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 $\hbar/2$ played by $k_BT/\Omega$; where $T$ and $\Omega$ 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. $\dot S\geq0$) 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.
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 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.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
Concepts from information theory are used to investigate the relation between measurement and irreversibility in finite, closed classical systems having known Hamiltonians. The effects of measurement upon the predictions of the observer are found for the cases of a single measurement, two simultaneous measurements, and two measurements at different times. The physical basis of coarse graining and the significance of the increase in the coarse-grained entropy are discussed. An evolution equation is obtained describing the probability distribution of the measured quantities, and it is shown that this equation is non-Markoffian. Despite this non-Markoffian behavior, the distribution may behave irreversibly and may approach equilibrium. Thus, irreversible behavior is obtained without continual coarse graining. The condition for irreversible behavior is given for the case of a single measurement. The thermodynamic entropy is defined for general nonequilibrium situations.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A = M + B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S + M + B . The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix 𝒟 ^ ( t f ) of S + A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from 𝒟 ^ ( t f ), then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value A i of the macroscopic M. This is compatible with the principles of quantum mechanics. Born’s rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications A i of M in terms of the initial state of S. Von Neumann’s reduction amounts to updating of information due to selection of A i . We advocate the terms q -probabilities and q -correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.
While some of the deepest results in nature are those that give explicit bounds between important physical quantities, some of the most intriguing and celebrated of such bounds come from fields where there is still a great deal of disagreement and confusion regarding even the most fundamental aspects of the theories. For example, in quantum mechanics, there is still no complete consensus as to whether the limitations associated with Heisenberg's Uncertainty Principle derive from an inherent randomness in physics, or rather from limitations in the measurement process itself, resulting from phenomena like back action. Likewise, the second law of thermodynamics makes a statement regarding the increase in entropy of closed systems, yet the theory itself has neither a universally-accepted definition of equilibrium, nor an adequate explanation of how a system with underlying microscopically Hamiltonian dynamics (reversible) settles into a fixed distribution. Motivated by these physical theories, and perhaps their inconsistencies, in this thesis we use dynamical systems theory to investigate how the very simplest of systems, even with no physical constraints, are characterized by bounds that give limits to the ability to make measurements on them. Using an existing interpretation, we start by examining how dissipative systems can be viewed as high-dimensional lossless systems, and how taking this view necessarily implies the existence of a noise process that results from the uncertainty in the initial system state. This fluctuation-dissipation result plays a central role in a measurement model that we examine, in particular describing how noise is inevitably injected into a system during a measurement, noise that can be viewed as originating either from the randomness of the many degrees of freedom of the measurement device, or of the environment. This noise constitutes one component of measurement back action, and ultimately imposes limits on measurement uncertainty. Depending on the assumptions we make about active devices, and their limitations, this back action can be offset to varying degrees via control. It turns out that using active devices to reduce measurement back action leads to estimation problems that have non-zero uncertainty lower bounds, the most interesting of which arise when the observed system is lossless. One such lower bound, a main contribution of this work, can be viewed as a classical version of a Heisenberg uncertainty relation between the system's position and momentum. We finally also revisit the murky question of how macroscopic dissipation appears from lossless dynamics, and propose alternative approaches for framing the question using existing systematic methods of model reduction.