A density tensor hierarchy for open system dynamics: retrieving the noise

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.