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arXiv:1302.1844

Geometry of quantum dynamics and a time-energy uncertainty relation for mixed states

Ole Andersson, Hoshang Heydari

Published Feb 7, 2013• 3 citations
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Abstract

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. $1/\hbar$ 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.

DOI
10.48550/arxiv.1302.1844

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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.
Geometric effects make evolution time vary for different evolution curves that connect the same two quantum states. Thus, it is important to be able to control along which path a quantum state evolve to achieve maximal speed in quantum calculations. In this paper we establish fundamental relations between Hamiltonian dynamics and Riemannian structures on the phase spaces of unitarily evolving finite-level quantum systems. In particular, we show that the Riemannian distance between two density operators equals the infimum of the energy dispersions of all possible evolution curves connecting the two density operators. This means, essentially, that the evolution time is a controllable quantity. The paper also contains two applied sections. First, we give a geometric derivation of the Mandelstam-Tamm estimate for the evolution time between two distinguishable mixed states. Secondly, we show how to equip the Hamiltonians acting on systems whose states are represented by invertible density operators with control parameters, and we formulate conditions for these that, when met, makes the Hamiltonians transport density operators along geodesics.
The concept of quantum acceleration limit has been recently introduced for any unitary time evolution of quantum systems under arbitrary nonstationary Hamiltonians. While Alsing and Cafaro used the Robertson uncertainty relation in their derivation, employed the Robertson–Schrödinger uncertainty relation to find the upper bound on the temporal rate of change of the speed of quantum evolutions. In this paper, we provide a comparative analysis of these two alternative derivations for quantum systems specified by an arbitrary finite-dimensional projective Hilbert space. Furthermore, focusing on a geometric description of the quantum evolution of two-level quantum systems on a Bloch sphere under general time-dependent Hamiltonians, we find the most general conditions needed to attain the maximal upper bounds on the acceleration of the quantum evolution. In particular, these conditions are expressed explicitly in terms of two three-dimensional real vectors, the Bloch vector that corresponds to the evolving quantum state and the magnetic field vector that specifies the Hermitian Hamiltonian of the system. For pedagogical reasons, we illustrate our general findings for two-level quantum systems in explicit physical examples characterized by specific time-varying magnetic field configurations. Finally, we briefly comment on the extension of our considerations to higher-dimensional physical systems in both pure and mixed quantum states.
The Hamiltonian and evolution time of a quantum unitary system completely determine a unitary operator, which turns an initial state ${\ensuremath{\rho}}_{0}$ into its time-evolved state ${\ensuremath{\rho}}_{t}$ and may also turn another state ${\ensuremath{\sigma}}_{0}$ into its time-evolved state ${\ensuremath{\sigma}}_{t}$. Although the two pairs of states $({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{t})$ and $({\ensuremath{\sigma}}_{0},{\ensuremath{\sigma}}_{t})$ are governed by the same Hamiltonian and experience equal evolution periods, their state distances are not necessarily equal to each other. Specifically, the evolution operator contains information about the evolution other than that reflected by two specific states connected through it. As a matter of fact, two pairs of states with different state distances may have equal path distances. Here we propose a geometric measure for the path distance contained in a quantum unitary evolution, which depends only on the unitary operator itself and is independent of any pair of states connected through it. This path distance meets the basic requirements for a good measure of distance and is valid during the whole evolution process, no matter how long the evolution time is. This path distance is bounded by the path length of the evolution from above and by the Bures angle between the two states connected through it from below. It has potential applications in quantum information processing, such as providing another way to study the optimal control between two quantum states. We finally present the form of Hamiltonian to realize a unitary operator on demand.
In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schrödinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrödinger, and vice versa.
In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schr\"{o}dinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schr\"{o}dinger, and vice versa.