The initial system-bath state via the maximum-entropy principle

TL;DR

This paper develops a maximum-entropy based method to determine the initial system-bath state in quantum systems, improving upon the traditional uncorrelated assumption, with implications for understanding non-completely-positive dynamics.

Contribution

It introduces a maximum-entropy approach to estimate initial correlated system-bath states, providing explicit formulas for weak interactions and quantifying system-bath correlations.

Findings

The initial state often has minimal system-bath correlation.

Explicit correction formulas improve initial state modeling.

Deviations from predictions indicate hidden bath control.

Abstract

The initial state of a system-bath composite is needed as the input for prediction from any quantum evolution equation to describe subsequent system-only reduced dynamics or the noise on the system from joint evolution of the system and the bath. The conventional wisdom is to write down an uncorrelated state as if the system and the bath were prepared in the absence of each other; yet, such a factorized state cannot be the exact description in the presence of system-bath interactions. Here, we show how to go beyond the simplistic factorized-state prescription using ideas from quantum tomography: We employ the maximum-entropy principle to deduce an initial system- bath state consistent with the available information. For the generic case of weak interactions, we obtain an explicit formula for the correction to the factorized state. Such a state turns out to have little correlation between the system and the bath, which we can quantify using our formula. This has implications, in particular, on the subject of subsequent non-completely-positive dynamics of the system. Deviation from predictions based on such an almost uncorrelated state is indicative of accidental control of hidden degrees of freedom in the bath.