The concept of a two-photon coherent state is introduced for applications in quantum optics. It is a simple generalization of the well-known minimum-uncertainty wave packets. The detailed properties of two-photon coherent states are developed and distinguished from ordinary coherent states. These two-photon coherent states are mathematically generated from coherent states through unitary operators associated with quadraticHamiltonians. Physically they are the radiation states of ideal two-photon lasers operating far above threshold, according to the self-consistent-field approximation. The mean-square quantum noise behavior of these states, which is basically the same as those of minimum-uncertainty states, leads to applications not obtainable from coherent states or one-photon lasers. The essential behavior of two-photon coherent states is unchanged by small losses in the system. The counting rates or distributions these states generate in photocount experiments also reveal their difference from coherent states.

The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its Holevo information is not superadditive, and that a coherent-state encoding achieves capacity. The capacity of far-field, free-space optical communications is given as an example.PACS numbers: 03.67. Hk,89.70.+c,05.40.Ca A principal goal of quantum information theory is evaluating the information capacities of important communication channels. At present-despite the many efforts that have been devoted to this endeavor and the theoretical advances they have produced [1]-exact capacity results are known for only a handful of channels. In this paper we consider the lossy bosonic channel, and we develop an exact result for its classical capacity C, i.e., the number of bits that it can communicate reliably per channel use. The lossy bosonic channel consists of a collection of bosonic modes that lose energy en route from the transmitter to the receiver. Typical examples are free space or optical fiber transmission, in which photons are employed to convey the information. The classical capacity of the lossless bosonic channel-whose transmitted states arrive undisturbed at the receiver-was derived in [2,3]. When there is loss, however, the received state is in general different from the transmitted state, and quantum mechanics requires that there be an accompanying quantum noise source. In [4] a first step toward the capacity of such channels was given by considering only separable encoding procedures. Here, on the contrary, it is proven that the optimal encoding is indeed separable. We obtain the value of C in the presence of loss when the quantum noise source is in the vacuum state, i.e., when it injects the minimum amount of noise into the receiver. Our derivation proceeds by developing an upper bound for C and then showing that this bound coincides with the lower bound on C reported in [5,6]. Our upper bound results from comparing the capacity of the lossy channel to that of the lossless channel whose average input energy matches the average output energy constraint for the lossy case [7]. This argument is analogous to the derivation of the classical capacity of the erasure channel [8]. The lower bound comes from calculating the Holevo information for appropriately coded coherent-state inputs. Thus, because the two bounds coincide, we not only have the capacity of the lossy bosonic channel, but we also know that capacity can be achieved by transmitting coherent states.

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