从经典的 Frenet-Serret 装置到量子力学演化的曲率和扭转。第一部分:静态哈密顿

IF 2.1 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL International Journal of Geometric Methods in Modern Physics Pub Date : 2024-02-29 DOI:10.1142/s0219887824501524
Paul M. Alsing, Carlo Cafaro
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引用次数: 0

摘要

众所周知,三维欧几里得空间中空间曲线的 Frenet-Serret 装置决定了曲线的局部几何。特别是,Frenet-Serret 装置规定了重要的几何不变式,包括曲线的曲率和扭转。量子信息科学领域也承认,在巧妙地操纵编码物理系统量子信息的量子态时,低复杂度和高效率是必须实现的基本特征。在本文中,我们从几何角度提出了如何量化由动态演化的状态矢量追踪的量子曲线的弯曲和扭曲。具体地说,我们提出了一种量子版的 Frenet-Serret 装置,该装置适用于投影希尔伯特空间中的量子轨迹,该轨迹由在指定薛定谔方程的静态哈密尔顿下单元化演化的平行传输纯量子态追踪。我们提出的恒定曲率系数由切线向量|T〉到状态向量|Ψ〉的协变导数的平方给出,是量子曲线弯曲程度的有效度量。而我们提出的恒定扭转系数,是以切线向量|T〉的协变导数投影的大小平方来定义的,与|T〉和|Ψ〉都正交。扭转系数为量子曲线的扭转提供了方便的度量。值得注意的是,我们证明了我们提出的曲率系数和扭转系数与文献中已有的系数不谋而合,尽管引入的方式完全不同。有趣的是,我们不仅确定了零曲率对应于投影希尔伯特空间量子传输过程中的单位大地效率,还发现曲率和扭转的概念有助于揭示量子理论的统计结构。事实上,前一个概念本质上可以用峰度概念来定义,而后一个概念的正向性则可以看作是对著名的皮尔逊不等式的重述,它同时涉及数理统计中的峰度和偏度概念。最后,我们不仅举例说明了不可能产生扭转的单量子比特时间无关哈密顿演化的非零曲率,还讨论了扩展到双量子比特静态哈密顿的物理应用,这些哈密顿会产生由具有不同纠缠程度的量子态(从可分离态到最大纠缠的贝尔态)追踪的既有非零曲率又有非消失扭转的曲线。在附录 C 中,我们研究了在量子海森堡汉密尔顿演化下三个量子比特|GHZ〉和|W〉态的不同曲率和扭转特性。
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From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians

It is known that the Frenet–Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet–Serret apparatus specifies important geometric invariants, including the curvature and the torsion of a curve. It is also acknowledged in quantum information science that low complexity and high efficiency are essential features to achieve when cleverly manipulating quantum states that encode quantum information about a physical system.

In this paper, we propose a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by dynamically evolving state vectors. Specifically, we propose a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced by a parallel-transported pure quantum state evolving unitarily under a stationary Hamiltonian specifying the Schrödinger equation. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector |T to the state vector |Ψ and represents a useful measure of the bending of the quantum curve. Our proposed constant torsion coefficient, instead, is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector |T, orthogonal to both |T and |Ψ. The torsion coefficient provides a convenient measure of the twisting of the quantum curve. Remarkably, we show that our proposed curvature and torsion coefficients coincide with those existing in the literature, although introduced in a completely different manner. Interestingly, not only we establish that zero curvature corresponds to unit geodesic efficiency during the quantum transportation in projective Hilbert space, but we also find that the concepts of curvature and torsion help enlighten the statistical structure of quantum theory. Indeed, while the former concept can be essentially defined in terms of the concept of kurtosis, the positivity of the latter can be regarded as a restatement of the well-known Pearson inequality that involves both the concepts of kurtosis and skewness in mathematical statistics. Finally, not only do we present illustrative examples with nonzero curvature for single-qubit time-independent Hamiltonian evolutions for which it is impossible to generate torsion, but we also discuss physical applications extended to two-qubit stationary Hamiltonians that generate curves with both nonzero curvature and nonvanishing torsion traced by quantum states with different degrees of entanglement, ranging from separable states to maximally entangled Bell states. In the Appendix C, we examine the different curvature and torsion characteristics of the three qubit |GHZ and |W states under evolution by a quantum Heisenberg Hamiltonian.

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CiteScore
3.40
自引率
22.20%
发文量
274
审稿时长
6 months
期刊介绍: This journal publishes short communications, research and review articles devoted to all applications of geometric methods (including commutative and non-commutative Differential Geometry, Riemannian Geometry, Finsler Geometry, Complex Geometry, Lie Groups and Lie Algebras, Bundle Theory, Homology an Cohomology, Algebraic Geometry, Global Analysis, Category Theory, Operator Algebra and Topology) in all fields of Mathematical and Theoretical Physics, including in particular: Classical Mechanics (Lagrangian, Hamiltonian, Poisson formulations); Quantum Mechanics (also semi-classical approximations); Hamiltonian Systems of ODE''s and PDE''s and Integrability; Variational Structures of Physics and Conservation Laws; Thermodynamics of Systems and Continua (also Quantum Thermodynamics and Statistical Physics); General Relativity and other Geometric Theories of Gravitation; geometric models for Particle Physics; Supergravity and Supersymmetric Field Theories; Classical and Quantum Field Theory (also quantization over curved backgrounds); Gauge Theories; Topological Field Theories; Strings, Branes and Extended Objects Theory; Holography; Quantum Gravity, Loop Quantum Gravity and Quantum Cosmology; applications of Quantum Groups; Quantum Computation; Control Theory; Geometry of Chaos.
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