FEATURES OF COMPLEXITY

RONNIE KOSLOFF

Institute of Chemistry and the Fritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem, Israel

CONTENTS

• I. Introduction
• II. The Emergence of Classical Dynamics from the Underlying Quantum Laws
  • A. Insight from Quantum Control Theory
• III. The Emergence of Thermodynamical Phenomena
  • A. The Quantum Otto Cycle
  • B. Quantum Dynamics of the Working Medium
  • C. Quantum Thermodynamics
  • D. The Quantum Tricycle
  • E. The Third Law of Thermodynamics
• IV. Perspective
• Acknowledgments
• References
• Discussion

I. INTRODUCTION

A complex dynamical system is associated with a description which requires a number of variables comparable to the number of particles. If we accept that quantum mechanics is the basic theory of matter, we are faced with the dilemma of the emergence of dynamical complexity. One of the main pillars of quantum mechanics is the superposition principle. As a result, the theory is completely linear. Dynamical complexity is typically associated with nonlinear phenomena. Complexity can be quantified as the ratio between the number of variables required to describe the dynamics of the system to the number of degrees of freedom. Chaotic dynamics is such an example where this ratio is close to one [1]. Classical mechanics is generically nonlinear and therefore chaotic dynamics emerges. Typical classical systems of even a few degrees of freedom can become extremely complex. The complexity can be associated with positive Kolmogorov entropy [2]. In contrast, quantum mechanics is regular. Strictly, closed quantum systems have zero Kolmogorov entropy [3]. How do these two fundamental theories which address dynamical phenomena have such striking differences? The issue of the emergence of classical mechanics from quantum theory is therefore a non-resolved issue despite many years of study (cf. Figure 1.1).

Figure 1.1 The relation between complexity, the number of particles and temperature in the physical world. Complexity is measured by the ratio of the number of variables required to describe the dynamics of a system compared to the number of degrees of freedom.

Thermodynamics is a rule-based theory with a very small number of variables. The theory of chaos has been invoked to explain the emergence of simplicity from the underlying complex classical dynamics. Chaotic dynamics leads to rapid loss of the ability to keep track of the systems trajectory. As a result, a coarse grain picture of self-averaging reduces the number of variables. Following this viewpoint, complexity is created in the singular transition between quantum and classical dynamics. When full chaos dominates, thermodynamics takes over.

Quantum thermodynamics is devoted to the study of thermodynamical processes within the context of quantum dynamics. This leads to an alternative direct route linking quantum mechanics and thermodynamics. This link avoids the indirect route to the theory through classical mechanics. The study is based on the thermodynamic tradition of learning by example. In this context, it is necessary to establish quantum analogues of heat engines. These studies unravel the intimate connection between the laws of thermodynamics and their quantum origin [4-27]. The key point is that thermodynamical phenomena can be identified at the level of an individual small quantum device [28].

II. THE EMERGENCE OF CLASSICAL DYNAMICS FROM THE UNDERLYING QUANTUM LAWS

A. Insight from Quantum Control Theory

Quantum Control focuses on guiding quantum systems from initial states to targets governed by time-dependent external fields [29, 30]. Two interlinked theoretical problems dominate quantum control: the first is the existence of a solution and the second is how to find the control field. Controllability addresses the issue of the conditions on the quantum system which enable control. The typical control targets are state-to-state transformations or optimising a pre-specified observable. A more demanding task is implementing a unitary transformation on a subgroup of states. Such an implementation is the prerequisite for quantum information processing. The unitary transformation connects the initial wavefunction which encodes the computation input to the final wave function which encodes the computation output. Finding a control filed for this task can be termed the quantum compiler problem. The existence of a solution for a unitary transformation is assured by the theorem of complete controllability [31-33]. In short, a system is completely controllable if the combined Hamiltonians of the control and system span a compact Lie algebra. Moreover complete controllability implies that all possible state-to-state transformations are guarantied.

Finding a control field that implements the task is a complex inversion problem. Given the target unitary transformation at final time T what is the control field that generates it?

where , and e(t) is the control field. The methods developed to solve the inversion problem could be classified as global, such as Optimal Control Theory (OCT) [34-36], or local, for example, Local Control [37-39]. OCT casts the inversion task into an optimisation problem which is subsequently solved by an iterative approach. The number of iterations required to converge to a high fidelity solution is a measure of the complexity of the inversion.

How difficult is it to solve for the quantum compiler for a specific unitary transformation? This task scales at least factorially with the size of the transformation.The rational is based on the simultaneous task of generating N - 1 state-to-state transformations which constitute the eigenfunctions of the target unitary transformation. To set the relative phase of these transformations, a field that drives a superposition state to the final target time has to be found. All these individual control fields have to be orthogonal to all the other transformations, thus the scaling becomes N! more difficult than finding the field that generates an individual state-to-state transformation [40]. This scaling fits the notion that the general quantum compiler computation problem has to be hard in the class of NP problems [41]. If it would be an easy task, a unitary transformation could solve in one step all algorithmic problems.

The typical control Hamiltonian can be divided into an uncontrolled part the drift Hamiltonian, and a control Hamiltonian composed from an operator sub-algebra:

where aj(t) is the control field for the operator and the set of operators form a closed small Lie sub-algebra. This model includes molecular systems controlled by a dipole coupling to the electromagnetic field. Complete controllability requires that the commutators of and span the complete algebra U(N) where N = n2 - 1 and n is the size of the Hilbert space of the system. If is part of the control algebra, the system is not completely controllable, that is, there are state-to-state transitions which cannot be accomplished. In the typical quantum control scenarios the size of the control sub-algebra is constant but the size of control space increases. For example, in coherent control of molecules by a light field, the three components of the dipole operator compose the control algebra. These operators are sufficient to completely control a vast number of degrees of freedom of the molecule.

Are systems still completely controllable in the more messy and complex real world? This task is associated with the control of an open quantum system where the controlled system is in contact with the environment. The theorems of controllability do not cover open quantum systems which remains an open problem. Coherent control which is based on interfering pathways is typically degraded by environmental noise or decoherence. Significant effort has been devoted to overcome this issue, mostly in the context of implementing gates for quantum computers. The remedy which is known as dynamical decoupling employs very fast control fields to reset the system on track [42-44].

We argue that there is a fundamental flaw in these remedies. Although the noise from the environment can be suppressed, the fast controls introduce a new source of noise originating from the controllers. The controller which generates the control field has to be fast in the timescale of the controlled system. This means that the noise introduced by the controller can be modeled as a delta correlated Gaussian noise. For the control algebra of Eq. (1.2) we obtain:

where ? is determined by the noise in the controls [45].

Equation (1.3) can have a different interpretation. It describes a system subject to simultaneous weak quantum measurement of the set of operators . Quantum measurement causes the collapse of the system to an eigenstate of the measured operator. A weak measurement is a small step in this direction. It achieves only a small amount of information on the system and induces only partial collapse. Equation (1.3) describes a continuous...