Probabilistic Unitary Formulation of Open Quantum System Dynamics*

We show explicitly that for any continuously evolving open quantum system, be it finite (d-dimensional) or countably infinite dimensional,  its dynamics can be described by a time-dependent Hamiltonian and probabilistic combinations of up to d-1 (d → ∞ for infinite dimensional case), instead of d²-1, time-dependent unitary operators, resulting in a quadratic improvement in simulation resources. Importantly, both types of operations must be initial state-dependent in general, and thus the simulation is tailored to that initial state. Such description is exact under all cases, and does not rely on any assumptions other than the continuity and differentiability of the density matrix. It turns out that upon generalizations, the formalism can also be used to describe general quantum channels, which may not be complete positive or even positive, and results in a Kraus-like representation. Experimentally, the formalism provides a scheme to control a quantum state to evolve along designed quantum trajectories, and can be particularly useful in quantum computing and quantum simulation scenes since only unitary resources are needed for implementation. Philosophically,  it provides us with a new perspective to understand the dynamics of open quantum systems and related problems such as decoherence and quantum measurement, i.e. the non-unitary evolution of quantum states can thereby be regarded as the combined effect of state-dependent deterministic evolutions and probabilistic applications of unitary operators.