  Given an ensemble of identical pure quantum states that depend on an unknown
parameter, recently it was shown that the quantum Fisher information can be
losslessly compressed into a subensemble with a much smaller number of samples.
However, generalization to mixed states leads to a technical challenge that is
formidable to overcome directly. In this work, we avoid such technicality by
unveiling the physics of a featured lossless post-selection measurement: while
the post-selected quantum state is unchanged, the parametric derivative of the
density operator is amplified by a large factor equal to the square root of the
inverse of the post-selection success probability. This observation not only
clarifies the intuition and essence of post-selected quantum metrology but also
allows us to develop a mathematically compact theory for the lossless
post-selection of mixed states. We find that if the parametric derivative of
the density operator of a mixed state, or alternatively the symmetric
logarithmic derivative, vanishes on the support of the density matrix, lossless
post-selection can be achieved with an arbitrarily large amplification factor.
We exemplify with the examples of superresolution imaging and unitary encoding
of mixed initial states. Our results are useful for realistic post-selected
quantum metrology in the presence of decoherence and of foundational interests
to several problems in quantum information theory.
