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Physica D Nonlinear Phenomena

Energy cost of information transmission (along the path to understanding)

Lev B. Levitin

Published Sep 1, 1998• 18 citations
Publisher

Abstract

It is shown that the minimum energy required to transmit a natural unit of information (nat) over a noisy channel is k T, where T is the effective noise temperature which depends on the interaction between the information career and the environment. The result follows from the entropy defect principle and general laws of quantum dynamics without any specific assumptions. © 1998 Published by Elsevier Science B.V. All rights reserved. 1.

DOI
10.1016/s0167-2789(98)00051-7

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It is shown that the minimum energy required to transmit a natural unit of information (nat) over a noisy channel is k T, where T is the effective noise temperature which depends on the interaction between the information career and the environment. The result follows from the entropy defect principle and general laws of quantum dynamics without any specific assumptions. © 1998 Published by Elsevier Science B.V. All rights reserved. 1.
Pinning down a precise understanding of information flow within physical interactions remains a central concern to fields such as stochastic thermodynamics and quantum information science. In both spheres a careful accounting of bits (or qubits) enables a deeper understanding of the physical nature of information. In this work we propose a measure of information flow as a saddle-point solution of the mutual information. This approach places a lower bound on the channel capacity between a particle and an interacting environment. The measure is given by P_{0}/2E_{0} in nats/sec, with P_{0} the average power flux between the particle and its environment, and E_{0} the initial average energy of the particle, all computed in a frame where the particle has zero average momentum. We use a communication theory lens to suggest an associated channel analogy, in which this bound is interpreted as a signal-to-noise ratio. We find that this measure can also quantify early-time information flow for a particle interacting with a thermal bath.
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.
Energy transfer and information transmission are two fundamental aspects of nature. They are seemingly unrelated, while recent findings suggest that a deep connection between them is to be discovered. This amounts to asking: Can we phrase the processes of transmitting classical bits equivalently as specific energy-transmitting tasks, thereby uncovering foundational links between them? We answer this question positively by showing that, for a broad class of classical communication tasks, a quantum dynamics' ability to transmit n bits of classical information is equivalent to its ability to transmit n units of energy in a thermodynamic task. This finding not only provides an analytical correspondence between information transmission and energy extraction tasks, but also quantifies classical communication by thermodynamics. Furthermore, our findings uncover the dynamical version of Landauer's principle, showing the strong link between transmitting information and energy. In the asymptotic regime, our results further provide thermodynamic meanings for the well-known Holevo-Schumacher-Westmoreland theorem in quantum communication theory.
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.
In this thesis we aim to analyze and quantify the energetic and information contents that can be extracted from a dynamical system subject to the external environment. The latter is usually assumed to be deleterious for the feasibility of specific control tasks, since it can be responsible for uncontrolled time-dependent changes of the system. However, if the effects of the random interaction with a noisy environment are properly modeled by the introduction of a given stochasticity within the dynamics of the system, then even noise contributions might be seen as control knobs. As a matter of fact, even a partial knowledge of the environment can allow to set the system in a dynamical condition in which the response is optimized by the presence of noise sources. In particular, we have investigated what kind of measurement devices can work better in noisy dynamical regimes and studied how to maximize the resultant information via the adoption of estimation algorithms. Moreover, we have shown the optimal interplay between quantum dynamics, environmental noise and complex network topology in maximizing the energy transport efficiency. Then, foundational scientific aspects, such as the occurrence of an ergodic property for the system-environment interaction modes of a randomly perturbed quantum system or the characterization of the stochastic quantum Zeno phenomena, have been analyzed by using the predictions of the large deviation theory. Finally, the energy cost in maintaining the system in the non-equilibrium regime due to the presence of the environment is evaluated by reconstructing the corresponding thermodynamics entropy production. In conclusion, the present thesis can constitute the basis for an effective resource theory of noise, which is given by properly engineering the interaction between a dynamical system and its external environment.