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Coherent state contellations and polar codes for Bosonic Gaussian channels

Felipe Gomes Lacerda

Published Mar 18, 2016• 0 citations
OpenAlex

Abstract

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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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.
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.
While the ultimate limits on the rate of quantum communication over noisy channels are well-established, very few efficient coding schemes achieving these limits are known. This contrasts sharply with transmitting classical data over classical channels, where polar codes and low-density parity-check codes both efficiently achieve the classical capacity of any noisy channel. In this work we construct an efficient entanglement generation scheme which achieves the coherent information for a broad class of channels. For degradable channels in the class, which includes truncated versions of the pure loss Bosonic channel, the resulting scheme achieves the quantum capacity. The construction makes use of polar codes for two kinds of classical-input, quantum-output (CQ) channels: those with commuting outputs and those whose outputs are heralded mixtures of pure states. The former case is amenable to decoding by classical polar decoding algorithms, the latter by the belief propagation with quantum messages (BPQM) algorithm. To our knowledge, this construction is the first explicit and efficiently decodable protocol that achieves the energy-constrained quantum capacity of the pure-loss Bosonic channel.
We construct explicit codes for transmitting classical or quantum information over the thermal-noise Bosonic Gaussian channel. For classical information transmission, our codes achieve the classical capacity, recently characterized by Giovannetti et al. [Nature Photonics 8, 796 (2014)], while in the quantum case they transmit at a rate given by the Gaussian coherent information. The construction is based on concatenating polar codes with a suitable discretization of the channel input into a finite constellation of coherent states. The codes have efficient encoders, but the complexity of decoding is unknown. Our results are based on a quantum version of Wu and Verd\'u's technique of determining the achievable rates of finite constellations for the classical Gaussian-noise channel [Allerton 2010, pp. 620].
We propose constellations of finitely-many coherent states for high-rate quantum and classical communication over the thermal noise Bosonic Gaussian channel. Our constructions are based on constellations for the classical additive white Gaussian noise (AWGN) channel, and we adapt the results of Wu and Verdú [Allerton 2010, pp. 620] for the AWGN to determine achievable rates of classical and quantum information transmission for the thermal noise channel. Several constellations allow classical rates approaching the classical capacity, recently determined by Giovannetti et al. [Nature Photonics 8, 796 (2014)], while in the quantum case the rates approach the Gaussian coherent information. The constellations can also be used for private transmission of classical information at the coherent information rate.
Polar coding, introduced 2008 by Arıkan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for 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 Pauli channels and qubit erasure channels, we use classical polar codes to construct a coding scheme that asymptotically achieves a net transmission rate equal to the coherent information using efficient encoding and decoding operations and code construction. Our codes generally require preshared entanglement between sender and receiver, but for channels with a sufficiently low noise level we demonstrate that the rate of preshared entanglement required is zero.
We construct new polar coding schemes for the transmission of quantum or private classical information over arbitrary quantum channels. In the former case, our coding scheme achieves the symmetric coherent information, and in the latter, the symmetric private information. Both schemes are built from a polar coding construction capable of transmitting classical information over a quantum channel. Appropriately merging two such classical-quantum schemes, one for transmitting amplitude information and the other for transmitting phase, leads to the new private and quantum coding schemes, similar to the construction for Pauli and erasure channels of Renes et al. The encoding is entirely similar to the classical case, and thus efficient. The decoding can also be performed by successive cancellation, as in the classical case, but no efficient successive cancellation scheme is yet known for arbitrary quantum channels. An efficient code construction is unfortunately still unknown. Generally, our two coding schemes require entanglement or secret-key assistance, respectively, but we extend two known conditions under which the needed assistance rate vanishes. Finally, although our results are formulated for qubit channels, we show how the scheme can be extended to multiple qubits. This then demonstrates a near-explicit coding method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels, which individually have zero quantum capacity can have a nonzero quantum capacity when used together.