Partner formula for an arbitrary moving mirror in $1+1$ dimensions

In the information loss problem in black hole evaporation, investigation of the purification partner of a Hawking particle is crucial. It is a well-known fact that 3+1 dimensional spherically symmetric gravitational collapse can be approximately described by 1+1 dimensional moving mirror models. Since a detected particle in field theory is defined by what a particle detector observes, the diversity of detector designs yields a variety of particles and their partners. We provide a formula of generalized partners of detected particles emitted out of mirrors in arbitrary motion in a free massless scalar field theory. Using the formula, we directly demonstrate information storage about pre-thermal information in a pure state of a Hawking particle and its partner. The partner form drastically changes depending on the detailed designs of particle detectors for the Hawking radiation. In a case of the Hawking particle and its partner sensitive to information about pre-thermal era, spatial configurations of the partner has long tails in a stage where only zero-point fluctuation is emitted out of the mirror.