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Physical Review Letters

Geometry of quantum evolution

J. Anandan, Yakir Aharonov

Published Oct 1, 1990• 794 citations
Publisher

Abstract

For an arbitrary quantum evolution, it is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space. It is the distance along this curve measured by the Fubini-Study metric. This gives a new time-energy uncertainty principle. New geometric meanings to time as measured by a clock and the transition probability during a quantum measurement are also obtained.

DOI
10.1103/physrevlett.65.1697

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For an arbitrary quantum evolution, it is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space. It is the distance along this curve measured by the Fubini-Study metric. This gives a new time-energy uncertainty principle. New geometric meanings to time as measured by a clock and the transition probability during a quantum measurement are also obtained.
We show that the quadratic short time behavior of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by nonlinear and nonunitary evolution equations, within the collapse mechanism of the wave function during measurement process.
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.
In this paper we establish important relations between Hamiltonian dynamics and Riemannian structures on phase spaces for unitarily evolving finite level quantum systems in mixed states. We show that the energy dispersion (i.e. 1/ℏ1/\hbar1/ℏ times the path integral of the energy uncertainty) of a unitary evolution is bounded from below by the length of the evolution curve. Also, we show that for each curve of mixed states there is a Hamiltonian for which the curve is a solution to the corresponding von Neumann equation, and the energy dispersion equals the curve's length. This allows us to express the distance between two mixed states in terms of a measurable quantity, and derive a time-energy uncertainty relation for mixed states. In a final section we compare our results with an energy dispersion estimate by Uhlmann.
We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.
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.