Qubit Channels Which Require Four Inputs to Achieve Capacity:   Implications for Additivity Conjectures
  An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3-state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacity and optimal inputs is presented and its relation to a relative entropy optimization discussed. 
Masahito Hayashi, Hiroshi Imai, Keiji Matsumoto, Mary Beth Ruskai,   Toshiyuki Shimono