Efficient Quantum Polar Coding

Abstract

Polar coding, introduced 2008 by Arikan, is the first efficiently encodable and decodable coding scheme that provably achieves the Shannon bound for the rate of information transmission over classical discrete memoryless channels (in the asymptotic limit of large block sizes). Here we study the use of polar codes for the transmission of quantum information. Focusing on the case of qubit channels we construct a coding scheme which, using some pre-shared entanglement, asymptotically achieves a net transmission rate equal to the coherent information using efficient encoding and decoding operations and code construction. Furthermore, for channels with sufficiently low noise level, no pre-shared entanglement is required.