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Physical Review A

Quantum limit for information transmission

Marcelo Schiffer

Published May 1, 1991• 11 citations
Publisher

Abstract

In this paper, we give two independent and rigorous derivations for the quantum bound on the information transmission rate proposed independently by Bekenstein [Phys. Rev. Lett. 46, 623 (1981)] and Bremermann .ul2 [Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, edited by L. M. LeCam and J. Neyman (University of California Press, Berkeley, 1967], preceded by a heuristic argument showing why such a bound must hold. In both approaches, information carriers are quanta for some field. The first method resembles the microcanonical approach to statistical mechanics, where the strategy of overestimating the real number of states by relaxing the indistinguishability of quanta was adopted. The second is based entirely upon maximum-entropy methods. Amazingly enough, the results obtained by these physically unrelated premises turn out to be identical, namely, that the single (noiseless) channel capacity is I${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{\mathrm{max}}$=E/2\ensuremath{\pi}\ensuremath{\Elzxh} bits ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$. It is further shown that, in a finite time \ensuremath{\tau}, no information can ever be conveyed unless the energy threshold 2\ensuremath{\pi}\ensuremath{\Elzxh}/\ensuremath{\tau} is reached, allowing the reinterpretation of the time-energy uncertainty in informational-theoretic language.

DOI
10.1103/physreva.43.5337

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Physical Review A•May 1, 1988•26 citations
In this paper, we give two independent and rigorous derivations for the quantum bound on the information transmission rate proposed independently by Bekenstein [Phys. Rev. Lett. 46, 623 (1981)] and Bremermann .ul2 [Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, edited by L. M. LeCam and J. Neyman (University of California Press, Berkeley, 1967], preceded by a heuristic argument showing why such a bound must hold. In both approaches, information carriers are quanta for some field. The first method resembles the microcanonical approach to statistical mechanics, where the strategy of overestimating the real number of states by relaxing the indistinguishability of quanta was adopted. The second is based entirely upon maximum-entropy methods. Amazingly enough, the results obtained by these physically unrelated premises turn out to be identical, namely, that the single (noiseless) channel capacity is I\ifmmode˙\else˙\fimax{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{\mathrm{max}}\ifmmode˙\else˙\fimax​=E/2\ensuremath{\pi}\ensuremath{\Elzxh} bits s\ensuremath−1{\mathrm{s}}^{\mathrm{\ensuremath{-}}1}s\ensuremath−1. It is further shown that, in a finite time \ensuremath{\tau}, no information can ever be conveyed unless the energy threshold 2\ensuremath{\pi}\ensuremath{\Elzxh}/\ensuremath{\tau} is reached, allowing the reinterpretation of the time-energy uncertainty in informational-theoretic language.
The capacity CCC of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power PPP. The limits arise ultimately from the Einstein relation that a field quantum at frequency fff has energy E=hfE=hfE=hf. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field---i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is ${C}_{\mathrm{WB}}=(\frac{\ensuremath{\pi}}{\mathrm{ln}2})\sqrt{\frac{2P}{3h}}$ bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity CWB{C}_{\mathrm{WB}}CWB​ are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.
Information must take up space, must weigh, and its flux must be limited. Quantum limits on communication and information storage leading to these conclusions are described here. Quantum channel capacity theory is reviewed for both steady state and burst communication. An analytic approximation is given for the maximum signal information possible with occupation number signal states as a function of mean signal energy. A theorem guaranteeing that these states are optimal for communication is proved. A heuristic "proof" of the linear bound on communication is given, followed by rigorous proofs for signals with specified mean energy, and for signals with given energy budget. And systems of many parallel quantum channels are shown to obey the linear bound for a natural channel architecture. The time-energy uncertainty principle is reformulated in information language by means of the linear bound. The quantum bound on information storage capacity of quantum mechanical and quantum field devices is reviewed. A simplified version of the analytic proof for the bound is given for the latter case. Solitons as information caches are discussed, as is information storage in one-dimensional systems. The influence of signal self-gravitation on communication is considered. Finally, it is shown that acceleration of a receiver acts to block information transfer.
Information must take up space, must weigh, and its flux must be limited. Quantum limits on communication and information storage leading to these conclusions are described here. Quantum channel capacity theory is reviewed for both steady state and burst communication. An analytic approximation is given for the maximum signal information possible with occupation number signal states as a function of mean signal energy. A theorem guaranteeing that these states are optimal for communication is proved. A heuristic "proof" of the linear bound on communication is given, followed by rigorous proofs for signals with specified mean energy, and for signals with given energy budget. And systems of many parallel quantum channels are shown to obey the linear bound for a natural channel architecture. The time-energy uncertainty principle is reformulated in information language by means of the linear bound. The quantum bound on information storage capacity of quantum mechanical and quantum field devices is reviewed. A simplified version of the analytic proof for the bound is given for the latter case. Solitons as information caches are discussed, as is information storage in one-dimensional systems. The influence of signal self-gravitation on communication is considered. Finally, it is shown that acceleration of a receiver acts to block information transfer.
The Bekenstein bound posits a maximum entropy for matter with finite energy confined to a spatial region. It is often interpreted as a fundamental limit on the information that can be stored by physical objects. In this work, we test this interpretation by asking whether the Bekenstein bound imposes constraints on a channel's communication capacity, a context in which information can be given a mathematically rigorous and operationally meaningful definition. We study specifically the Unruh channel that describes a stationary Alice exciting different species of free scalar fields to send information to an accelerating Bob, who is confined to a Rindler wedge and exposed to the noise of Unruh radiation. We show that the classical and quantum capacities of the Unruh channel obey the Bekenstein bound that pertains to the decoder Bob. In contrast, even at high temperatures, the Unruh channel can transmit a significant number of zero-bits , which are quantum communication resources that can be used for quantum identification and many other primitive protocols. Therefore, unlike classical bits and qubits, zero-bits and their associated information processing capability are generally not constrained by the Bekenstein bound. However, we further show that when both the encoder and the decoder are restricted, the Bekenstein bound does constrain the channel capacities, including the zero-bit capacity.
The Bremermann-Bekenstein bound is a fundamental bound on the maximal rate with which information can be transmitted. However, its derivation relies on rather weak estimates and plausibility arguments, which make the application of the bound impractical in the laboratory. In this paper, we revisit the bound and extend its scope by explicitly accounting for the back action of quantum measurements and refined expressions of the quantum speed limit. Our result can be interpreted as an upper bound on the maximal rate of quantum learning, and we show that the Bremermann-Bekenstein bound follows as a particular limit. Our results are illustrated, by first deriving a tractable expression from time-dependent perturbation theory, and then evaluating the bound for two time-dependent systems -- the harmonic oscillator and the Pöschl-Teller potential.
The quantum limitations on the maximum communication rate (capacity) possible through a single noiseless channel with signals of finite duration are investigated. They may be summarized in the characteristic information function defined here. In the absence of dispersion this description is a Lorentz-invariant one, and applies also in the presence of exterior gravitational fields. For long duration the characteristic information function corresponds to the standard quantum-channel-capacity formula of Gordon, of Lebedev and Levitin, and of Pendry. For finite signal duration it proves useful to distinguish between heralded and self-heralding signals, according to whether their arrival is anticipated or not. The two types have different characteristic functions. These are calculated here for occupation-number signal states in channels where the carrier quantum field may be represented by independent modes. The Gordon-Lebedev-Levitin-Pendry formula provides an upper bound to the exact results for both types of signals. The linear bound on communication rate of Bremermann and the present author bounds the capacity only for self-heralding signals. However, it is the better estimator of capacity for self-heralding signals with modest information content.