E6and the bipartite entanglement of three qutrits
Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information theory. Examples include: (1) $N=2$ STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is $[SL(2){]}^{3}$ and (2) $N=8$ black holes and the tripartite entanglement of seven qubits where the common symmetry is ${E}_{7}\ensuremath{\supset}[SL(2){]}^{7}$. Here we present another example: $N=8$ black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where the common symmetry is ${E}_{6}\ensuremath{\supset}[SL(3){]}^{3}$. Both the black hole (or black string) entropy and the entanglement measure are provided by the Cartan cubic ${E}_{6}$ invariant. Similar analogies exist for magic $N=2$ supergravity black holes in both four and five dimensions.