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Bidirectional Classical Stochastic Processes with Measurements and Feedback

G. E. Hahne

Published Jan 1, 2005• 1 citation
OpenAlex

Abstract

A measurement on a quantum system is said to cause the "collapse" of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there is feedback from the output of a measurement to the input, and is intended to give a model for quantum-mechanical processes that occur along a space-like reaction coordinate. The classical system can be thought of in physical terms as two counterflowing probability streams, which stochastically exchange probability currents in a way that the net probability current, and hence the overall probability, suitably interpreted, is conserved. The proposed formalism extends the . mathematics of those stochastic processes describable with linear, single-step, unidirectional transition probabilities, known as Markov chains and stochastic matrices. It is shown that a certain rearrangement and combination of the input and output of two stochastic matrices of the same order yields another matrix of the same type. Each measurement causes the partial collapse of the probability current distribution in the midst of such a process, giving rise to calculable, but non-Markov, values for the ensuing modification of the system's output probability distribution. The paper concludes with an analysis of a classical probabilistic version of the so-called grandfather paradox.

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A measurement on a quantum system is said to cause the "collapse" of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there is feedback from the output of a measurement to the input, and is intended to give a model for quantum-mechanical processes that occur along a space-like reaction coordinate. The classical system can be thought of in physical terms as two counterflowing probability streams, which stochastically exchange probability currents in a way that the net probability current, and hence the overall probability, suitably interpreted, is conserved. The proposed formalism extends the . mathematics of those stochastic processes describable with linear, single-step, unidirectional transition probabilities, known as Markov chains and stochastic matrices. It is shown that a certain rearrangement and combination of the input and output of two stochastic matrices of the same order yields another matrix of the same type. Each measurement causes the partial collapse of the probability current distribution in the midst of such a process, giving rise to calculable, but non-Markov, values for the ensuing modification of the system's output probability distribution. The paper concludes with an analysis of a classical probabilistic version of the so-called grandfather paradox.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.
We present topological characterization of classical stochastic processes described by discrete-time Markov chains on lattices. We point out that point-gap topology of stochastic matrices entails two distinct physical consequences that hinge on the choice of the reference point. The point-gap topology around a generic reference point is related to the direction of transport, and nontrivial topology around the origin of the complex spectrum of a stochastic matrix implies non-Markovianity caused by, e.g., feedback control. On the basis of this characterization, we identify the topological origin of directed transport in a classic experiment of Maxwell's demon [S. Toyabe et al., Nat. Phys. 6, 988 (2010)] and find the topological nature of feedback control beyond thermodynamic interpretation. We demonstrate that a topologically enforced non-Markovian classical stochastic process can be simulated by a Markovian quantum master equation, indicating a topological form of quantum advantage.
Abstract Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.
Quantum mechanics contains certain novel mathematical concepts. Among these are complex numbers, Hilbert spaces with their unitary and self-adjoint operators, states represented by complex vectors, superpositions of states, collapse of wave functions, Born's rule for probabilities and others. If we accept that quantum mechanics is probabilistic, then these concepts can be derived and they become secondary. In this work, we begin with what we call a conditional stochastic process, which is based on real numbers and probabilities. As we shall see, such processes are defined by three simple axioms. We then use conditional stochastic processes to derive quantum mechanics by employing a correspondence called a dictionary. We also show that the converse holds. That is, beginning with a quantum system, we employ the dictionary to derive a conditional stochastic process.
The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (c) 2001 American Institute of Physics.