Probabilistic Unitary Formulation of Open Quantum System Dynamics

We show that all non-relativistic quantum processes, whether open or closed, are either unitary or probabilistic unitary, i.e., probabilistic combination of unitary evolutions. This means that for open quantum systems, its continuous dynamics can always be described by the Lindblad master equation with all jump operators being unitary. We call this formalism the probabilistic unitary formulation of open quantum system dynamics. This formalism is shown to be exact under all cases, and does not rely on any assumptions other than the continuity and differentiability of the density matrix. Moreover, it requires as few as $d-1$ jump operators, instead of $d^2-1$, to describe the open dynamics in the most general case, where $d$ is the dimension of Hilbert space of the system. Importantly, different from the conventional Lindblad master equation, this formalism is state-dependent, meaning that the Hamiltonian, jump operators, and rates, in general all depend on the current state of the density matrix. Hence one needs to know the explicit expression of the density matrix in order to write down the probabilistic unitary master equation explicitly. Experimentally, the formalism provides a scheme to control a quantum state to evolve along designed non-unitary quantum trajectories, and can be potentially useful in quantum computing and quantum control scenes since only unitary resources are needed for implementation.