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arXiv:1506.04999 · Journal of Mathematical Physics

Achieving the Holevo bound via a bisection decoding protocol

Matteo Rosati, Vittorio Giovannetti

Published Jun 1, 2016• 22 citations
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Abstract

We present a new decoding protocol to realize transmission of classical information through a quantum channel at asymptotically maximum capacity, achieving the Holevo bound and thus the optimal communication rate. At variance with previous proposals, our scheme recovers the message bit by bit, making use of a series of “yes-no” measurements, organized in bisection fashion, thus determining which codeword was sent in log2 N steps, N being the number of codewords.

DOI
10.1063/1.4953690

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We present a new decoding protocol to realize transmission of classical information through a quantum channel at asymptotically maximum capacity, achieving the Holevo bound and thus the optimal communication rate. At variance with previous proposals, our scheme recovers the message bit by bit, making use of a series of “yes-no” measurements, organized in bisection fashion, thus determining which codeword was sent in log2 N steps, N being the number of codewords.
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.
We obtain strict upper bounds on the bit transmission rate for communication of Classical bit codewords over Quantum channels. Albeit previous arguments in arXiv: 1804.01797 which have demonstrated that lower bounds can be shown to hold for the bit transmission rate without the presence of significant noise over the channel shared by Alice and Bob for the purposes of encoding, decoding, transmission and authentication, the author suggests that upper bounding the bit transmission rate could be of use towards classifying paradoxical aspects of communication protocols, as well as constructing error correcting codes which are resilient to noise. The upper bound that is obtained in this work for the bit transmission rate, as a converse result, is dependent upon the natural logarithm of the size of each player's alphabet, as well as smaller alphabets, which can be leveraged for simultaneously realizing Quantum advantage for maximizing error correction and minimizing false acceptance. Crucially, the upper bound to the bit transmission rate is dependent upon a pruning procedure, which seeks to determine whether letters from player's alphabets can be removed so that prospective Quantum advantage, in order for Alice and Bob to implement error correction protocols with high probability, despite the fact that there is more noise over the channel between Alice and Bob in comparison to that between Bob and Eve.
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to encode and decode information under a single-channel use. We prove that the optimal capacity of single-channel Gaussian communication is achieved by one of two well-known protocols, i.e., coherent-state communication or squeezed-state communication, depending on the energy (number of photons) as well as the characteristics of the channel. Our result suggests that the coherent-state scheme known to achieve the ultimate information-theoretic capacity is not a practically optimal scheme for the case of using a finite number of channels. We find that overall the squeezed-state communication is optimal in a small-photon-number regime whereas the coherent-state communication performs better in a large-photon-number regime.
We study the problem of decoding classical information encoded on quantum states at the output of a quantum channel, with particular focus on increasing the communication rates towards the maximum allowed by Quantum Mechanics. After a brief introduction to the main theoretical formalism employed in the rest of the thesis, i.e., continuous-variable Quantum Information Theory and Quantum Communication Theory, we consider several decoding schemes. First, we treat the problem from an abstract perspective, presenting a method to decompose any quantum measurement into a sequence of easier nested measurements through a binary-tree search. Furthermore we show that this decomposition can be used to build a capacity-achieving decoding protocol for classical communication on quantum channels and to solve the optimal discrimination of some sets of quantum states. These results clarify the structure of optimal quantum measurements, showing that it can be recast in a more operational and experimentally-oriented fashion. Second, we consider a more practical approach and describe three receiver structures for coherent states of the electromagnetic field with applications to single-mode state discrimination and multi-mode decoding at the output of a quantum channel. We treat the problem bearing in mind the technological limitations faced nowadays in the field of optical communications: we evaluate the performance of general decoding schemes based on such technology and report increased performance of two schemes, the first one employing a non-Gaussian transformation and the second one employing a code tailored to be read out easily by the most common detectors. Eventually we characterize a large class of multi-mode adaptive receivers based on common technological resources, obtaining a no-go theorem for their capacity.
We obtain a lower bound on the maximum number of qubits, Qn, ε(N), which can be transmitted over n uses of a quantum channel N, for a given non-zero error threshold ε. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the Petz recovery map (also known as the transpose channel). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which Qn, ε(N) scales as n.