# Minimal control power of controlled dense coding and genuine tripartite entanglement

## TL;DR

This paper studies the minimal control power in controlled dense coding for three-qubit states, identifying bounds and linking MCP to genuine tripartite entanglement, with the standard GHZ state maximizing MCP.

## Contribution

It derives bounds for minimal control power in controlled dense coding and connects MCP to the degree of genuine tripartite entanglement, highlighting the standard GHZ state as optimal.

## Key findings

- Standard GHZ state maximizes MCP among GHZ states.
- MCP is zero for biseparable or fully separable states.
- Upper bound of MCP achieved only by the standard GHZ state.

## Abstract

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.