Generalised state space geometry in Hermitian and non-Hermitian quantum systems

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 the unitary evolution. In this work, we explore 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 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.

• The paper presents a generalized metric framework integrating a modified Fubini-Study tensor to analyze state space geometry in both Hermitian and non-Hermitian quantum systems.
• It employs a biorthogonal formalism to distinguish complex-valued metrics and Berry curvature terms, elucidating the quantum phase contributions.
• The framework underpins quantum natural gradient descent, paving the way for advanced optimization in variational quantum eigensolvers and machine learning applications.

Generalised State Space Geometry in Hermitian and Non-Hermitian Quantum Systems

Introduction

In classical information geometry, probability distributions are equipped with the Fisher-Rao metric and dual $\alpha$-connections, defining distance and parallel transport on the distribution manifold. This geometric framework is less straightforward in quantum systems, where the natural metric arises from a Hermitian tensor on the complex projective space of pure states. This is known as the Fubini-Study (FS) metric, which is invariant under unitary evolution. This work investigates possible non-Hermitian generalizations of this framework by modifying the FS metric to explore the geometric properties of pure quantum states, potentially inducing dual connections aligned with classical probability functions but influenced by quantum phase.

State Space Geometry

The classical Fisher-Rao metric provides a measure of statistical distinguishability on the parameter manifold of classical probability distributions. Similarly, for quantum states, a Hermitian structure yields the quantum metric tensor (QMT), derived from the FS metric, that provides a notion of distance on the manifold of pure quantum states. The QMT involves the statistical averages and derivatives of the wave function in terms of both amplitude and phase, reflecting contributions from classical and quantum origins, respectively.

Generalizing beyond traditional Hermitian systems, this work introduces a modified FS tensor that can exist in systems governed by non-Hermitian Hamiltonians. The proposal includes a biorthogonal formalism identifying four potential tensor types: complex-valued metric and Berry curvature terms. Each of these components serves a specific role in characterizing the state space.

Non-Hermitian Quantum Systems

Non-Hermitian quantum mechanics, extending the conventional Hermitian paradigms, deals with Hamiltonians whose eigenvalues may not be real, though they can be essential for describing open quantum systems or systems with gain and loss. Within this framework, the FS tensor's complex nature necessitates considering both real and imaginary parts separately. In such systems, four different geometric tensors emerge, including two associated with Berry curvature-like phenomena that capture the system's topological and geometrical features.

Crucially, the work elucidates the quantum natural gradient descent in this extended geometric framework, showcasing how each part of the non-Hermitian FS tensor can guide optimization and measurement outcomes.

Implementation Considerations

The realization of these theoretical constructs into practical applications requires careful handling of computational resources and consideration of the system's dynamics, especially involving quantum phase effects or non-Hermitian elements. Efficient algorithms leveraging the geometric properties established in this framework could be developed for quantum applications like natural gradient-based optimization in machine learning or variational quantum eigensolvers for non-Hermitian systems.

Conclusion

This work suggests a generalized metric framework for understanding quantum state space geometry beyond the constraints of Hermiticity. By introducing a duality akin to classical $\alpha$-connections and exploring the role of a complex FS tensor in optimizing quantum systems, it broadens the application scope of quantum information geometry into new realms where non-Hermiticity plays a critical role. Further exploration will involve refining the metrics and connections obtained from this framework and examining their utility in identifying and leveraging topological features inherent in non-Hermitian quantum mechanics.