2007/04/30 by Stephen L. Adler, Stephen L Adler · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Quantum Information and Cryptography #Quantum many-body systems #Spectroscopy and Quantum Chemical Studies #astro-ph #cond-mat.stat-mech #hep-ph #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/1751-8113/40/30/020
published as J.Phys.A40:8959-8990,2007 · TeX, 58 pages v2: minor editing and typos corrected; v3: some significant revisions and shortening; v4: has final corrections in published version
openalex publication_date 2007/07/12 · arxiv created 2007/07/16 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
We develop a density tensor hierarchy for open system dynamics that recovers information about fluctuations (or 'noise') 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. Whereas all members of the classical noise hierarchy are system observables, 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 spatially isotropic ensemble of spin-1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle and for 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 Schrödinger 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. The descent relations for our various examples are checked in a series of appendices.