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arXiv:1012.0106 · Physical Review Letters

Sequential Projective Measurements for Channel Decoding

Seth Lloyd, Vittorio Giovannetti, Lorenzo Maccone

Published Jun 20, 2011• 41 citations
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Abstract

We study the transmission of classical information in quantum channels. We present a decoding procedure that is very simple but still achieves the channel capacity. It is used to give an alternative straightforward proof that the classical capacity is given by the regularized Holevo bound. This procedure uses only projective measurements and is based on successive "yes-no" tests only.

DOI
10.1103/physrevlett.106.250501

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We study the transmission of classical information in quantum channels. We present a decoding procedure that is very simple but still achieves the channel capacity. It is used to give an alternative straightforward proof that the classical capacity is given by the regularized Holevo bound. This procedure uses only projective measurements and is based on successive "yes-no" tests only.
We present a decoding procedure to transmit classical information in a quantum channel which, saturating asymptotically the Holevo bound, achieves the optimal rate of the communication line. In contrast to previous proposals, it is based on performing a sequence of projective yes-no measurements which in NNN steps determines which codeword was sent by the sender (NNN being the number of the codewords). Our analysis shows that, as long as NNN is below the limit imposed by the Holevo bound, the error probability can be sent to zero asymptotically in the length of the codewords.
The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present a new simplified expression which gives an upper bound for the Shannon capacity of a channel, and which bears a striking resemblance to the well-known Holevo bound.
The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present a new simplified expression which gives an upper bound for the Shannon capacity of a channel, and which bears a striking resemblance to the well-known Holevo bound.
We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.
In this article we show that the Holevo capacity of a classical quantum channel, can be reached not only by a POVM but by von Neumann measurement too. Furthermore we show two use of this fact 1) We can generalize the classical compound channel to quantum environment and with this new measure technique we can give an optimal coding/decoding algorithm to it. 2) We give an algorithm, with it a classical system can be decoded by quantum apparatus in linear time, which is a surprising result, because best known optimal classical decoding algorithm need n*log(n) time. This means that if a quantum machine can perform arbitrarily von Neumann measurement with only two possible outcome, then it can solve non-polynomial classical problems in linear time.