Decoherence, entanglement negativity, and circuit complexity for an open quantum system

In this paper, we compare the saturation timescales for complexity, linear entropy, and entanglement negativity for two open quantum systems. Our first model is a coupled harmonic oscillator, where we treat one of the oscillators as the bath. The second one is a type of Caldeira-Leggett model, where we consider a one-dimensional free scalar field as the bath. Using these open quantum systems, we discovered that both the complexity of purification and the complexity from operator-state mapping is always saturated for a completely mixed state. More explicitly, the saturation timescale for both types of complexity is smaller than the saturation timescale for linear entropy. On top of this, we found that the saturation timescale for linear entropy and entanglement negativity is of the same order for the Caldeira-Leggett model.