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Transactions of the Institute of Measurement and Control

Quantum mechanics and dynamic programming

H.H. Rosenbrock, Zhengtao Ding

Published Mar 1, 2008• 2 citations
Publisher

Abstract

This brief paper continues a development in earlier publications by Rosenbrock, in which results in quantum mechanics were obtained by adding a random disturbance to Hamilton's principle. The disturbance is complex, and the resulting variational problem is solved by dynamic programming, which computes the causal relations needed to accomplish a purpose. It is shown here that this procedure leads to many results that agree with those obtained in the standard theory of quantum mechanics, but also to some that are not readily obtained in that theory. For example, certain changes at a time t m to the end condition x f , which will be reached at t f >t m , can instantaneously change some aspects of the particle's behaviour at t m while it is still in transit. This accounts for Einstein's `spooky action at a distance'. An approach is also given to interference and entanglement, which is simpler than the standard account and may therefore be useful in quantum control and quantum computation.

DOI
10.1177/0142331207083484

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This brief paper continues a development in earlier publications by Rosenbrock, in which results in quantum mechanics were obtained by adding a random disturbance to Hamilton's principle. The disturbance is complex, and the resulting variational problem is solved by dynamic programming, which computes the causal relations needed to accomplish a purpose. It is shown here that this procedure leads to many results that agree with those obtained in the standard theory of quantum mechanics, but also to some that are not readily obtained in that theory. For example, certain changes at a time t m to the end condition x f , which will be reached at t f >t m , can instantaneously change some aspects of the particle's behaviour at t m while it is still in transit. This accounts for Einstein's `spooky action at a distance'. An approach is also given to interference and entanglement, which is simpler than the standard account and may therefore be useful in quantum control and quantum computation.
Abstract An earlier development of some results in quantum mechanics from a stochastic variational principle is extended in several directions. An outline is first given of the methods of control theory upon which the development is based, and earlier results are briefly described. Extensions are then given to relativistic systems, to Dirac’s equation, and to elementary quantum field theory. The aim thoughout is to show that results in the standard theory can be obtained in a uniform way from an extended form of Hamilton’s principle, which has the advantage of conciseness and a relatively close relationship to the classical theory. The wave function appears as a modified form of the optimal cost function, and the photon can be identified with a singularity in the electromagnetic field. Interference is explained by optimization of an expected value, the ensemble over which the expectation is taken being dependent upon the information available.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
All physical quantum systems are in contact with the external world which is inevitably a source of noise. It is therefore necessary to take into account this dissipation when designing controls that accomplish useful tasks in quantum information processing. This thesis is on that overlap of open quantum systems and control theory; it looks at what dynamics can happen with the use of external driving, and how this driving can be chosen to accomplish a desired goal. The first result looks at how a given quantum dissipator can be manipulated, using coherent controls, into replicating the action of a different type of noise. This results in no-go theorems for how noise can be transformed based on its isotropy, with applications in simulating open systems. Another way of doing simulations is to use only unitary dynamics over the system and a finite dimensional ancilla, and it is proved that there always exists a dilation Hamiltonian that replicates the noisy dynamics continuously in time. This also highlights the fact that adding controls on noise can result in a different evolution than if the controls were done on the under- lying system-environment level. A conjecture is introduced and studied which states that both approaches are equivalent in the special case of the controls commuting with the Lindbladian. A way around this difficulty is to use a quantum system to compute in situ which controls are best for achieving a desired task on itself. This problem is studied in the context of reaching entangling gates on a quantum simulator in order to upgrade it into a quantum computer. The experimental cost of doing so is found to be polynomial in the number of qubits in simulations. The same underlying principle is also used to find error correcting codes tailored to the dissipation in a system.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.