Discrete classical vs. continuous quantum data in abstract quantum mechanics

Abstract

``Quantum'' stands for for the concepts (both operational and formal)
which had to be added to classical physics in order to understand
otherwise unexplainable observed phenomena such as the structure of
the spectral lines in atomic spectra.  While the basic part of
classical mechanics deals with the (essentially) reversible
dynamics, quantum required adding the notions of ``measurement'' and
(possibly non-local) ``correlations'' to the discussion.  Crucially,
all this comes with a ``probabilistic calculus''.  The corresponding
mathematical formalism was considered to have reached maturity in
[von Neumann 1932], but there are some manifest problems with that
formalism:

(i) While measurements are applied to physical systems, application
of their formal counterpart (i.e. a self-adjoint linear operator) to
the vector representing that state of the system in no way reflects
how the state changes during the act of measurement.  Analogously,
the composite of two self-adjoint operators has no physical
significance while in practice measurements can be effectuated
sequentially. More generally, the formal types in von Neumann's
formalism do not reflect the nature of the corresponding underlying
concept at all!

(ii) Part of the problem regarding the measurements discussed above
is that in the von Neumann formalism there is no place for storage,
manipulation and exchange of the classical data obtained from
measurements.  Protocols such as quantum teleportation involving
these cannot be given a full formal description.

(iii) The behavioral properties of quantum entanglement which for
example enable continuous data exchange using only finitary
communication are hidden in the formalism.

In [Abramsky and Coecke 2004] we addressed all these problems, and in
addition provided a purely categorical axiomatization of quantum
mechanics. The concepts of the abstract quantum mechanics are
formulated relative to a strongly compact closed category with
biproducts (of which the category FdHilb of finite dimensional
Hilbert spaces and linear maps is an example).  Preparations,
measurements, either destructive or not, classical data exchange are
all morphisms in that category, and their types fully reflect their
kinds. Correctness properties of standard quantum protocols can be
abstractly proven.

Surprisingly, in this seemingly purely qualitative setting even the
quantitative Born rule arises, that is the rule which tells you how
to calculate the probabilities.  Indeed, each such category has as
endomorphism Hom of the tensor unit an abelian semiring of
`scalars', and a special subset of these scalars will play the role
of weights: each scalar induces a natural transformation which
propagates through physical processes, and when a `state' undergoes
a `measurement', the composition of the corresponding morphisms
gives rise to the weight. Hence the probabilistic weights live
within the category of processes.

J. von Neumann. Mathematische Grundlagen der Quantenmechanik.
Springer-Verlag (1932). English translation in   Mathematical
Foundations of Quantum Mechanics.  Princeton University Press (1955).

S. Abramsky and B. Coecke. A categorical semantics of quantum
protocols. In the proceedings of LiCS'04 (2004). An extended version
is available at arXiv:quant-ph/0402130   A more reader friendly
version entitled `Quantum information flow, concretely, abstractly'
is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf