  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 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.
