  We ask how quantum theory compares to more general physical theories from the
point of view of dimension. To do so, we first give two model independent
definition of the dimension of physical systems, based on measurements and on
the capacity of storing information. While both definitions are equivalent in
classical and quantum mechanics, they are in general different in generalized
probabilistic theories. We discuss in detail the case of a theory known as
'boxworld', and show that such a theory features systems with a dimension
mismatch. This dimension mismatch can be made arbitrarily large by using an
amplification procedure. Furthermore, we show that the dimension mismatch of
boxworld has strong consequences on its power for performing
information-theoretic tasks, leading to the collapse of communication
complexity and to the violation of information causality. Finally, we discuss
the consequences of a dimension mismatch from the perspective of
thermodynamics, and ask whether this effect could break Landauer's erasure
principle and thus the second law.
