The capacity $C$ of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power $P$. The limits arise ultimately from the Einstein relation that a field quantum at frequency $f$ has energy $E=hf$. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field—i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is $C_{\mathrm{WB}}=\left(\frac{\pi }{\ln 2}\right)\sqrt{\frac{2P}{3h}}$ bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity $C_{\mathrm{WB}}$ are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.

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