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Quantum systems loose their properties due to information leaking intoenvironment. On the other hand, we perceive the outer world through theenvironment. We show here that there is a gap between what leaks into theenvironment and what can be extracted from it. We quantify this gap, using theprominent example of the Caldeira-Leggett model, by demonstrating thatinformation extraction is limited by its own lengthscale, calleddistinguishability length, larger than the celebrated thermal de Brogliewavelength, governing the decoherence. We also introduce a new integral kernel,called Quantum Fisher Information kernel, complementing the well-knowndissipation and noise kernels, and show a type of disturbance-information gaintrade-off, similar to the famous fluctuation-dissipation relation. Our resultsshow that the destruction of quantum coherences and indirect observationshappen at two different scales with a "gray zone" in between. This putsintrinsic limitations on capabilities of indirect observations.

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