We study a measurement framework motivated by considering macroscopic (i.e. large, active, and with finite temperature) measurement of microscopic (i.e. small and lossless) but classical dynamics. This unavoidably leads to “measurement back action” on the microscopic dynamics that nevertheless still allows for optimal filtering to minimize estimation error, but with tradeoffs between errors due to estimation and errors due to the back action. We focus on a simple case in which the deterministic effects of the measurement process are completely canceled by active control, and the remaining (coupled) stochastic back action and measurement noise is optimally filtered to minimize estimation error. This leads to a particularly interesting tradeoffs and limits on estimation and back action, analogous in many respects with the Heisenberg uncertainty principle but in an entirely classical framework.
We study a measurement framework motivated by considering macroscopic (i.e. large, active, and with finite temperature) measurement of microscopic (i.e. small and lossless) but classical dynamics. This unavoidably leads to “measurement back action” on the microscopic dynamics that nevertheless still allows for optimal filtering to minimize estimation error, but with tradeoffs between errors due to estimation and errors due to the back action. We focus on a simple case in which the deterministic effects of the measurement process are completely canceled by active control, and the remaining (coupled) stochastic back action and measurement noise is optimally filtered to minimize estimation error. This leads to a particularly interesting tradeoffs and limits on estimation and back action, analogous in many respects with the Heisenberg uncertainty principle but in an entirely classical framework.
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.
In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics, and use the results to derive limitations of classical measurements. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear system is dissipative if, and only if, it can be approximated by a linear lossless system over arbitrarily long time intervals. Hence lossless systems are in this sense dense in dissipative systems. A linear active system can be approximated by a nonlinear lossless system that is charged with initial energy. As a by-product, we obtain mechanisms explaining the Onsager relations from time-reversible lossless approximations, and the fluctuation-dissipation theorem from uncertainty in the initial state of the lossless system. The results are applied to measurement devices and are used to quantify limits on the so-called observer effect, also called back action, which is the impact the measurement device has on the observed system. In particular, it is shown that deterministic back action can be compensated by using active elements, whereas stochastic back action is unavoidable and depends on the temperature of the measurement device.
In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear macroscopic system is dissipative if and only if it can be approximated by a linear lossless microscopic system, over arbitrarily long time intervals. As a by-product, we obtain mechanisms explaining Johnson-Nyquist noise as initial uncertainty in the lossless state, as well as measurement back action and a trade-off between process and measurement noise.
In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear macroscopic system is dissipative if and only if it can be approximated by a linear lossless microscopic system, over arbitrarily long time intervals. As a by-product, we obtain mechanisms explaining Johnson-Nyquist noise as initial uncertainty in the lossless state as well as measurement back action and a trade off between process and measurement noise.
Description: This work investigates a fundamental but often overlooked limitation of physical measurement: the finite duration required to perform any measurement. We demonstrate that the impossibility of instantaneous measurement imposes intrinsic limits on the assignment of exact numerical values to physical quantities, even within purely classical systems. By developing a unified temporal framework, the paper shows that classical dynamical uncertainty arising from system evolution during the measurement interval shares a common origin with quantum measurement back-action. A generalized uncertainty relation is derived that incorporates both classical dynamical effects and quantum measurement noise, revealing a temperature-dependent crossover between classical and quantum regimes. The theory predicts a measurable classical–quantum transition in back-action noise and proposes a concrete experimental test using cavity optomechanical systems. Practical implications for precision metrology, quantum-enhanced sensing, and the operational interpretation of physical quantities are discussed. These results suggest that measurement precision is inherently temporal rather than instantaneous, providing new insight into the foundations of measurement in both classical and quantum physics. Keywords:measurement theory; quantum measurement; classical dynamics; uncertainty relations; back-action; optomechanics; metrology; temporal averaging
Measurement-induced back-action, a direct consequence of the Heisenberg uncertainty principle, is the defining feature of quantum measurements. We use quantum measurement theory to analyze the recent experiment of Safavi-Naeini et al. [Phys. Rev. Lett. 108, 033602 (2012)], and show that the results of this experiment not only characterize the zero-point fluctuation of a near-ground-state nanomechanical oscillator, but also demonstrate the existence of quantum back-action noise---through correlations that exist between sensing noise and back-action noise. These correlations arise from the quantum coherence between the mechanical oscillator and the measuring device, which build up during the measurement process, and are key to improving sensitivities beyond the standard quantum limit.