Minimal control power of controlled dense coding and genuine tripartite entanglement

We investigate minimal control power (MCP) for controlled dense coding defined by the channel capacity. We obtain MCPs for extended three-qubit Greenberger-Horne-Zeilinger (GHZ) states and generalized three-qubit $W$ states. Among those GHZ states, the standard GHZ state is found to maximize the MCP and so does the standard $W$ state among the $W$-type states. We find the lower and upper bounds of the MCP and show for pure states that the lower bound, zero, is achieved if and only if the three-qubit state is biseparable or fully separable. The upper bound is achieved only for the standard GHZ state. Since the MCP is nonzero only when a three-qubit entanglement exists, this quantity may be a good candidate to measure the degree of genuine tripartite entanglement.