How does the Entropy of Entanglement generalize to tripartite and multipartite pure states?

Question

Entropy of Entanglement is a very beautiful measure of entanglement.

I wonder how this concept generalizes to three-parite entangled states, such as $$ \left| \psi \right\rangle=\frac{1}{\sqrt{2}}\left(\left| 0,0,0 \right\rangle + \left| 1,1,1 \right\rangle \right) $$ or for higher-dimensional systems $$ \left| \psi \right\rangle=\frac{1}{\sqrt{3}}\left(\left| 0,0,0 \right\rangle + \left| 1,1,1 \right\rangle + \left| 2,2,2 \right\rangle \right) $$ or for asymmetrically entangled pure states $$ \left| \psi \right\rangle=\frac{1}{\sqrt{3}}\left(\left| 0,0,0 \right\rangle + \left| 1,0,1 \right\rangle + \left| 2,1,0 \right\rangle \right). $$

Question 1: What is the Entropy of Entanglement of these three states?

Question 2: If Entropy of Entanglement cannot be calculated for pure, tripartite states, are there related measures that I can apply to these three-partite pure states?

Thank you very much!

Answer 1

For a bipartite state $\vert\psi\rangle_{AB}$, the entropy of entanglement (EoE) is defined as $$ E(\vert\psi\rangle) = S(\mathrm{tr}_B\vert\psi\rangle\langle\psi\vert)\ , $$ with $S(\rho)=-\mathrm{tr}(\rho\log\rho)$ the von Neumann entropy.

Why is the entropy of entanglement such a great measure for the entanglement in a pure bipartite state? The reason is that it uniquely quantifies the entanglement in a bipartite state in an asymptotic setting -- just as in classical information theory the Shannon entropy quantifies the information content (=compressibility) of a data souce. Specifically, given many copies $N_1$ of a state $\vert\psi_1\rangle$, they can be asymptotically converted with arbitrary accuracy to $N_2$ copies of another state $\vert\psi_2\rangle$, and back, using only local operations and classical communication (LOCC), as long as the ratio $$ \frac{N_1}{N_2}\to \frac{E(\vert\psi_2\rangle)}{E(\vert\psi_1\rangle)}\ , $$ or equivalently $$ N_1E(\vert\psi_1\rangle)\approx N_2E(\vert\psi_2\rangle)\ . $$

Unfortunately, for systems consisting of three parties or more, this is no longer possible: A seminal result by Dür, Vidal, and Cirac is that for three qubits, there exist inequivalent classes of entangled states, which cannot be converted into each other with LOCC at all (with the GHZ and the W state as prominent representatives). Thus, there cannot exist a single number which measures the entanglement in tripartite (or more comlex) states in a meaningful way.