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Physical Review A

Quantized geometry associated with uncertainty and correlation

Sumiyoshi Abe

Published Dec 1, 1993• 48 citations
Publisher

Abstract

The geometric structure of the law of quantum-state evolution is studied. The Fubini-Study metric induced on the quantum evolution submanifold of the projective Hilbert space is shown to be completely expressed by the uncertainties and correlations of various generators of evolutions. The Riemannian connection is expressed as a quantum-mechanical expectation value of a certain Hermitian operator. It is discussed that the metric carries some of quantum numbers contained in a given reference state, in general, and consequently the geometry is inherently quantized. These results are demonstrated by the simple examples of the squeezed coherent state, displaced number state, squeezed number state, and generalized coherent spin state.

DOI
10.1103/physreva.48.4102

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The geometric structure of the law of quantum-state evolution is studied. The Fubini-Study metric induced on the quantum evolution submanifold of the projective Hilbert space is shown to be completely expressed by the uncertainties and correlations of various generators of evolutions. The Riemannian connection is expressed as a quantum-mechanical expectation value of a certain Hermitian operator. It is discussed that the metric carries some of quantum numbers contained in a given reference state, in general, and consequently the geometry is inherently quantized. These results are demonstrated by the simple examples of the squeezed coherent state, displaced number state, squeezed number state, and generalized coherent spin state.
The state space of a quantum mechanical system is a complex projective space, the space of rays in the associated Hilbert space. The state space comes equipped with a natural Riemannian metric (the Fubini-Study metric) and a compatible symplectic structure. The operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry. It is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analysis of quantum dynamics, particularly in connection with state vector reduction. A promising approach to understanding reduction, studied recently by a number of authors, involves the use of nonlinear stochastic dynamics to modify the ordinary linear Schrodinger evolution. Here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse. In this picture, the conventional Schrodinger evolution, which corresponds to the unitary flow associated with a Killing vector of the Fubini-Study metric, is replaced by a more general stochastic flow on the state manifold. In the simplest example of such a flow, the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the Hamiltonian. The conservation of energy is represented by the requirement that the actual process followed by the expectation of the Hamiltonian, as the state evolves, should be a martingale. This requirement implies the existence of a nonlinear term in the drift vector of the state process, which is always oriented opposite to the direction of increasing energy uncertainty. As a consequence, the state vector necessarily collapses to an energy eigenstate, and a martingale argument can be used to show that the probability of collapse to a given eigenstate, from any particular initial state, is, in fact, given by precisely the usual quantum mechanical probability.
This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.
The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the SU(1,1)\text{SU}(1,1)SU(1,1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the SU(1,1)\text{SU}(1,1)SU(1,1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
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.