  Small, controllable quantum systems, known as quantum probes, have been
proposed to estimate various parameters characterizing complex systems such as
the environments of quantum systems. These probes, prepared in some initial
state, are allowed to interact with their environment, and subsequent
measurements reveal information about different quantities characterizing the
environment such as the system-environment coupling strength, the cutoff
frequency, and the temperature. These estimates have generally been made by
considering only the way that the probe undergoes decoherence. However, we show
that information about the environment is also imprinted on the probe via the
probe and environment correlations that exist before the probe state
preparation. This information can then be used to improve our estimates for any
environment. We apply this general result to the particular case of a two-level
system probe undergoing pure dephasing, due to a harmonic oscillator
environment, to show that a drastic increase in the quantum Fisher information,
and hence the precision of our estimates, can indeed be obtained. We also
consider applying periodic control pulses to the probe to show that with a
combination of the two - the effect of the control pulses as well as the
initial correlations - the quantum Fisher information can be increased by
orders of magnitude.
