Building upon a recent analysis of the measurement process in Hamiltonian mechanics, this article investigates the Bayesian epistemology of classical physics -- the landscape of accessible probability distributions over phase space. I prove a thermodynamic limitation on the information that can be obtained about a classical system by means of observations: A direct analogue of the Robertson-Schrödinger quantum uncertainty relation controls the acquisition of information at the classical microscale. Central to this theorem is the notion of the "quality" of a measuring probe; a temperature-dependent strictly positive quantity that serves as a figure of merit of the probe, and that plays the role of $1/\hbar$ in the classical uncertainty relation. This study sets the stage for a new area of research into resource theories of classical measurement, in which high-quality measurements and high-information states of knowledge are the limited resources.
Building upon a recent analysis of the measurement process in Hamiltonian mechanics, this article investigates the Bayesian epistemology of classical physics -- the landscape of accessible probability distributions over phase space. I prove a thermodynamic limitation on the information that can be obtained about a classical system by means of observations: A direct analogue of the Robertson-Schrödinger quantum uncertainty relation controls the acquisition of information at the classical microscale. Central to this theorem is the notion of the "quality" of a measuring probe; a temperature-dependent strictly positive quantity that serves as a figure of merit of the probe, and that plays the role of 1/ℏ in the classical uncertainty relation. This study sets the stage for a new area of research into resource theories of classical measurement, in which high-quality measurements and high-information states of knowledge are the limited resources.
Quantum theory famously entails the existence of incompatible measurements, pairs of system observables which cannot be simultaneously measured to arbitrary precision. Incompatibility is widely regarded to be a uniquely quantum phenomenon, linked to failure to commute of quantum operators. Even in the face of deep parallels between quantum commutators and classical Poisson brackets, no connection has been established between the Poisson algebra and any intrinsic limitations to classical measurement. Here I examine measurement in classical Hamiltonian physics as a process involving the joint evolution of an object-system and a finite-temperature measuring apparatus. Instead of the ideal measurement capable of extracting information without disturbing the system, I find a Heisenberg-like precision-disturbance relation: Measuring an observable leaves all Poisson-commuting observables undisturbed but inevitably disturbs all non-Poisson-commuting observables. In this classical uncertainty relation the role of h-bar is played by an apparatus-specific quantity, q-bar. While this is not a universal constant, the analysis suggests that q-bar takes a finite positive value for any apparatus that can be built. (Specifically: q-bar vanishes in the model only in the unreachable limit of zero absolute temperature.) I show that a classical version of Ozawa's model of quantum measurement [Ozawa, Phys. Rev. Lett. 60, 385 (1988)0031-900710.1103/PhysRevLett.60.385], originally proposed as a means to violate Heisenberg's relation, does not violate the classical relation. If this result were to generalize to all models of measurement, then incompatibility would prove to be a feature not only of quantum, but of classical physics too. Put differently: The approach presented here points the way to studying the (Bayesian) epistemology of classical physics, which was until now assumed to be trivial. It now seems possible that it is nontrivial and bears a resemblance to the quantum formalism. The present findings may be of interest to researchers working on foundations of quantum mechanics, particularly for ψ-epistemic interpretations. More practically, there may be applications in the fields of precision measurement, nanoengineering, and molecular machines.
Measurement in classical physics is examined here as a process involving the joint evolution of object-system and measuring apparatus. For this, a model of measurement is proposed which lends itself to theoretical analysis using Hamiltonian mechanics and Bayesian probability. At odds with a widely-held intuition, it is found that the ideal measurement capable of extracting finite information without disturbing the system is ruled out (by the third law of thermodynamics). And in its place a Heisenberg-like precision-disturbance relation is found, with the role of ℏ/2 played by kBT/Ω; where T and Ω are a certain temperature and frequency characterizing the ready-state of the apparatus. The proposed model is argued to be maximally efficient, in that it saturates this Heisenberg-like inequality, while various modifications of the model fail to saturate it. The process of continuous measurement is then examined; yielding a novel pair of Liouville-like master equations -- according to whether the measurement record is read or discarded -- describing the dynamics of (a rational agent's knowledge of) a system under continuous measurement. The master equation corresponding to discarded record doubles as a description of an open thermodynamic system. The fine-grained Shannon entropy is found to be a Lyapunov function (i.e. S˙≥0) of the dynamics when the record is discarded, providing a novel H-theorem suitable for studying the second law and non-equilibrium statistical physics. These findings may also be of interest to those working on the foundations of quantum mechanics, in particular along the lines of attempting to identify and unmix a possible epistemic component of quantum theory from its ontic content. More practically, these results may find applications in the fields of precision measurement, nanoengineering and molecular machines.
Quantum information science is at the interface of physics and information theory, with applications including quantum computing and communication. Within this science, a resource theory is a detailed study of a particular class of physical processes, under which the possible state transitions are restricted. Given any initial physical state, the possible final states are also restricted. Initial states with a wealth of possible final states can then be considered more valuable than less versatile initial states, leading to a notion of "resourcefulness" of states. The broad goals of a resource theory include (1) to identify, characterize, and quantify resources; (2) to find an efficient way to determine whether an arbitrary instance of resource conversion is possible; and (3) to relate practical applications to resource conversion. In this thesis, we will take the resource-theoretic approach to understanding thermodynamics. The mathematical methods used in this work will enable us to also develop resource theories for measurement uncertainty and reference frames; in fact, the resource-theoretic treatment of thermodynamics is a combination of these two. Our approach is based on a model called thermal operations, applicable on individual quantum-mechanical systems out of thermal equilibrium. We study this model in detail, resulting in a better understanding of the resource of thermal inequilibrium. In a simplified version of the model where quantum coherence is neglected, we find that the conditions for state transitions are already more complex than the classical second law of thermodynamics. We extend the model to quantum systems controlled by classical circuitry. We also extend it to incorporate quantum coherence in the low-temperature limit. Measurement uncertainty is an important element in quantum mechanics. But its current understanding is through particular measures of uncertainty, such as the variance. We apply the resource theory approach to develop a measure-independent formalism for uncertainty, even in the case where the measurement is on part of a composite system with quantum correlations. Under physical processes restricted by some symmetry (e.g. rotational symmetry), asymmetric states act as resourceful reference frames. In this resource theory, we find the rate at which large quantities of resources can be interconverted through measurements.
The principle of superposition is one of the main building blocks of quantum physics and has tremendous consequences both for our fundamental understanding of nature and for technological applications. In particular, the existence of coherent superposition leads to the concept of unavoidable quantum uncertainty. The role played by this "coherent" uncertainty within thermodynamics, as well as its relationship to classical lack of knowledge, is the main subject of this thesis. In Part I we study thermodynamic limitations of processing quantum coherence within a resource-theoretic framework. Using the time-translation symmetry that arises from the first law of thermodynamics, we find constraints on possible manipulations of coherence and prove their irreversibility due to the second law. We also generalise to the quantum domain Szilard's concept of converting information into work. Namely, we show how, in the presence of a heat bath, coherence of a system can be exploited to perform mechanical work. Finally, we analyse the effect that coherence has on the structure of the thermodynamic arrow of time, i.e., on the set of states into which a given state can freely evolve under thermodynamic constraints. In Part II we focus on the interplay between quantum and classical uncertainty manifested in uncertainty relations. We show that separating the total uncertainty into these two distinct components leads to a new type of "fixed-entropy" uncertainty relation. We also analyse how classical ignorance affects the structure of states that minimise the unavoidable uncertainty arising from the noncommutativity of two observables. Finally, we study error-disturbance trade-off relations and, by proving that quantum uncertainty can be simultaneously maximised for any two observables, we clarify the unphysical nature of state-dependent relations.
Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory describable. The Cramer-Rao inequality serves that purpose. For a particle hidden from direct observation, the position uncertainty determined by the enclosing boundaries leads to thermodynamics in a straightforward extension of the presented formalism. The introduction of uncertainty in classical mechanics formulation enables the translation of mechanical laws into the wide ranging conceptual framework of information theory. The boundaries between classical mechanics, thermodynamics and quantum mechanics are defined in terms of informational changes associated with the system evolution. As a direct application of the proposed formulation upper bounds for the rate of information transfer are derived.
General characterization of physical measurements is discussed within the framework of a classical information theory. Uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems controlled under general kinds of equations of motion. We have treated only a mean-square error of the Gauss(-like) distributions in this report. Lessening of the Kennard-Robertson type uncertainties due to boundary conditions are quantitatively discussed using the information entropy.