Complementarity between decoherence and information retrieval from the environment

We address the problem of the fundamental limitations of information extraction from the environment in open quantum systems. We derive a model-independent, hybrid quantum-classical solution of open dynamics in the recoilless limit, which includes environmental degrees of freedom. Specifying to the celebrated Caldeira-Leggett model of hot thermal environments, ubiquitous in everyday situations, we reveal the existence of a new lengthscale, called the distinguishability length, different from the well-known thermal de Broglie wavelength that governs the decoherence. Interestingly, a new integral kernel, called quantum Fisher information kernel, appears in the analysis. It complements the well-known dissipation and noise kernels and satisfies disturbance-information gain type of relations, similar to the famous fluctuation-dissipation relation. Our results complement the existing treatments of the Caldeira-Legget model from a non-standard and highly nontrivial perspective of information dynamics in the environment. This leads to a full picture of how the open evolution looks like from both the system and the environment points of view, as well as sets limits on the precision of indirect observations.

Figure 1

The environment decoheres the central system at a lengthscale ${\lambda }_{\text{dec}}$ (equal to the thermal de Broglie wavelength ${\lambda }_{\text{dB}}$ in the studied example) as a result of dynamical buildup of correlations and information leakage into the environment. However, not all of that information is accessible, the retrieval is limited by its own resolution, the distinguishability length ${\lambda }_{\text{dist}}$. Since ${\lambda }_{\text{dist}}\gg {\lambda }_{\text{dec}}$, part of the information stays bounded in the environment. Decoherence and distinguishability are complementary to each other as reflected by information gain versus disturbance type of a relation: ${\lambda }_{\text{dist}}{\lambda }_{\text{dec}}=$ const. The accompanying timescales satisfy $t_{\text{dist}}/t_{\text{dec}}\sim {({\lambda }_{\text{dist}}/{\lambda }_{\text{dec}})}^2$, so that reaching a given information retrieval resolution takes a much longer time than reaching the same decoherence resolution.