Softmetric based channel decoding for photon counting receivers
Abstract
We address photonnumberassisted, polarizationbased, binary communication systems equipped with photon counting receivers. In these channels information is encoded in the value of polarization phaseshift but the carrier has and additional degree of freedom, i.e. its photon distribution, which may be exploited to implement binary inputmultiple output (BIMO) channels also in the presence of a phasediffusion noise affecting the polarization. Here we analyze the performances of these channels, which approach capacity by means of iteratively decoded error correcting codes. In this paper we use softmetricbased low density parity check (LDPC) codes for this purpose. In order to take full advantage of all the information available at the output of a photon counting receiver, soft information is generated in the form of loglikelihood ratios, leading to improved frame error rate (FER) and bit error rate (BER) compared to binary symmetric channels (BSC). We evaluate the classical capacity of the considered BIMO channel and show the potential gains that may be provided by photon counting detectors in realistic implementations.
I Introduction
In binary optical communication, the logical information is encoded onto two different states of the radiation field. After the propagation, the receiver should perform a measurement, aimed at discriminating the two signals. Currently, most of the longdistance amplificationfree optical classical communication schemes employ relatively weak laser sources leading to small mean photon count values at the receiver. The same is true for quantumenhanced secure cryptographic protocols. In fact, laser radiation, which is described by coherent states, preserves its Poissonian photonnumber statistics and polarization also in the presence of losses. On the other hand, operating in the regime of low number of detected photons gives rise to the problem of discriminating the signals by quantumlimited measurements [1, 2, 3]. Indeed, the binary discrimination problem for coherent states has been thoroughly investigated, both for its fundamental interest and for practical purposes [4, 5, 6, 7, 8, 9]. It should be mentioned however that in order to exploit the phase properties of coherent states, one should implement phase sensitive receivers [10, 11] with nearly optimal performances also in the presence of dissipation and noise [7, 12]. This is a challenging task, since it is generally difficult, and sometimes impossible, to have a suitable and reliable phase reference in order to implement this kind of receiver [13].
The simplest choice for a detection scheme involving radiation is given by detectors which simply reveal the presence or the absence of radiation (on/off detectors) with acceptable deadtime values and dark count rates. A natural evolution of such schemes would be to employ photon counting receivers. Indeed, development of photon counters has been extensively pursued in the last decades, as well as of methods to extract the photon distribution by other schemes [14, 15, 16, 17, 18]. Given that one could use photon counting detectors for weakenergy optical communications, a question arises on whether and how such detectors may be employed to improve the system performance. A possible way to answer this question is to determine the capacity of the corresponding optical channels, and the achievable residual Bit Error Rate (BER) and Frame Error Rate (FER) of practical communication schemes over these channels. A photon counting detector is clearly able to extract more information than a simple on/off detector. The practical consequence is that a photon counting detector allows one to generate a meaningful loglikelihood (i.e. a softmetric), as opposed to a hardmetric allowed by a hard (or on/off) detector. Furthermore, softmetrics lead to improved performances when exploited by powerful iteratively decoded forward error correcting codes.
Recently, a simple polarizationbased communication scheme involving weak coherent optical signals and lowcomplexity photon counting receivers has been presented [1], and its performances have been analyzed based on an equivalent Binary Symmetric Channel (BSC) model of the overall scheme. In this paper, we extend the scheme of [1] and model the effect of the photon distribution of the coherent signals as a time varying Binary InputMultiple Output (BIMO) channel. In particular, we employ softmetric based Low Density Parity Check (LDPC) codes for transmission over the BIMO channel to approach capacity using iteratively decoded error correcting codes and investigate the potential improvements that may be obtained in terms of classical capacity and residual BER using photon counting receivers [27]. It is worth noting that recently photoncounting detectors have been proposed to enhance the discrimination of weak optical signal in the case of ary coherent state discrimination [28, 29]: in these cases, however, a suitable feedback scheme or the use of squeezing are required.
The receiver introduced in [1] is based on an optical setup for oneparameter qubit gate optimal estimation [2, 21, 19]. In this scheme, the qubit is encoded in the polarization degree of freedom of a light beam, whose intensity (photon) degree of freedom has been prepared in a coherent state, and the oneparameter gate corresponds to a polarization transformation. In the ideal case, orthogonal polarization states can be perfectly discriminated. However, in a realistic scenario and especially in freespace communication, nondissipative (diffusive) noise affecting light polarization disturbs the orthogonality of the states at the receiver, thus requiring suitable detection and strategy for discrimination. It is worth noting that coherent states preserve their fundamental properties when propagating in purely lossy channels, suffering only attenuation, thus only the noise affecting the polarization is detrimental. Remarkably, since our receiver is phaseinsensitive, the scheme works as well as when phasediffusion noise is affecting the channel. This also holds in the case of phaserandomized coherent states [22] which can be easily generated, characterized and manipulated [23] and are useful for enhancing security in decoy state quantum key distribution [24, 25].
The paper is organized as follows; the physical system is described in Section II, where the corresponding channel model and loglikelihood metric are also defined. The associated channel capacity is evaluated in Section IV, while the achievable residual frame and bit error rate obtained with LDPC coding is presented in Section V. Section VI concludes the paper with some final remarks.
Ii The physical channel
The channel we are going to investigate corresponds to the optical setup schematically depicted in Fig. 1. The information bit is encoded onto the polarization degree of freedom of a light beam prepared in a coherent state , initially linearly polarized at with respect to the axis, i.e.:
where and denote horizontal and vertical polarization states with respect to the axis. The encoding rule for the bit is applied to the qubit by means of the polarization rotation , being the Pauli matrix. Due to the analogy with the phaseshift encoding, from now on we will refer to as “phase shift”. In order to follow the scheme proposed in Refs. [1] and in view of a possible experimental verification reported in [21, 19], we assume that the encoding rule for the bit given in Table I.
The polarization rotation (phase shift) may be easily implemented by means of a KDP crystal driven by a high voltage generator, and corresponds to a change of the polarization from linear to elliptical.
At the detection stage information is retrieved by intensity measurement, in a scheme involving a HalfWave Plate (HWP), a Polarizing Beam Splitter (PBS) and two photon counters. This scheme has been experimentally tested to achieve oneparameter qubit gate optimal estimation [21, 19]. Furthermore, several examples of detectors now used by the quantum optics community, can be used as photon counters [15, 16, 17, 21, 22, 23]. The outcomes of the measurement are thus pairs of integer numbers , where is the number of detected photons in the reflected and transmitted beam, respectively. Notice that the total number of detected photons is varying shot by shot. We assume that no photon is lost at the beam splitter. The number of photons in the coherent carrier is a Poisson distributed random variable with mean value . Also the two beams after the PBS are coherent states and the joint probability of obtaining the outcome is the product of two factorized Poisson distributions. The mean values depend on the polarization phaseshift, i.e. on the bit value. Upon denoting by the mean photon number in the reflected or transmitted beam when the imposed phaseshift is , we have:
The probability of the event is thus given by:
(1) 
The overall scheme is suitable for working with weak optical signals, where the value of is typically small. The relevant observation to be made here is that the information is retrieved by photon counting, and therefore the discrete bit value , encoded in the polarization qubit, is mapped at the detection stage onto pairs of integer numbers. The considered scheme can be modeled as shown in Fig. 2, i.e. with an equivalent binaryinput/multipleoutput channel that receives the binary random variable as input, and generates the two random variables as outputs. In particular, for a given number of detected photons, there are pairs such that . The availability of multiple outputs, whose likelihood can be exploited for softinformation processing, is a crucial characteristic of the described scheme.
If propagation of the light beam occurs in an environment, which perturbs the polarization but preserves the energy, then the state impinging onto the PBS has no longer a welldefined polarization (phase): If the initial state is , where refers to the polarization qubit, the phasediffusion noise affects the polarization according to the map [21]:
(2) 
where represents the density matrix of the degraded polarization qubit and is a normal distribution of the variable with zero mean and standard deviation . From the physical point of view, Eq. (2) follows from a Master equation approach [30] which represents a dynamics in which the quantum state of light undergoes an energy conserving ÒscatteringÓ affecting the polarization. Overall, this corresponds to applying a random polarization rotation (or phase shift) of the input polarization distributed according to . The probabilities of the outcomes are still given by Eq. (1), however with the mean photon numbers modified to:
(3)  
(4) 
Iii Evaluation of the LogLikelihood Ratios
Softdecoding algorithms are typically based on the use of LogLikelihoodRatios (LLR). In our particular case, the LLR values associated to the channel model of Fig. 2 can be evaluated as:
(5) 
where,
(6) 
is the probability that the transmitted bit was “” given the outcomes . Using Bayes theorem, Eq. (5) may be rewritten as:
(7) 
Finally, using Eq. (1) we arrive at:
(8) 
where,
(9) 
for the chosen encoding. The system described up to this point represents, for a given , a BIMO Discrete Memoryless Channel (DMC) [20] with binary input and outputs , where is a Poisson distributed random variable. In the next Section we will evaluate the capacity of this channel.
Iv Evaluation of capacity
A sufficient statistic for detection with photon counting detectors is the difference photocurrent at the output, i.e. . Since the two random variables and are Poisson distributed, the outcome of is Skellam distributed, namely:
(10) 
where is the modified Bessel function of the first kind, such that:
(11) 
Upon denoting by the input binary variable, the relevant figure of merit to evaluate the channel capacity is the mutual information:
where,
is the Shannon entropy of the input alphabet, () being the apriori probability of sending the bit () and is the conditional entropy:
and,
(12) 
is the overall probability of the outcome , irrespective of the input bit.
Our BIMO DMC is neither symmetric nor weakly symmetric. Recall that a DMC is said to be symmetric if the rows (and the columns) of the channel transition probability matrix are permutations of each other. If, on the other hand, the rows are permutations of each other and the column sums are equal but the columns are not permutations of each other, the DMC is said to be weakly symmetric. It can be shown that for symmetric or weakly symmetric channels uniform probability on input maximizes the mutual information thus yielding capacity. However, it can be easily shown that the input probability distribution maximizing the mutual information in the BIMO case above is the uniform one, i.e. . The channel capacity is thus given by:
(13) 
Our goal is now to compare the capacity of the present photon counting receiver channel to that of the equivalent binary symmetric channel resulting from the detection of optical signals by on/off receiver, which just discriminates the presence or the absence of radiation (i.e., performs hard decoding). The transition probability of the equivalent BSC associated with the considered photon counting receiver (i.e. the raw BER, denoted in the following as QBER) can be obtained as:
QBER  (14)  
(15) 
Essentially, assuming is true, a detection error occurs for a hard decision detector if . In case , the detector can toss a fair coin and assign a decoded bit arbitrarily, in which case the probability of error is .
In our case, in the limit , we can write:
and,
When “ is transmitted and it is mapped to ”, we get from Eqs. (3) and (4):
analogously when “1 is transmitted and it is mapped to ” :
where,
and,
After some manipulation we have:
The final expression of the conditional entropy as a function of the two parameters and is:
Note that capacity is achieved with . The results are shown in Fig. 3 and 4. The capacity of the BIMO DMC compared to that of the equivalent BSC obtained in case of on/off detection is shown in Fig. 3 as a function of the mean photon number and for , while in Fig. 4 the BIMO DMC is compared to the equivalent BSC for as a function of the phase diffusion parameter . It is possible to observe that a higher capacity can be obtained by the BIMO DMC with respect to the equivalent BSC, that may possibly lead to improved BER improvement when an error correction code is applied to the two channels. This aspect is indeed investigated in the next section. The capacity improvement offered by the photon counting detector decreases as increases, in particular for low values of , as it can be observed by both Fig. 3 and Fig. 4.
V Ber Performance in Presence of Fec
This section investigates the performance obtainable with Forward Error Correction (FEC) codes applied to the scheme of Figs. 1 and 2. The bits codeword of a systematic FEC code with code rate is generated concatenating information bits and redundancy bits so that and .
A systematic low density parity check code has been selected as test FEC code, due to its capacity achieving performance (albeit at very large block lengths) and low complexity iterative decoding structure, and a simulation analysis has been performed to assess the potential performance improvements obtainable using the softmetric of Eq. (8). Three different quantum channel models have been considered, all with the same equivalent uncoded raw BER value, that will be denoted as QBER. The simulation results are shown in Fig. 5, where each pair of BERFER curves depicts the residual bit error rate and frame error rate after channel decoding. The following parameters have been considered:

, , ,

, , ,

, , .
The black curves in Fig. 5 labeled ”QBSC” are associated with an equivalent BSC with binary input , binary output and transition probability QBER derived from Eq. (15) (i.e. a receiver that does not use the additional information derived from the knowledge of and and simply performs on/off detection) with LLR values [26]:
The blue curves labeled as “QAWGN” represent the performance obtainable over a fictitious Additive White Gaussian Noise (AWGN) channel model with a Signal to Noise Ratio (SNR) selected in order to achieve an uncoded bit error probability QBER with a binary antipodal scheme. The curves labeled as “QBIMO” represent the main result and are obtained transmitting through the BIMO DMC quantum channel model shown in Fig. 2 with equivalent uncoded bit error probability QBER and using as input softmetrics for the LDPC decoder, the LLR values generated via photon counting according to Eq. (8).
As it is apparent from the results for the photon counting receiver, the BER and FER performance largely improve when the BIMO DMC and the LLR metrics from Eq. (8) are employed instead of the simpler BSC metrics. As an example, in the upper plot in Fig. 5 for the BIMO DMC with softmetric processing offers almost three orders of magnitude improvement in BER with respect to the BSC model and the associated hardmetric processing. We must note that the curves labeled as “QAWGN” must only be used as reference, since with the small number of photons we considered in our simulations the AWGN channel model would not be appropriate.
A comparison among the residual FER and BER values obtainable with the considered channel models for LDPC codes with code rates (center plot) and (bottom plot) is shown in Fig. 5. Also in these cases, both FER and BER values improve up to several orders of magnitude when using a photon counting receiver and the associated LLR values. Furthermore, we can observe that as the code rate increases, the “QBIMO” performances obtained with BIMO LLR metrics get closer to the “QAWGN” performances obtained with classic AWGN LLR metrics (although, as mentioned before, the AWGN model is not applicable in case of low number of received photons).
Fig. 6 compares the BER values obtained with the BSC and the BIMO channel models for different code rates, showing that, as expected, for higher rates, a lower QBER value is required before significant coding gains can be observed.
¿From Fig. 7, we can observe that for high values of (i.e. at low values of QBER) the BIMO DMC model can be approximated with an AWGN model, while the AWGN model approximation may be unreliable at high QBER (low ) values, in particular at lower code rate values.
Finally, Fig. 8 shows the residual BER obtained on the BIMO channel by LDPC codes with code rate for different values of .
Vi Conclusions
In this paper a photonnumberassisted, polarizationbased binary transmission scheme equipped with a lowcomplexity photon counting receiver has been considered, analyzing both its capacity and its BER performance in presence of capacity achieving low density parity check codes. Different channel models applicable to the considered transmission scheme have been compared, proposing a time varying binaryinput/multipleoutput model and evaluating its LLR metrics and channel capacity. It has been shown how the BIMO channel model outperforms the corresponding BSC model, by taking full advantage of the additional information offered by the photon counting detector. It was also shown that, as expected, the advantage offered by the photon counting detector deceases as the mean photon number increases, and that the BIMO model can be approximated by an AWGN model at low values of QBER, i.e. for high values of .
Acknowledgment
This work was supported by MIUR (grant FIRB “LiCHIS”  RBFR10YQ3H) and NATO (SfP project 984397 ”Secure Quantum Communications”).
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Marina Mondin is Associate Professor at Dipartimento di Elettronica, Politecnico di Torino. Her current interests are in the area of signal processing for communications, modulation and coding, simulation of communication systems, and quantum communication. She holds two patents. She has been Associate Editor for IEEE Transactions on Circuits and SystemsI from 2010 to 2013 and has been in the 2012 TCAS committee for the selection of the Darlington and GuilleminCauer Best Paper Awards. She is currently coordinating the NATO SfP project “Secure Communication Using Quantum Information Systems”. She is author of more than 150 publications. 
Fred Daneshgaran received the Ph.D. degree in electrical engineering from University of California, Los Angeles (UCLA), and since 1997 has been a full Professor with the ECE Department at California State University, Los Angeles (CSLA). From 2006 he serves as the chairman of the ECE department. Dr. Daneshgaran is the founder of Euroconcepts, S.r.l, a R&D company specializing in the design of advanced communication links and software radio that operated from 2000 to 2010. From 1999 to 2001 he was the Chief Scientist and member of the management team, for TechnoConcepts, Inc. where he directed the development of a prototype software defined radio system, managed the hardware and software teams and orchestrated the entire development process. He is the director of the fiber and nonlinear optics research laboratory at CSLA, and served as the Associate Editor of the IEEE Trans. On Wireless Comm. in the areas of modulation and coding, multirate and multicarrier communications, broadband wireless communications, and software radio, from 2003 to 2009. He has served as a member of the Technical Program Committee (TPC) on numerous conferences. Most recent contributions include IEEE WCNC 2014, CONWIRE 2012, ISCC 2011 to 2014, and PIMRC 2011. 
Inam Bari Inam Bari obtained his BS in Telecommunication Engineering from the National University of Computer and Emerging Science (NUCESFAST), Pakistan in 2007, and was awarded bronze medal. In 2008, he was awarded a full 5 years MS leading to PhD scholarship by the Higher Education Commission of Pakistan. He obtained his MS and PhD degrees from Politecnico di Torino, Italy, and is currently Assistant Professor at NUCESFAST, Peshawar, Pakistan. 
Maria Teresa Delgado received her BS Degree in Electrical Engineering at the Universidad Central de Venezuela in Caracas, Venezuela in 2006, and her MS and Ph.D. degrees in Telecommunication Engineering from Politecnico di Torino, Italy, in 2008 and 2012. Her interests are in the area of signal processing for telecommunications, coding, simulation of communication systems, quantum cryptography and physical layer security for wireless and quantum communication systems. She is currently researcher at Istituto Superiore Mario Boella, Turin, Italy. 
Stefano Olivares received the Ph.D. degree in Physics from the University of Milan, Milano, Italy, and is currently a Researcher at the Department of Physics, University of Milan, Italy. He is a theoretician and his interests include quantum information, quantum estimation, quantum optics, quantum interferometry and quantum computation. His main contributions are in the fields of quantum estimation of states and operations, generation and application of entanglement, quantum information, communication and decoherence. Although his research activity is mainly theoretical, he is an active collaborator in many experimental groups. He is author of about 100 publications. 
Matteo G. A. Paris received his Ph.D. in physics from University of Pavia, and is currently professor of quantum information and quantum optics at the Department of Physics of the University of Milan. His main contributions are in the fields of quantum estimation of states and operations, quantum tomography, generation, characterization and application of entanglement, quantum interferometry, nonclassical states and open quantum systems. In these fields he is author of about 250 publications in international journals. From 2013 he is editorinchief of Quantum measurements and quantum metrology. 