# Entanglement of formation for two qutrits

For a pure bipartite state $|\psi\rangle\in\mathbb C^d\otimes \mathbb C^d$, the entanglement is given by $E(|\psi\rangle) = S(\mathrm{tr}_B|\psi\rangle\langle\psi|)$, with $S(\rho)=-\mathrm{tr}(\rho\log\rho)$ the von Neumann entropy. For an arbitrary bipartite mixed state $\rho$, the Entanglement of Formation (EoF) is defined as the minimum average entanglement of an ensemble of pure states that represents the given mixed state. For two qubits, and explicit formula for the Entanglement of Formation is known.

I would like to compute the Entanglement of Formation for an arbitrary mixed state of two qutrits. Are there any known results for non-trivial upper and lower bounds for the entanglement of formation of two qutrits?

• I edited your question for clarity; let me know if this is ok. – Norbert Schuch Jan 2 '17 at 13:25
• Regarding the question, have you looked at SDP relexations? There are also results for special families such as isotropic states which give bounds. – Norbert Schuch Jan 2 '17 at 13:26
• I don't quite see why the (edited) version of the question has been closed as unclear what you're asking. (This was different for the unedited one, which however only acquired two close votes IIRC.) I think this is a very clear question, though it is not entirely clear if there is a simple answer. (I was going to try to to write one, though.) – Norbert Schuch Jan 2 '17 at 21:17
• @NorbertSchuch Two qutrits... in what state? Separable? Maximally entangled? The OP is ill-posed as currently phrased. – Emilio Pisanty Jan 3 '17 at 8:17
• @NorbertSchuch that's a very different (and much better) question now. – Emilio Pisanty Jan 3 '17 at 15:21