  We introduce a density tensor hierarchy for open system dynamics, that
recovers information about fluctuations lost in passing to the reduced density
matrix. For the case of fluctuations arising from a classical probability
distribution, the hierarchy is formed from expectations of products of pure
state density matrix elements, and can be compactly summarized by a simple
generating function. For the case of quantum fluctuations arising when a
quantum system interacts with a quantum environment in an overall pure state,
the corresponding hierarchy is defined as the environmental trace of products
of system matrix elements of the full density matrix. Only the lowest member of
the quantum noise hierarchy is directly experimentally measurable. The unit
trace and idempotence properties of the pure state density matrix imply descent
relations for the tensor hierarchies, that relate the order $n$ tensor, under
contraction of appropriate pairs of tensor indices, to the order $n-1$ tensor.
As examples to illustrate the classical probability distribution formalism, we
consider a quantum system evolving by It\^o stochastic and by jump process
Schr\"odinger equations. As examples to illustrate the corresponding trace
formalism in the quantum fluctuation case, we consider collisional Brownian
motion of an infinite mass Brownian particle, and the weak coupling Born-Markov
master equation. In different specializations, the latter gives the hierarchies
generalizing the quantum optical master equation and the Caldeira--Leggett
master equation. As a further application of the density tensor, we contrast
stochastic Schr\"odinger equations that reduce and that do not reduce the state
vector, and discuss why a quantum system coupled to a quantum environment
behaves like the latter.
