Compression-Aware Entanglement Efficiency Rate via Multiple Hamming Weighted States

We propose a notion of compression-aware entanglement efficiency that accounts for the communication cost associated with distributing entangled qubits. Our analysis focuses on a class of pure bipartite quantum states constructed as amplitude-weighted superpositions of Dicke states. By optimizing the amplitude distribution and the choice of Hamming weights, we demonstrate that it is possible to maximize the ratio of entanglement entropy to the number of qubits that must be transmitted. This approach offers a principled way to quantify and enhance the efficiency of entanglement distribution in quantum communication protocols. The proposed symmetric states are highly compressible, and for $n=2,3,4,$ and $6$ qubits, they achieve the optimal entanglement efficiency rates.