  The theoretical basis of the phenomenon of effective and exact dimensional
reduction, or holographic correspondence, is investigated in a wide variety of
physical systems. We first derive general inequalities linking quantum systems
of different spatial (or spatio-temporal) dimensionality, thus establishing
bounds on arbitrary correlation functions. These bounds enforce an {\em
effective} dimensional reduction and become most potent in the presence of
certain symmetries. {\em Exact} dimensional reduction can stem from a duality
that (i) follows from properties of the local density of states, and/or (ii)
from properties of Hamiltonian-dependent algebras of interactions. Dualities of
the first type (i) are illustrated with large-$n$ vector theories whose local
density of states may remain invariant under transformations that change the
dimension. We argue that a broad class of examples of dimensional reduction may
be understood in terms of the functional dependence of observables on the local
density of states. Dualities of the second type (ii) are obtained via {\em bond
algebras}, a recently developed algebraic tool. We apply this technique to
systems displaying topological quantum order, and also discuss the implications
of dimensional reduction for the storage of quantum information.
