arXiv:1307.0014v2 [quant-ph] 8 Jan 2014 Optimal Binary Codes and Measurements for Classical Communication over Qubit Channels Nicola Dalla Pozza, 1, ∗ Nicola Laurenti, 1, † and Francesco Ticozzi 1, 2, ‡ 1 Dipartimento di Ingegneria dell’Informazione, Universit` a di Padova, via Gradenigo 6/B, 35131 Padova, Italy 2 Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA (Dated: July 11, 2018) We propose constructive approaches for the optimization of binary classical communication over a general noisy qubit quantum channel, for both the error probability and the classical capacity functionals. After showing that the optimal measurements are always associated to orthogonal pro- jections, we construct a parametrization of the achievable transition probabilities via the coherence vector representation. We are then able to rewrite the problem in a form that can be solved by standard, efficient numerical algorithms and provides insights on the form of the solutions. PACS numbers: 03.67.Hk I. INTRODUCTION Every communication system relies on a physical layer into which encode information for delivery. The role of the quantum properties of a media in communication protocols was first investigated by the pioneering work by Helstrom [1], and the study has subsequently evolved into a rich branch of quantum information, see e.g. [2, 3]. Most notably, the results of quantum information the- ory have provided rigorous definitions and results for the (classical and quantum) capacity of quantum channels, see e.g. [4] for an introductory review of early results, and [5] for more recent developments. In this work, we consider the problem of transmitting classical information over a quantum channel that is not ideal, namely, it is described by a Completely-Positive, Trace-Preserving (CPTP) map [4, 6]. We aim at find- ing optimal input states and output measurements with respect to some performance index. We focus on the binary case, namely where two “symbol” states can be transmitted and two “detection” measurement outcomes are considered. An effectively implementable solution of this problem, beside having an obvious relevance per se, would be also instrumental to real-time optimization of communications over time-varying channels and the im- provement of the key generation rate of well established quantum cryptography protocols [7, 8]. In the literature, two different functionals are typically used to evaluate the quality of a classical digital commu- nication system: symbol error probability and channel capacity. Adopting the first functional for classical communica- tion over quantum channels leads to a problem that is closely related to optimal discrimination. The quantum binary discrimination problem, namely the problem of distinguishing two given quantum states with maximal * Electronic address: nicola.dallapozza@dei.unipd.it † Electronic address: nicola.laurenti@dei.unipd.it ‡ Electronic address: ticozzi@dei.unipd.it probability, has been addressed by Helstrom, and the form of the optimal measurement operators, as well as the maximal probability of correct discrimination, have been found analytically for every pair of input states [1]. Finding the optimal measurement for the discrimina- tion problem is equivalent to the optimization of the re- ceiver for a quantum binary ideal communication channel with respect to the error probability. When a non-ideal channel is considered, and the input states are fixed, the optimal measurement operators are the projectors that solve the discrimination problem for the corresponding output states (see e.g. [4]). When a subsequent opti- mization with respect to the input states is aimed, the number of variables involved in the optimization proce- dure is still considerable. Therefore, we solve the problem of optimization with respect to both the states and measurements by deriving necessary conditions for the optimality and by proposing an efficient numerical procedure. On the other hand, the capacity of a classical binary channel represents the maximum amount of information that can be reliably sent from the transmitter to the re- ceiver per use of the channel when only two symbols are employed. It corresponds to the maximum of the mutual information between input and output, computed over all possible a priori distribution and coding of the source. We here assume that source bits are encoded into pairs of quantum states, and this encoding as well as the de- coding protocol are memoryless so that the cascade of the encoder/quantum channel/decoder is equivalent to a classical binary memoryless channel. The channel there- fore is completely characterized by the transition prob- abilities. We then consider the classical capacity of the binary channel we obtain. Note that this in general different from the Shannon capacity C Shan , the one-letter capacity C χ and the full capacity C of the qubit channel, that are the maximum amount of information that can be sent through the quantum channel with respectively separable quantum states and separable measurement operators, separable input states and joint (possibly entangled) measurement operators, and possibly non-separable states and joint