Quantum mechanics nominally works in a space of infinite dimensions. In practice, however, the computer-based solution of quantum-mechanical problems relies on a restriction of that space to a high but finite number of dimensions. This is important since computational quantum-mechanical simulations have a huge presence in today's physical science. In many instances, the finite-dimensional space needs to evolve during a simulation. In this work, the authors extend the formalism used for finite spaces to that of evolving spaces. Here, well-known equations acquire a new meaning in the language of the curved spaces familiar to general relativity. The authors provide a novel geometric approach to calculating tensorially well-defined derivatives of quantum-mechanical state and operator representations with respect to arbitrary external parameters. Contributions due to basis function evolution play the role of the Christoffel symbol terms of Riemannian geometry. The curvature of the resulting geometry is equivalent to Berry's curvature, but generalized for possibly nonorthogonal, smoothly varying basis functions. This work provides insights into dynamical quantum simulations that enable the development of improved simulation algorithms. It also opens up new scientific horizons, as is ever the case when establishing formal links between different subfields of physics.
Quantum mechanics nominally works in a space of infinite dimensions. In practice, however, the computer-based solution of quantum-mechanical problems relies on a restriction of that space to a high but finite number of dimensions. This is important since computational quantum-mechanical simulations have a huge presence in today's physical science. In many instances, the finite-dimensional space needs to evolve during a simulation. In this work, the authors extend the formalism used for finite spaces to that of evolving spaces. Here, well-known equations acquire a new meaning in the language of the curved spaces familiar to general relativity. The authors provide a novel geometric approach to calculating tensorially well-defined derivatives of quantum-mechanical state and operator representations with respect to arbitrary external parameters. Contributions due to basis function evolution play the role of the Christoffel symbol terms of Riemannian geometry. The curvature of the resulting geometry is equivalent to Berry's curvature, but generalized for possibly nonorthogonal, smoothly varying basis functions. This work provides insights into dynamical quantum simulations that enable the development of improved simulation algorithms. It also opens up new scientific horizons, as is ever the case when establishing formal links between different subfields of physics.
Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.
Abstract We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations.
In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems.
Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.
Quantum mechanics traditionally describes the state of a system by a vector in Hilbert space and the dynamics by a unitary evolution governed by a Hamiltonian. Recent advances in differential geometry applied to quantum theory have revealed that quantum states can carry geometric structures such as Berry phases and curvature tensors. In this paper we introduce the notion of a quantum state curvature field, denoted Kψ(x), which is defined pointwise on the parameter space of a family of states {ψ(x)}. We propose a protocol called Quantum State Curvature Measurement (QSCM) that reconstructs Kψ by observing continuous quantum evolution along a path in parameter space. The method relies on measuring the infinitesimal phase accumulation and the deviation from parallel transport, which together encode the curvature. We discuss the mathematical foundation of Kψ, its relation to the Berry curvature, and the experimental feasibility of implementing QSCM on current quantum platforms. Finally, we outline the potential implications of discovering new quantum geometric degrees of freedom, including possible modifications to the foundational structure of quantum theory.
The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.