Irreducibility of n-ary quantum information

Quantum information is radically different from classical information in that the quantum formalism (Hilbert space) makes necessary the introduction of irreducible “nits,” n being an arbitrary natural number (bigger than one); not just bits. As pointed out many times by Landauer and others (e.g., [1, 2]) the formal concept of information is tied to physics, at least as far as applicability is a concern. Thus it should come as no surprise that quantum mechanics requires fundamentally new concepts of information as compared to the ones appropriate for classical physics. And indeed, research into quantum information and computation theory has exploded in the last decade, bringing about a wealth of new ideas and formalisms. There seems to be one issue, which, despite notable exceptions (e.g., [3, Footnote 6]), has not yet been acknowledged widely: the principal irreducibility of n-ary quantum information associated with the n-dimensionality of Hilbert space. A physical configuration allowing for n possible outcomes has to be encoded quantum mechanically by an n-dimensional Hilbert space. Any single one of the n basis vector corresponds to a onedimensional subspace spanned by that basis vector which in turn corresponds to the following physical proposition: “the physical system is in a pure state corresponding to the basis vector.” In more operational terms, a particle can be prepared in a single one of n possible states. This particle then carries the information to “be in a single one from n different states.” Subsequent measurements