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arXiv:quant-ph/9909010 · Physical Review A

Back reaction of clocks and limitations on observability in closed systems

A. Casher, Benni Reznik

Published Sep 1, 2000• 10 citations
arXivHTMLPDF

Abstract

Measurements are ordinarily described with respect to absolute ``Newtonian'' time. In reality, however, the switching-on of the measuring device at the instance of the measurement requires a timing device. Hence the classical time t must be replaced by a suitable quantum time variable \ensuremath{\tau} of a physical clock. The main issue raised in this paper is that while doing so, we can no longer neglect the back reaction due to the measurement on the clock. For a von Neumann measurement model, this back reaction yields a bound on the accuracy of the measurement. When this bound is violated the result of a measurement is generally not an eigenvalue of the observable, and furthermore, the state of the system after the measurement is generally not a pure state. We argue that as a consequence, a subclass of observables in a closed system cannot be realized by a von Neumann--like measurement.

DOI
10.1103/physreva.62.042104

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Measurements are ordinarily described with respect to absolute ``Newtonian'' time. In reality, however, the switching-on of the measuring device at the instance of the measurement requires a timing device. Hence the classical time t must be replaced by a suitable quantum time variable \ensuremath{\tau} of a physical clock. The main issue raised in this paper is that while doing so, we can no longer neglect the back reaction due to the measurement on the clock. For a von Neumann measurement model, this back reaction yields a bound on the accuracy of the measurement. When this bound is violated the result of a measurement is generally not an eigenvalue of the observable, and furthermore, the state of the system after the measurement is generally not a pure state. We argue that as a consequence, a subclass of observables in a closed system cannot be realized by a von Neumann--like measurement.
The problem of time is a deep paradox in our physical description of the world. According to Aristotle's relational theory, time is a measure of change and does not exist on its own. In contrast, quantum mechanics, just like Newtonian mechanics, is equipped with a master clock that dictates the evolution of a system. This clock is infinitely precise and tacitly supplied free of charge from outside physics. Not only does this absolute time make it notoriously difficult to make a consistent theory of quantum gravity, it is also the underlying problem in establishing the second law. Indeed, contrary to our experience, the Wheeler-deWitt equation --a canonical quantization of general relativity-- predicts a static universe. Similarly, when simply concerned with the dynamics of a closed quantum system, there is no second law because the Von Neumann entropy is invariant under unitary transformations. Here we are mainly concerned with the latter problem and we show that it can be resolved by attributing a minimal amount of resources to the measurement of time. Although there is an absolute time in quantum mechanics, an observer can only establish a time by measuring a clock. For a local measurement, the minimal entropy production is equal to the number of ticks. This lower bound is attained by a black hole.
We show that, in the context of von Neumann measurement theory, the condition of strong measurement is the classical limit of the measuring device, and vice versa. This means that a strong measurement can be performed only by using a measuring device prepared in a classical limit state, and vice versa. Since it is the strong measurement that conforms with the postulate that the outcome of a single measurement is one of the eigenvalues of the corresponding operator, the aforementioned postulate is valid only in the classical limit for a von Neumann measurement.
Processes such as quantum computation, or the evolution of quantum cellular automata are typically described by a unitary operation implemented by an external observer. In particular, an interaction is generally turned on for a precise amount of time, using a classical clock. A fully quantum mechanical description of such a device would include a quantum description of the clock whose state is generally disturbed because of the back-reaction on it. Such a description is needed if we wish to consider finite sized autonomous quantum machines requiring no external control. The extent of the back-reaction has implications on how small the device can be, on the length of time the device can run, and is required if we want to understand what a fully quantum mechanical treatment of an observer would look like. Here, we consider the implementation of a unitary by a finite sized device which we call the "Quasi-Ideal clock", and show that the back-reaction on it can be made exponentially small in the device's dimension with only a linear increase in energy. As a result, an autonomous quantum machine need only be of modest size and or energy. We are also able to solve a long-standing open problem by using a finite sized quantum clock to approximate the continuous evolution of an Idealised clock. The result has implications on the equivalence of different paradigms of quantum thermodynamics, some which allow external control and some which only allow autonomous thermal machines.
The one-time and two-time measurements are studied by employing the idea of Aharonov et al. that the strength of a measurement depends on the state of the measuring device in the context of von Neumann measurement theory. A measurement is classified as either a strong or a weak measurement depending on whether the interaction between the system and the measuring device is infinitely large or negligibly small. A single strong measurement yields one of the eigenvalues of the operator while a single weak measurement, in addition to the eigenvalues, has infinite other possible outcomes. Nevertheless, the expectation values of the strong and the weak one-time measurements are the same. However, the expectation values of the two-time measurements depend on the strength of the first measurement. For the strong case the result is consistent with the conventional result without having to introduce the state collapsing. For the weak case the result is the real part of the expectation value of the time-ordered two-time operator. The fact that this expectation value cannot be factorized indicates that the entanglement between the system and the measuring device introduced by a weak measurement cannot be neglected.
Time remains one of the least well-understood concepts in physics, most notably in quantum mechanics.A central goal is to find the fundamental limits of measuring time.One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly.Here, we explore these questions by introducing a model of time measurements that is complete and autonomous.Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium-a prerequisite for any system to function as a clock-powered by minimal resources, namely, two thermal baths at different temperatures.Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock's performance in terms of its accuracy and resolution.Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power.More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.
An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'. Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time. We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation. The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad terms. Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed. Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock. An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis for testing such models.