Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.
Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.
Abstract One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher–Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states, the Fubini–Study tensor, which is preserved under unitary evolution. In this work, we explore how the modification of the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini–Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.
We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e., the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kähler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher-Rao metric tensor is recovered in the Abelian case, that the Fubini-Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures-Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini-Study and the Bures-Helstrom metric tensor.
A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point ω=0 and g/Γ=1/2; the physical manifold is non-Kähler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring Kähler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.