Dynamics of initially entangled open quantum systems

Linear, trace and hermiticity preserving, maps of density matrices that describe the evolution of open quantum systems initially entangled to parts of their environment are studied. Complete positivity is an additional property that is often attributed to maps describing the evolution of open quantum dynamics. If there is initial entanglement the dynamical maps are found to be not completely positive. They are not even positive unless the domain of action of the maps is restricted. The initial entanglement of the system means that only subset of states, called the compatibility domain, are allowed and it gives a physical reason for restricting the domain of action of the map. The maps we obtain are shown to be positive on the compatibility domain. An example for two initially entangled qubits is worked out in detail. The maps are obtained first as maps between expectation values of observables of the system and then generalized to maps between basis matrices. The maps are also studied as affine transformations on the space of states of the system. An operator sum representation similar to that of completely positive maps is constructed for the maps obtained here and a parameterization of the maps given. The reasons commonly cited for stipulating that open quantum dynamics be described exclusively in terms of completely positive maps are analyzed and found inconclusive in light of the understanding gained here. We find that positive as well as not positive maps are good candidates for describing open quantum evolution.