Modified Hermitian Tensors Generalise Connections in Quantum State Geometry

The geometry of quantum states governs how systems evolve and respond to change, but extending established geometrical frameworks to encompass more complex quantum behaviours presents a significant challenge. Kunal Pal from the Asia Pacific Center for Theoretical Physics, along with collaborators, investigates how the fundamental geometry of quantum states changes when systems move beyond standard Hermitian descriptions, exploring the implications for non-Hermitian quantum mechanics. This research demonstrates that a generalised geometrical structure can be consistently defined for these more complex systems, revealing a duality between different geometrical connections analogous to that found in classical probability theory. By classifying the possible geometrical tensors arising in non-Hermitian dynamics, and identifying the associated Berry curvature, the team provides a new framework for understanding and optimising the behaviour of these systems, potentially impacting areas such as quantum technologies and materials science.

Structures commonly used in classical geometry 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. This research explores how modifying 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. The work demonstrates that it is indeed possible to construct a family of connections that are dual to each other.

Information Geometry, Quantum Systems, and Networks

This extensive list of references details research related to quantum mechanics, information geometry, and related fields including condensed matter physics and neural networks, highlighting several key themes and current research directions. The compilation centers on information geometry, with references to foundational works by Amari and Molitor, indicating a focus on applying differential geometry to study probability distributions and information processing. Numerous references point to quantum mechanics and quantum information theory, with a strong emphasis on non-Hermitian systems, a rapidly growing area of physics. A significant portion of the references deal specifically with non-Hermitian quantum mechanics, PT-symmetry, and related phenomena, particularly relevant to condensed matter physics and quantum optics.

Research in condensed matter physics features prominently, with references to topological insulators, exciton-polaritons, and various materials, suggesting investigations into non-Hermitian effects within these systems. The list also includes references to topological physics, indicating an interest in topological phases of matter, and to Amari and others, suggesting the application of information geometry and related concepts to neural networks and machine learning, bridging the gap between quantum mechanics and machine learning. The focus on non-Hermitian systems often implies a study of open quantum systems, those interacting with their environment, and PT-symmetry is a recurring theme within non-Hermitian physics. This compilation reflects highly interdisciplinary research, drawing from mathematics, physics, and computer science, and points to current research trends in physics and related fields, with a strong emphasis on non-Hermitian physics, topological materials, and quantum machine learning. It leans heavily towards theoretical research, with a strong emphasis on mathematical frameworks and concepts, and is remarkably comprehensive, covering a wide range of relevant topics, including many recent publications, indicating a snapshot of very recent research activity. In summary, this bibliography points to research at the forefront of theoretical physics, information geometry, and their applications to emerging technologies like quantum computing and machine learning, demonstrating the growing interconnectedness of these fields.

Quantum Geometry Beyond Hermitian Inner Products

Researchers have developed a new framework for understanding the geometry of quantum states, extending concepts from classical information geometry to the quantum realm. This work addresses a key challenge: the standard rules of classical geometry do not directly translate to quantum systems due to the unique mathematical structure governing quantum states. The team discovered that by generalizing the way inner products are defined, they could create a consistent geometric framework for quantum systems, even those evolving under non-Hermitian dynamics. The core of this advancement lies in recognizing that the usual Hermitian inner product, which dictates how quantum states are compared, isn’t the only possibility.

By introducing a more general tensor structure, researchers were able to define a family of connections that exhibit a duality mirroring classical information geometry, relating different geometric perspectives on quantum states in a way analogous to well-established classical principles. Specifically, the team identified four distinct types of tensor structures that can arise when a quantum system is governed by a non-Hermitian Hamiltonian, offering a comprehensive classification of possible geometric configurations. A striking result is the ability to separate classical and quantum contributions to the geometry of quantum states. By utilizing the polar decomposition of the wavefunction, essentially breaking it down into amplitude and phase components, the researchers demonstrated that the geometry can be understood as a combination of familiar classical effects and uniquely quantum phenomena, providing a clearer understanding of how quantum effects influence the geometric properties of states. Furthermore, this framework allows for the definition of a complex-valued Berry curvature, and reveals its connection to a quantum natural gradient descent optimization problem, offering potential applications in quantum control and algorithm design. The team’s approach not only extends the reach of information geometry to quantum systems but also provides a powerful new lens for exploring the fundamental geometric properties of quantum states and their dynamics.

Gauge Invariant Quantum Geometries Demonstrated

This work explores generalisations of information geometry to quantum systems, specifically investigating how modifications to the underlying tensor structure affect the geometry of pure states. Researchers demonstrate the construction of a family of connections that are dual to each other, even when the system is governed by a non-Hermitian Hamiltonian, and identify the corresponding complex-valued metric and Berry curvature. These connections are linked to classical probability distributions modified by the presence of a non-trivial phase. Importantly, the initial connections developed were not gauge-invariant, meaning their form changed with certain transformations.

The team addressed this by constructing gauge-invariant combinations of connections, which preserve physical meaning and allow for a meaningful comparison to classical information geometry. These gauge-invariant connections satisfy a relationship reminiscent of the classical ±α duality, though proper symmetrisation is required due to the non-symmetric nature of the underlying tensor. The authors acknowledge that the quantities involved are not fully symmetric, and further work is needed to fully explore the implications of this asymmetry. They also note that the connections developed are formal, and future research could investigate their application to specific physical systems and optimisation problems, building on the demonstrated generalisation of natural gradient descent to the non-Hermitian case.