Bound information in the environment: Environment learns more than it will reveal

Quantum systems loose their properties due to information leaking into environment. On the other hand, we perceive the outer world through the environment. We show here that there is a gap between what leaks into the environment and what can be extracted from it. We quantify this gap, using the prominent example of the Caldeira-Leggett model, by demonstrating that information extraction is limited by its own lengthscale, called distinguishability length, larger than the celebrated thermal de Broglie wavelength, governing the decoherence. We also introduce a new integral kernel, called Quantum Fisher Information kernel, complementing the well-known dissipation and noise kernels, and show a type of disturbance-information gain trade-off, similar to the famous fluctuation-dissipation relation. Our results show that the destruction of quantum coherences and indirect observations happen at two different scales with a "gray zone" in between. This puts intrinsic limitations on capabilities of indirect observations.Comment: 10 pages, 1 figur