Two-center black holes, qubits, and elliptic curves

We relate the $U$-duality invariants characterizing two-center extremal black-hole solutions in the $stu$, $s{t}^2$, and $t^3$ models of $N=2$, $d=4$ supergravity to the basic invariants used to characterize entanglement classes of four-qubit systems. For the elementary example of a D0D4-D2D6 composite in the $t^3$ model we illustrate how these entanglement invariants are related to some of the physical properties of the two-center solution. Next we show that it is possible to associate elliptic curves to charge configurations of two-center composites. The hyperdeterminant of the hypercube, a four-qubit polynomial invariant of order 24 with 2 894 276 terms, is featuring the $j$ invariant of the elliptic curve. We present some evidence that this quantity and its straightforward generalization should play an important role in the physics of two-center solutions.