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Register a new userCoherent-state constellations and polar codes for thermal Gaussian channels
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Optical communication channels are ultimately quantum-mechanical in nature, and we must therefore look beyond classical information theory to determine their communication capacity as well as to find efficient encoding and decoding schemes of the highest rates. Thermal channels, which arise from linear coupling of the field to a thermal environment, are of particular practical relevance; their classical capacity has been recently established, but their quantum capacity remains unknown. While the capacity sets the ultimate limit on reliable communication rates, it does not promise that such rates are achievable by practical means. Here we construct efficiently encodable codes for thermal channels which achieve the classical capacity and the so-called Gaussian coherent information for transmission of classical and quantum information, respectively. Our codes are based on combining polar codes with a discretization of the channel input into a finite constellation of coherent states. Encoding of classical information can be done using linear optics.
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We suggest a new protocol for the information reconciliation stage of quantum key distribution based on polar codes. The suggested approach is based on the blind technique, which is proved to be useful for low-density parity-check (LDPC) codes. We show that the suggested protocol outperforms the blind reconciliation with LDPC codes, especially when there are high fluctuations in quantum bit error rate (QBER).
A secret can be an encrypted message or a private key to decrypt the ciphertext. One of the main issues in cryptography is keeping this secret safe. Entrusting secret to one person or saving it in a computer can conclude betrayal of the person or destruction of that device. For solving this issue, secret sharing can be used between some individuals which a coalition of a specific number of them can only get access to the secret. In practical issues, some of the members have more power and by a coalition of fewer of them, they should know about the secret. In a bank, for example, president and deputy can have a union with two members by each other. In this paper, by using Polar codes secret sharing has been studied and a secret sharing scheme based on Polar codes has been introduced. Information needed for any member would be sent by the channel which Polar codes are constructed by it.
As with classical information, error-correcting codes enable reliable transmission of quantum information through noisy or lossy channels. In contrast to the classical theory, imperfect quantum channels exhibit a strong kind of synergy: there exist pairs of discrete memoryless quantum channels, each of zero quantum capacity, which acquire positive quantum capacity when used together. Here we show that this superactivation phenomenon also occurs in the more realistic setting of optical channels with attenuation and Gaussian noise. This paves the way for its experimental realization and application in real-world communications systems.
Polar codes are introduced for discrete memoryless broadcast channels. For $m$-user deterministic broadcast channels, polarization is applied to map uniformly random message bits from $m$ independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region. For two-user noisy broadcast channels, polar implementations are presented for two information-theoretic schemes: i) Covers superposition codes; ii) Martons codes. Due to the structure of polarization, constraints on the auxiliary and channel-input distributions are identified to ensure proper alignment of polarization indices in the multi-user setting. The codes achieve rates on the capacity boundary of a few classes of broadcast channels (e.g., binary-input stochastically degraded). The complexity of encoding and decoding is $O(n*log n)$ where $n$ is the block length. In addition, polar code sequences obtain a stretched-exponential decay of $O(2^{-n^{beta}})$ of the average block error probability where $0 < beta < 0.5$.
Quantum error-correcting codes are used to protect qubits involved in quantum computation. This process requires logical operators, acting on protected qubits, to be translated into physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $mathbb{C}^{N times N}$ as a partial $2m times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As a corollary of these results, we prove that for an $[![ m,k ]!]$ stabilizer code every logical Clifford operator has $2^{r(r+1)/2}$ symplectic solutions, where $r = m-k$, up to stabilizer degeneracy. The desired physical circuits are then obtained by decomposing each solution into a product of elementary symplectic matrices, that correspond to elementary circuits. This enumeration of all physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper discusses a proof of concept synthesis for the $[![ 6,4,2 ]!]$ CSS code. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall LCS (logical circuit synthesis) algorithm, can be found at: https://github.com/nrenga/symplectic-arxiv18a
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