Propagation of initial uncertainties to Arthurs-Kelly inequality

The generalized version of the Arthurs-Kelly inequality is derived when the initial state is a tripartite separable state. When each initial substate obeys the minimal uncertainty, the generalized version reduces to the well-known inequality, i.e. twice of the Heisenberg uncertainty. If the initial probe state is entangled, it is shown that the generalized version of the Arthurs-Kelly inequality can be violated. We show the violation explicitly by introducing a special example.