Strings, black holes, the tripartite entanglement of seven qubits, and the Fano plane

Recently it has been observed that the group $E_7$ can be used to describe a special type of quantum entanglement of seven qubits partitioned into seven tripartite systems. Here we show that this curious type of entanglement is entirely encoded into the discrete geometry of the Fano plane. We explicitly work out the details concerning a qubit interpretation of the $E_7$ generators as representatives of tripartite protocols acting on the 56-dimensional representation space. Using these results we extend further the recently studied analogy between quantum information theory and supersymmetric black holes in four-dimensional string theory. We point out that there is a dual relationship between entangled subsystems containing three and four tripartite systems. This relationship is reflected in the structure of the expressions for the black hole entropy in the $N=4$ and $N=2$ truncations of the $E_{7(7)}$ symmetric area form of $N=8$ supergravity. We conjecture that a similar picture based on other qubit systems might hold for black hole solutions in magic supergravities.

Figure 1

The Fano plane with seven copies of the two-dimensional complex vector space $V$ corresponding to qubits $A$, $B$, $C$, $D$, $E$, $F$, and $G$ attached to its vertices.

Figure 2

The Fano plane with the associated qubits labeled by elements $\sigma$ of ${\mathbf{Z}}_2^3$.

Figure 3

The Fano plane with the tripartite spaces ${\mathcal{H}}_{\sigma }$, $\sigma \in {\mathbf{Z}}_2^3$ associated to its lines.

Figure 4

The multiplication table of the octonions as represented by the oriented Fano plane.

Figure 5

The dual Fano plane. To its points now we attached tripartite states from the spaces ${\mathcal{H}}_{\sigma }$, $\sigma \in {{\mathbf{Z}}_{\mathbf{2}}}^3-(000)$ with their complex amplitudes labeled as in Ref. 8. See also Eq. (41).