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.