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arXiv:quant-ph/9703013

On Reliability Function of Quantum Communication Channel

M. V. Burnashev, Alexander Semenovich Holevo

Published Mar 8, 1997• 42 citations
arXivHTMLPDF

Abstract

The reliability function gives the rate of exponential convergence to zero of the error probability in a communication channel. In this paper bounds for the reliability function of a quantum pure state channel are given, reminiscent of the corresponding classical bounds. This in particular suggests an alternative proof of the coding theorem for quantum noiseless channel, which would make no use of the notion of typical subspace. Example of binary quantum channel is considered in some detail.

DOI
10.48550/arxiv.quant-ph/9703013

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The reliability function gives the rate of exponential convergence to zero of the error probability in a communication channel. In this paper bounds for the reliability function of a quantum pure state channel are given, reminiscent of the corresponding classical bounds. This in particular suggests an alternative proof of the coding theorem for quantum noiseless channel, which would make no use of the notion of typical subspace. Example of binary quantum channel is considered in some detail.
Information theory tells us that if the rate of sending information across a noisy channel were above the capacity of that channel, then the transmission would necessarily be unreliable. For classical information sent over classical or quantum channels, one could, under certain conditions, make a stronger statement that the reliability of the transmission shall decay exponentially to zero with the number of channel uses, and the proof of this statement typically relies on a certain fundamental bound on the reliability of the transmission. Such a statement or the bound has never been given for sending quantum information. We give this bound and then use it to give the first example where the reliability of sending quantum information at rates above the capacity decays exponentially to zero. We also show that our framework can be used for proving generalized bounds on the reliability.
In information theory the reliability function and its bounds, describing the exponential behavior of the error probability, are the most important quantitative characteristics of the channel performance. From a general point of view, these bounds provide certain measures of distinguishability of a given set of states. In an earlier paper we introduced quantum analogs of the random coding and the expurgation lower bounds for the case of pure signal states. Here we discuss the general case, in particular, we prove the previously conjectured expurgation bound and find the quantum cutoff rate in the case of arbitrary mixed signal states.
In information theory, the reliability function and its bounds, describing the exponential behavior of the error probability, are important quantitative characteristics of the channel performance. From a more general point of view, these bounds provide certain measures of distinguishability of a given set of classical states. In a previous paper, quantum analogs of the random coding and expurgation lower bounds for the case of pure signal states were introduced. Here we discuss the case of general quantum states, in particular, we prove the expurgation bound conjectured previously and find the quantum cutoff rate for arbitrary mixed signal states.
Contents I. Introduction II. General considerations § 1. Quantum communication channel § 2. Entropy bound and channel capacity § 3. Formulation of the quantum coding theorem. Weak conversion III. Proof of the direct statement of the coding theorem § 1. Channels with pure signal states § 2. Reliability function § 3. Quantum binary channel § 4. Case of arbitrary states with bounded entropy IV. c-q channels with input constraints § 1. Coding theorem § 2. Gauss channel with one degree of freedom § 3. Classical signal on quantum background noise Bibliography
The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel’s entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.
In this paper, the sphere packing bound of Fano, Shannon, Gal- lager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability func- tion obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo (21). Thus, for pure state channels, the reliability function at high rates is now ex- actly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.