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

In the information loss problem in black hole evaporation, the investigation of the purification partner of a Hawking particle is crucial. It is a well-known fact that the 3+1 dimensional spherically symmetric gravitational collapse can be mimicked 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 an arbitrary motion for any Gaussian state in a free massless scalar field theory. Using our formula, we directly demonstrate information storage about initial phase information in a pure state of a detected particle and its partner. The form of the partner drastically changes depending on the detailed designs of particle detectors for Hawking radiation. In the case of a detected particle and its partner sensitive to information about initial phase, spatial configurations of the partner has long tails in a stage where only zero-point fluctuation is emitted out of the mirror.