Mary Beth Ruskai, Tufts University
Abstract
Subtitle: A numerical project that needs HELP
After Shor's proof of equivalence of additivity conjectures, attention shifted from capacity and entanglement of formation to the seemingly easier questions of minimal output entropy. Despite existence proofs for non-additivity in high dimensions, explicit examples remain elusive. It may be that the violations for minimal output entropy are so small and require such large dimensions, that numerical searches won't find them.
For Holevo capacity, they are reasons to believe that some non-unital qubit channels exhibit non-additivity for multiple copies. Moreover, by using a max-min expression for the capacity in terms of the relative entropy one avoids the needs to optimize over the entire input ensemble. This makes numerical testing of a modest number of copies feasible.
Moreover, a good candidate for such tests are the 3-state and 4-state qubit channels. These non-unital channels have the counter-intuitive property that -- despite acting on qubits -- optimal encoding of classical information uses strings of 0, 1, 2 rather than 0, 1.
You do NOT need to be an expert on channel capacity to work on this. Good numerical skills with matrices of modes size are what is needed. The algorithm can be written down in fairly straightforward language, and some notes requiring no knowledge of capacity or quantum information are available.