The capacity of hybrid quantum memory

It is a basic but fundamental result that all forms of perfect
classical memory, traditionally measured in bits, are equivalent except
for total capacity. There is also a restricted quantum version of this
question: Finite-dimensional Hilbert spaces are also all equivalent except
for their dimension. However, there is also a mutual generalization of this
question if one considers in any finite quantum system, the information
that is left after noise has been applied for an infinite amount of time.
I will argue that a general system of this type can be modeled by a
finite-dimensional C^*-algebra with both quantum and classical features.
The total capacity of this type of hybrid quantum memory is no longer
a scalar. The question of when one such memory is larger than another
has an interesting answer in terms of bin packing and $p$-norms.