Measurement of a Quantum System Coupled to Independent Heat-Bath and Pointer Modes

We present an exact derivation of a process in which a microscopic measured system interacts with heat-bath and pointer modes of a measuring device, via a coupling involving a general Hermitian operator $\mathrm{\Lambda }$ of the system. In the limit of strong interaction with these modes, over a small time interval, we derive the exact effective many-body density matrix of the measured system plus pointer. We then discuss the interpretation of the dynamics considered as the first stage in the process of quantum measurement, eventually involving the wave-function collapse due to interactions with “the rest of the universe.” We establish that the effective density matrix represents the required framework for the measured system and the pointer part of the measuring device to evolve into a statistical mixture described by direct-product states such that the system is in each eigenstate of $\mathrm{\Lambda }$ with the correct quantum-mechanical probability, whereas the expectation values of pointer-space operators retain amplified information of the system’s eigenstate.