  Symmetry plays a central role in many areas of modern physics. Here we show
that it also underpins the dual particle and wave nature of quantum systems. We
begin by noting that a classical point particle breaks translational symmetry
whereas a wave with uniform amplitude does not. This provides a basis for
associating particle nature with asymmetry and wave nature with symmetry. We
derive expressions for the maximum amount of classical information we can have
about the symmetry and asymmetry of a quantum system with respect to an
arbitrary group. We find that the sum of the information about the symmetry
(wave nature) and the asymmetry (particle nature) is bounded by log(D) where D
is the dimension of the Hilbert space. The combination of multiple systems is
shown to exhibit greater symmetry and thus more wavelike character. In
particular, a class of entangled systems is shown to be capable of exhibiting
wave-like symmetry as a whole while exhibiting particle-like asymmetry
internally. We also show that superdense coding can be viewed as being
essentially an interference phenomenon involving wave-like symmetry with
respect to the group of Pauli operators.
