  We develop a unified, information theoretic interpretation of the
number-phase complementarity that is applicable both to finite-dimensional
(atomic) and infinite-dimensional (oscillator) systems, with number treated as
a discrete Hermitian observable and phase as a continuous positive operator
valued measure (POVM). The relevant uncertainty principle is obtained as a
lower bound on {\it entropy excess}, $X$, the difference between the entropy of
one variable, typically the number, and the knowledge of its complementary
variable, typically the phase, where knowledge of a variable is defined as its
relative entropy with respect to the uniform distribution. In the case of
finite dimensional systems, a weighting of phase knowledge by a factor $\mu$
($> 1$) is necessary in order to make the bound tight, essentially on account
of the POVM nature of phase as defined here. Numerical and analytical evidence
suggests that $\mu$ tends to 1 as system dimension becomes infinite. We study
the effect of non-dissipative and dissipative noise on these complementary
variables for oscillator as well as atomic systems.
