Multiedge-type low-density parity-check codes for continuous-variable quantum key distribution

Abstract

Continuous-variable quantum key distribution (QKD) utilizes an ensemble of coherent states of light to distribute secret encryption keys between two parties. An essential ingredient of the QKD protocol is highly efficient information reconciliation. To achieve highly efficient reconciliation, error-correcting codes with a low channel coding rate are inevitable in the most common schemes of multilevel coding and multistage decoding (MLC-MSD) and multidimensional reconciliation. Multiedge-type (MET) low-density parity-check (LDPC) codes are well suited for highly efficient reconciliation at low rates. Here, we calculate the optimal channel coding rates in the MLC-MSD scheme for reverse reconciliation, introduce the concept of generalized extrinsic information transfer charts for MET-LDPC codes, which constitute a simple and fast asymptotic analysis tool, and present a set of MET-LDPC codes with asymptotic efficiency >97% for channel coding rates 0.1, 0.05, 0.02, and 0.01. We believe that our codes will find wide application in implementations of continuous-variable quantum key distribution based on Gaussian modulation.

Explicit asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation

“…In the second expression, we write Y to denote a discretized version of Y , since otherwise the conditional entropy is ill-defined. Provided that the reconciliation protocol fully exploits soft-information, meaning that the discretization is sufficiently precise, then high values of β between 95 and 98% are achievable [19,29,31] for a Gaussian modulation. Similarly, for a QPSK modulation, it is possible to easily reach 90% at arbitrarily low SNR.…”

Distributional Transform Based Information Reconciliation

“…Both SEC and MD reconciliation procedures present a way to extract bit sequences from continuous valued data so that an error correcting code could be applied, typically an LDPC code [10], [13], [15], [16]. A relatively recent alternative, proposed by Araújo and Assis [14] leverages a property of random variables with continuous distribution function widely used in Copula Theory [17], [18,Theorem 1.2.6].…”

“…One is the Sliced Error Correction (SEC) protocol [7], which consists of a set of slicing functions for Alice and a set of estimators on Bob's side. After slicing procedure has taken place, each emerging binary symmetric channel (BSC) can be treated separately with multilevel coding and multistage decoding (MLC-MSD), applying LDPC codes to realize error correction close to channel capacity [9], [10]. The protocol efficiency depends not only on the error correcting codes but also on the quantization efficiency.…”

“…Another method widely used is the multidimensional (MD) reconciliation, which applies d-dimensional rotations to simulate virtual channels close to BIAWGNC (Binary-input AWGN channel) [10]- [13]. It means that d uses of the physical channel are mapped to d approximated copies of a virtual BI-AWGNC.…”

Modulation leakage-free continuous-variable quantum key distribution

“…After the quantum symbol recovery, Alice and Bob performed information reconciliation (IR) based on a multi-dimensional (MD) scheme using a multi-edge-type low-density-parity-check (MET-LDPC) error correcting code [33], assuming they were connected through an authenticated classical channel. Table 1 summarizes the related parameters of the constructed MET-LDPC code, optimized for a binary input additive white Gaussian noise (BI-AWGN) channel.…”