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IRE Transactions on Information Theory

Minimum energy cost of an observation

F. Adler

Published Dec 1, 1955• 12 citations
Publisher

Abstract

The minimum energy expenditure required in performing basic observations and measurements is analyzed. The energy cost, in ergs per binary unit (bit) of information, is found for three fundamental cases using idealized experimental procedures: 1) the determination of the presence (or absence) of an input signal on an indicating instrument, 2) the measurement of a time interval and 3) the measurement of a distance. The variation of energy cost with the reliability and accuracy of the experiment is determined; it is found that with a suitable procedure the minimum value ofkT \ln2ergs per bit predicted by the Second Law (interpreted so as to include informational entropy) can be approached arbitrarily closely under conditions of small reliability and high accuracy. The present results are compared with those derivable from C. E. Shannon's equation for the capacity of a communication channel.

DOI
10.1109/tit.1955.1055139

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The minimum energy expenditure required in performing basic observations and measurements is analyzed. The energy cost, in ergs per binary unit (bit) of information, is found for three fundamental cases using idealized experimental procedures: 1) the determination of the presence (or absence) of an input signal on an indicating instrument, 2) the measurement of a time interval and 3) the measurement of a distance. The variation of energy cost with the reliability and accuracy of the experiment is determined; it is found that with a suitable procedure the minimum value ofkT \ln2ergs per bit predicted by the Second Law (interpreted so as to include informational entropy) can be approached arbitrarily closely under conditions of small reliability and high accuracy. The present results are compared with those derivable from C. E. Shannon's equation for the capacity of a communication channel.
The statistical definition of information is compared with Boltzmann's formula for entropy. The immediate result is that information I corresponds to a negative term in the total entropy S of a system. S=S0−I. A generalized second principle states that S must always increase. If an experiment yields an increase ΔI of the information concerning a physical system, it must be paid for by a larger increase ΔS0 in the entropy of the system and its surrounding laboratory. The efficiency ε of the experiment is defined as ε = ΔI/ΔS0≤1. Moreover, there is a lower limit k ln2 (k, Boltzmann's constant) for the ΔS0 required in an observation. Some specific examples are discussed: length or distance measurements, time measurements, observations under a microscope. In all cases it is found that higher accuracy always means lower efficiency. The information ΔI increases as the logarithm of the accuracy, while ΔS0 goes up faster than the accuracy itself. Exceptional circumstances arise when extremely small distances (of the order of nuclear dimensions) have to be measured, in which case the efficiency drops to exceedingly low values. This stupendous increase in the cost of observation is a new factor that should probably be included in the quantum theory.
In 1961, Landauer [R. Landauer, IBM J. Res. Develop. 5, 183 (1961)] pointed out that resetting a binary memory requires a minimum energy of kBTln(2){k}_{B}Tln(2)kB​Tln(2). However, once written, any memory is doomed to lose its content if no action is taken. To avoid memory losses, a refresh procedure is periodically performed. We present a theoretical model and an experiment on a microelectromechanical system to evaluate the minimum energy required to preserve one bit of information over time. Two main conclusions are drawn: (i) in principle, the energetic cost to preserve information for a fixed time duration with a given error probability can be arbitrarily reduced if the refresh procedure is performed often enough, and (ii) the Heisenberg uncertainty principle sets an upper bound on the memory lifetime.
In most communication scenarios, sending a symbol encoded in a quantum state requires spending resources such as energy, which can be quantified by a cost of communication. A standard approach in this context is to quantify the performance of communication protocol by classical capacity, quantifying the maximal amount of information that can be transmitted through a quantum channel per single use of the channel. However, different figures of merit are also possible and a particularly well-suited one is the classical capacity per unit cost, which quantifies the maximal amount of information that can be transmitted per unit cost. I generalize this concept to account for the quantum nature of the information carriers and communication channels and show that if there exists a state with cost equal to zero, e.g., a vacuum state, the capacity per unit cost can be expressed by a simple formula containing maximization of the relative entropy between two quantum states. This enables me to analyze the behavior of photon information efficiency for general communication tasks and show simple bounds on the capacity per unit cost in terms of quantities familiar from quantum estimation theory. I calculate also the capacity per unit cost for general Gaussian quantum channels.
Considering a general microscopic model for a quantum measuring apparatus comprising a quantum probe coupled to a thermal bath, we analyze the energetic resources necessary for the realization of a quantum measurement, which includes the creation of system-apparatus correlations, the irreversible transition to a statistical mixture of definite outcomes, and the apparatus resetting. Crucially, we do not resort to another quantum measurement to capture the emergence of objective measurement results, but rather exploit the properties of the thermal bath which redundantly records the measurement result in its degrees of freedom, naturally implementing the paradigm of quantum Darwinism. In practice, this model allows us to perform a quantitative thermodynamic analysis of the measurement process. From the expression of the second law, we show how the minimal required work depends on the energy variation of the system being measured plus information-theoretic quantities characterizing the performance of the measurement – efficiency and completeness. Additionally, we show that it is possible to perform a thermodynamically reversible measurement, thus reaching the minimal work expenditure, and provide the corresponding protocol. Finally, for finite-time measurement protocols, we illustrate the increasing work cost induced by rising entropy production inherent in finite-time thermodynamic processes. This highlights an emerging trade-off between velocity of the measurement and work cost, on top of a trade-off between efficiency of the measurement and work cost. We apply those findings to bring new insights in the thermodynamic balance of the measurement-powered quantum engines.
Abstract The consideration of measuring instruments as macroscopic bodies leads to neglect of the microscopic processes that occur during measurements. This disregard is not justified in general cases. As an example of measurements using microscopic instruments, the scattering of a photon by an electron with electron interference at two slits (Compton effect) was used. The amount of information that can be obtained in such a process is inversely proportional to the wavelength of the incident photon. At large photon wavelengths (soft measurements), the pure state of the electron can be disrupted by an arbitrarily small extent; accordingly, the amount of information extracted in such an experiment is also arbitrarily small. It is shown that the energy price for a bit obtained in such a measurement tends toward a constant value for increasing the photon wavelength. Microscopic instruments can be used in situations where energy costs for measurements are important.