Quantum Decoherence at Finite Temperatures: Theory and Computations

The decoherence of a finite quantum system $S$ coupled to a finite quantum environment $E$ is considered, where the entirety $S$$+$$E$ is a closed quantum system. The entirety is prepared in a canonical thermal state at a finite temperature. By applying perturbation theory, we find closed form expressions for measures of decoherence and thermalization of $S$ in terms of the free energies of $S$ and $E$. Hence we have quantified how difficult it is to decohere a particular finite quantum system $S$ at a fixed temperature, the result being a function of the free energy of $S$. We have also quantified how potent a particular finite Hilbert space environment $E$ at a fixed temperature is at decohering a generic quantum system. To test these predictions, we performed both real and imaginary time calculations for the Schr{\"o}dinger equation for an entirety with up to 40 quantum spins. The large-scale calculations (vectors in Hilbert space with length up to $2^{40}$$\approx$$10^{12}$) validate our predictions for all temperatures. Preprint arXiv:1502.03996.