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arXiv:1611.05821 · Physical Review Letters

Universal Limit on Communication

Raphael Bousso A

Published Oct 2, 2017• 12 citations
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Abstract

I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order EΔt/ℏ, where E is the average energy of the signal, and Δt is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.

DOI
10.1103/physrevlett.119.140501

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I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order EΔt/ℏ, where E is the average energy of the signal, and Δt is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.
The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information. Moreover, restricting our attention to classical information, we bound the transmission distance between probability distributions by their entropic distance, which is a concave function of their Hellinger distance.
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
We show that the recently discovered universal upper bound on the thermal conductance of a single channel comprising particles obeying arbitrary fractional statistics is in fact a consequence of a more general universal upper bound, involving the averaged entropy and energy currents of a single channel connecting heat reservoirs with arbitrary temperatures and chemical potentials. The latter upper bound in turn leads, via Holevo's theorem, to a universal (i.e., statistics-independent) upper bound on the optimum capacity for classical information transmission down a single, wideband quantum channel.
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.
Quantum theory imposes fundamental limitations on the amount of information that can be carried by any quantum system. On the one hand, the Holevo bound rules out the possibility of encoding more information in a quantum system than in its classical counterpart, comprised of perfectly distinguishable states. On the other hand, when states are uniformly distributed in the state space, the so-called subentropy lower bound is saturated. How uniform quantum systems are can be naturally quantified by characterizing them as ttt-designs, with $t=\ensuremath{\infty}$ corresponding to the uniform distribution. Here we show the existence of a trade-off between the uniformity of a quantum system and the amount of information it can carry. To this aim, we derive a hierarchy of informational bounds as a function of ttt and prove their tightness for qubits and qutrits. By deriving asymptotic formulas for large dimensions, we also show that the statistics generated by any ttt-design with t>1t>1t>1 contains no more than a single bit of information, and this amount decreases with ttt. Holevo and subentropy bounds are recovered as particular cases for t=1t=1t=1 and $t=\ensuremath{\infty}$, respectively.