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Science

Minimal Energy Requirements in Communication

Rolf W. Landauer

Published Jun 28, 1996• 191 citations
Publisher

Abstract

The literature describing the energy needs for a communications channel has been dominated by analyses of linear electromagnetic transmission, often without awareness that this is a special case. This case leads to the conclusion that an amount of energy equal to kTln 2, where kT is the thermal noise per unit bandwidth, is needed to transmit a bit, and more if quantized channels are used with photon energies hnu > kT. Alternative communication methods are proposed to show that there is no unavoidable minimal energy requirement per transmitted bit. These methods are invoked as part of an analysis of ultimate limits and not as practical procedures.

DOI
10.1126/science.272.5270.1914

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The literature describing the energy needs for a communications channel has been dominated by analyses of linear electromagnetic transmission, often without awareness that this is a special case. This case leads to the conclusion that an amount of energy equal to kTln 2, where kT is the thermal noise per unit bandwidth, is needed to transmit a bit, and more if quantized channels are used with photon energies hnu > kT. Alternative communication methods are proposed to show that there is no unavoidable minimal energy requirement per transmitted bit. These methods are invoked as part of an analysis of ultimate limits and not as practical procedures.
It has been well known since the pioneering work of Claude Shannon in the 1940s that a message transmitted with optimal efficiency over a channel of limited bandwidth is indistinguishable from random noise to a receiver who is unfamiliar with the language in which the message is written. We derive some similar results about electromagnetic transmissions. In particular, we show that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given message is indistinguishable from blackbody radiation. The characteristic temperature of the radiation is set by the amount of energy used to make the transmission. If information is not encoded in the direction of the radiation, but only in its timing, energy, and polarization, then the most efficient format has the form of a one-dimensional blackbody spectrum.
An alternative physical way of communication, communication by the inherent background noise, is proposed which does not need net energy transfer in the information channel. The communicator devices do dissipate energy; however, they do not emit net energy into the channel, instead of that, they modulate the parameters of inherent spontaneous fluctuations in the channel. The method can use two different mechanisms, thermal noise (Johnson–Nyquist noise) for classical communication, and vacuum fluctuations/zero-point energy (quantum uncertainty noise) for quantum communication. The strongest advantage of the method is that it is apparently the most hidden (stealth) way of communication, because it is using the inherent background noise for communication. Therefore, it is extremely difficult or impossible to discover its presence. With proper wave-based arrangements and specific conditions, the sender and the receiver can easily detect eavesdropper activities, so that the eavesdropper is detected as soon as she extracts a single bit of information, thus the security of the method is comparable to the security of quantum communication/quantum key distribution schemes. Finally, concerning practical applications, environmental noise, out of the fundamental/inherent fluctuations, can also be used for this kind of communication provided that is sufficiently stationary.
The search for schemes that minimize the energy associated with the transmission of information is a longstanding fundamental issue in communication theory. In this paper we extend fundamental limits to the energy consumption per unit of information, as given by Shannon's theory, to the quantum domain. Unlike previous studies, we address a scenario where the signal may be manipulated in an arbitrary way while propagating from the transmitter to the receiver. This situation characterizes many realistic scenarios, such as multi-span fiber-optic communication systems. We obtain the ultimate quantum limit on the energy consumption in this scenario and propose a simple binary energy modulation scheme that approaches this limit within one order of magnitude for practically relevant values of spectral efficiency. Under the same conditions, the quantum energy consumption limit of the standard optically amplified coherent communication scheme is three orders of magnitude above the ultimate identified limit. Throughout the paper we consider transmission of classical information over a quantum channel.
Preventing signal detection in communication and active sensing requires careful control of transmission power. In fact, the square-root laws (SRL) for covert classical and quantum communication and sensing prescribe that the average output energy per channel use scales as 1/n1/\sqrt{n}1/n​ for nnn channel uses. \emph{Diffuse} and \emph{sparse} signaling achieve this. The former transmits signals whose energy decays as 1/n1/\sqrt{n}1/n​ over all nnn channel uses, which is convenient for mathematical analysis. The latter transmits constant-energy signals only approximately ∝n\propto\sqrt{n}∝n​ times out of nnn channel uses, remaining silent on the others. This offers significant practical advantages in compatibility with modern digital transmitters. Here, we study sparse signaling over the lossy thermal-noise bosonic channel, which is a quantum model of many practical channels (including optical, microwave, and radio-frequency). We characterize the input signal state that minimizes detectability. We find an unintuitive optimal quantum state structure: a mixture of just two consecutive photon-number states. In particular, in the low-brightness regime, the optimal signal state is a mixture of vacuum and a single photon. Since these states are generally suboptimal for both communication and active sensing, we explore the resulting trade-off and identify input-power thresholds for transitions between optimizing covertness and performance in communication and sensing tasks.
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.