1202.3817 (Tobias Fritz)

On infinite-dimensional state spaces    [PDF]

Tobias Fritz

It is well-known that the canonical commutation relation $[x,p]=i$ can be
realized only on an infinite-dimensional Hilbert space. While any finite set of
experimental data can also be explained in terms of a finite-dimensional
Hilbert space by approximating the commutation relation, Occam's razor prefers
the infinite-dimensional model in which $[x,p]=i$ holds on the nose. This
reasoning one will necessarily have to make in any approach which tries to
detect the infinite-dimensionality. One drawback of using the canonical
commutation relation for this purpose is that it has unclear operational
meaning. Here, we identify an operationally well-defined context from which an
analogous conclusion can be drawn: if two unitary transformations $U,V$ on a
quantum system satisfy the relation $V^{-1}U^2V=U^3$, then
finite-dimensionality entails the relation $UV^{-1}UV=V^{-1}UVU$; this
implication strongly fails in some infinite-dimensional realizations. This is a
result from combinatorial group theory for which we give a new proof. This
proof adapts to the consideration of cases where the assumed relation
$V^{-1}U^2V=U^3$ holds only up to $\eps$ and then yields a lower bound on the
dimension.

View original: http://arxiv.org/abs/1202.3817