$STU$ black holes as four-qubit systems

In this paper we describe the structure of extremal stationary spherically symmetric black-hole solutions in the $STU$ model of $D=4$, $N=2$ supergravity in terms of four-qubit systems. Our analysis extends the results of previous investigations based on three qubits. The basic idea facilitating this four-qubit interpretation is the fact that stationary solutions in $D=4$ supergravity can be described by dimensional reduction along the time direction. In this $D=3$ picture the global symmetry group $SL(2,\mathbb{R}{)}^{\times 3}$ of the model is extended by the Ehlers $SL(2,\mathbb{R})$ accounting for the fourth qubit. We introduce a four-qubit state depending on the charges (electric, magnetic, and Newman-Unti-Tamburino), the moduli, and the warp factor. We relate the entanglement properties of this state to different classes of black-hole solutions in the $STU$ model. In the terminology of four-qubit entanglement extremal black-hole solutions correspond to nilpotent, and nonextremal ones to semisimple states. In arriving at this entanglement-based scenario the role of the four algebraically independent four-qubit $SL(2,\mathbb{C})$ invariants is emphasized.