在主要评估中量子核无优势，不同模型差异不显著。还揭示相反结论产生原因，最后提出用于金融量子优势实证主张的协议标准。
Anatomy of the eigenstates distribution: a quest for a genuine multifractality
本文提出通用图解方法自洽计算一般Rosenzweig-Porter模型的分形维数，证明只有非均匀在位能量分布能导致真正的多重分形相，而统计均匀对角无序的随机矩阵模型无法承载该相。
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