Quantum Information Processing

, Volume 15, Issue 3, pp 1289–1308 | Cite as

Second-order coding rates for pure-loss bosonic channels

  • Mark M. WildeEmail author
  • Joseph M. Renes
  • Saikat Guha


A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon-number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest one, the pure-loss bosonic channel. In a second-order analysis of communication, one fixes the tolerable error probability and seeks to understand the back-off from capacity for a sufficiently large yet finite number of channel uses. We find a lower bound on the maximum achievable code size for the pure-loss bosonic channel, in terms of the known expression for its capacity and a quantity called channel dispersion. We accomplish this by proving a general “one-shot” coding theorem for channels with classical inputs and pure-state quantum outputs which reside in a separable Hilbert space. The theorem leads to an optimal second-order characterization when the channel output is finite-dimensional, and it remains an open question to determine whether the characterization is optimal for the pure-loss bosonic channel.


Second-order coding rates Pure-loss bosonic channel   Optical communication Spectral entropy 



We are grateful to Vincent Y. F. Tan, Marco Tomamichel, and Andreas Winter for helpful discussions related to this paper. The ideas for this research germinated in a research visit of S.G. and M.M.W. to J.M.R. at ETH Zurich in February 2013. S.G. and M.M.W. are grateful to Renato Renner’s quantum information group at the Institute of Theoretical Physics of ETH Zurich for hosting them during this visit. M.M.W. is grateful for the hospitality of the Quantum Information Processing Group at Raytheon BBN Technologies for subsequent research visits during August 2013 and April 2014. M.M.W. acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF under Award No. CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. J.M.R. acknowledges support from the Swiss National Science Foundation (through the National Centre of Competence in Research “Quantum Science and Technology” and Grant No. 200020-135048) and the European Research Council (Grant No. 258932). S.G. was supported by DARPA’s Information in a Photon (InPho) program, under Contract No. HR0011-10-C-0159.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics and Astronomy, Hearne Institute for Theoretical Physics, Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Institute for Theoretical PhysicsETH ZurichZurichSwitzerland
  3. 3.Quantum Information Processing GroupRaytheon BBN TechnologiesCambridgeUSA

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