  We propose an extension of the Schr\"odinger equation for a quantum system
interacting with environment. This equation describes dynamics of auxiliary
wave-functions $\mathbf{m}$, from which the system density matrix can be
reconstructed as $\hat{\rho} = \mathbf{m} \mathbf{m}^\dagger$. We formulate a
compatibility condition, which ensures that the reconstructed density satisfies
a given quantum master equation for the system density. The resulting
non-stochastic evolution equation preserves positive-definiteness of the system
density and is applicable to both Markovian and non-Markovian system-bath
treatments. Our formalism also resolves a long-standing problem of energy
non-conservation in the time-dependent variational principle applied to mixed
states of closed systems.
