Spatially overlapped partners in quantum field theory

In quantum field theory particles are physically defined as what Unruh-De Witt particle detectors observe. By detecting a particle mode A, a reduced density operator for a quantum state of A is constructed. Even if the entire quantum state of the quantum field is pure, the state of A is not pure but mixed due to entanglement between other subsystems. The partner mode B of the field is defined as a purification partner of A such that the AB system in a pure state. We show that, without any fine-tuning of the particle detector design of A, the weighting function of partner B has spatial overlap of that of A. We show a general formula of partner B associated with arbitrarily fixed A of a free field in a general Gaussian state. We demonstrate an example of memory effects in an expanding Freedman-Robertson-Walker universe.