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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!

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  • $\begingroup$ I disagree with your first sentence: EoF is an important concept for MIXED states. For pure states, EoF, distillable entanglement, etc., are all the same concept: The "entropy of entanglement". (Which is indeed a beautiful thing!) -- So is your question about pure or mixed states? $\endgroup$ – Norbert Schuch Oct 5 at 8:35
  • $\begingroup$ @NorbertSchuch Thanks, i changed the first sentence to avoid confusion. I am only interested in pure states, and want to see how this generalizes to more than two particles, especially in a higher state-space. Thank you! $\endgroup$ – NiceDean Oct 6 at 2:40
  • $\begingroup$ I think the short answer is that this concept does not generalize, at least not with the same simplicity and elegance: Three- and multi-partite systems are complicated. $\endgroup$ – Norbert Schuch Oct 6 at 11:31
  • $\begingroup$ @NorbertSchuch My specific question, as stated in the main text, is the entropy of entanglement of the three states. They are the most simplest examples of multiparty states, i cannot believe that this measure (or a related measure) is not applicable. Can you please provide references for your claims? $\endgroup$ – NiceDean Oct 6 at 18:12
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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.

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  • $\begingroup$ Thank you, i am aware of this paper and the result. I am interested in a value that generalizes, and it is not excluded by the result you posted. For example, 3*log2(2), 3*log2(3), log2(3*2*2) would be the sum of the entropy of each bipartition for the three example states. I wonder whether this quantity has a better interpretation than the sum of the entropies. I will add a bounty to this question, and you are welcome to extend your answer to adresse my question more directly. Thank you! $\endgroup$ – NiceDean Oct 7 at 23:31
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    $\begingroup$ How should one answer "What is the Entropy of Entanglement of these three states?" -- The point is exactly that the concept "Entropy of Entanglement", in the way it works for bipartite systems, does not apply for multipartite systems. This is the point of my answer. If you want to generalize it in some other way, you have to specify it. Otherwise, it is like asking "What is the taste of the above states?". -- Otherwise, there is a zoo of multipartite entanglement measures. Do you want all of them? One of them? What is your figure of merit for judging them? $\endgroup$ – Norbert Schuch Oct 8 at 9:57
  • $\begingroup$ hi norbert, thanks. i asked two unambiguous questions, for which's convincing answer i will award the bounty (if there will be such an answer). please feel free to extend you question, in case you know the answer for my two questions. thank you. $\endgroup$ – NiceDean Oct 9 at 2:39

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