  Although quantum logic by using exogenous approach has been proposed for
reasoning about closed quantum systems, an improvement would be worth to study
quantum logic based on density operators instead of unit vectors in the state
logic point of view. In order to achieve this, we build an exogenous quantum
operator logic(EQOL) based on density operators for reasoning about open
quantum systems. We show that this logic is sound and complete. Just as the
exogenous quantum propositional logic(EQPL), by applying exogenous approach,
EQOL is extended from the classical propositional logic, and is used to
describe the state logic based on density operators. As its applications, we
confirm the entanglement property about Bell states by reasoning and logical
argument, also verify the existence of eavesdropping about the basic BB84
protocol. As a novel type of mathematical formalism for open quantum systems,
we introduce an exogenous quantum Markov chain(EQMC) where its quantum states
are labelled using EQOL formulae. Then, an example is given to illustrate the
termination verification problem of a generalized quantum loop program
described using EQMC.
