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arXiv:1904.01559 · International Journal of Quantum Information

The quantum geometric tensor from generating functions

Javier Alvarez-Jimenez, J. David Vergara

Published Mar 1, 2019• 6 citations
arXivHTMLPDF

Abstract

We introduce a new method to compute the Quantum Geometric Tensor, this procedure allows us to compute the Quantum Information Metric and the Berry curvature perturbatively for a theory with an arbitrary interaction Hamiltonian. The calculation uses the generating function method, and it is illustrated with the harmonic oscillator with a linear and a quartic perturbation.

DOI
10.1142/s0219749919500175

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We introduce a new method to compute the Quantum Geometric Tensor, this procedure allows us to compute the Quantum Information Metric and the Berry curvature perturbatively for a theory with an arbitrary interaction Hamiltonian. The calculation uses the generating function method, and it is illustrated with the harmonic oscillator with a linear and a quartic perturbation.
Abstract The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek–Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.
The Uhlmann relative amplitude between two density matrices is shown to be a generating function, through which the quantum Fisher tensor that contains both the quantum Fisher information matrix and the mean Uhlmann curvature can be obtained via differentiation over system parameters. In the pure state limit, our generating function recovers that of the quantum geometric tensor proposed by Hetényi and Lévay, and also clarifies the fidelity and phase between two quantum states as the generating functions of the quantum metric and Berry curvature, respectively. A generic expression for the quantum Fisher tensor in terms of the Bloch representation of density matrices is derived, which facilitates the calculation of the tensor, mean Uhlmann curvature, and geometric properties derived from the quantum Fisher information matrix. Canonical ensembles of spins are adopted to demonstrate our formalism, which reveals a constant Ricci scalar, a vacuum Einstein equation, and a cosmological constant on the 3D Euclidean manifold of the magnetic field
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.
We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.
Abstract The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information.
We generalize the Quantum Geometric Tensor by replacing a Hamiltonian with a modular Hamiltonian. The symmetric part of the Quantum Geometric Tensor provides a Fubini-Study metric, and its anti-symmetric sector gives a Berry curvature. Our generalization dubbed Quantum Modular Geometric Tensor gives the metric and curvature of a Kinematic Space. We also use the result of the identity Virasoro block to relate the connected correlator of two Wilson lines to the two-point function of a modular Hamiltonian. This relation realizes a novel holographic entanglement formula for two intervals of a general separation. This result also provides a new interpretation to the connected correlators as the mutual information. Our study remains valid at the quantum level. Hence it provides an opportunity to explore the Kinematic Space in a generic case without conformal symmetry.