Strings, Black Holes, and Quantum Information

TL;DR

This work establishes a deep correspondence between extremal black holes in string theory and entanglement invariants of 2- and 3-qubit systems. It shows that axion-dilaton BH entropy maps to 2-qubit concurrence, while STU and, more generally, N=8 black hole entropy map to the 3-tangle and Cayley’s hyperdeterminant, linking to the quartic $E_{7(7)}$ invariant. The authors provide explicit charge-to-qubit dictionaries, discuss duality invariances, twistors, and classify BH solutions into small (zero classical entropy) and large (nonzero entropy) categories, with simple expressions for quantum-corrected entropy in terms of norms and 2-tangles for small BHs. Overall, the paper offers a unifying information-theoretic perspective on black hole entropy and suggests avenues for cross-pollination between quantum information and high-energy gravity, including connections to octonions and twistor geometry.

Abstract

We find multiple relations between extremal black holes in string theory and 2- and 3-qubit systems in quantum information theory. We show that the entropy of the axion-dilaton extremal black hole is related to the concurrence of a 2-qubit state, whereas the entropy of the STU black holes, BPS as well as non-BPS, is related to the 3-tangle of a 3-qubit state. We relate the 3-qubit states with the string theory states with some number of D-branes. We identify a set of "large" black holes with the maximally entangled GHZ-class of states and "small" black holes with separable, bipartite and W states. We sort out the relation between 3-qubit states, twistors, octonions, and black holes. We give a simple expression for the entropy and the area of stretched horizon of "small'' black holes in terms of a norm and 2-tangles of a 3-qubit system. Finally, we show that the most general expression for the black hole and black ring entropy in N=8 supergravity/M-theory, which is given by the famous quartic Cartan E_{7(7)} invariant, can be reduced to Cayley's hyperdeterminant describing the 3-tangle of a 3-qubit state.

Figures (10)

• Figure 1: The $2\times 2\times 2$ matrix corresponding to the quantum state (\ref{['funct']}).
• Figure 2: The $2\times 2\times 2$ matrix corresponding to supergravity black holes Behrndt:1996hu.
• Figure 3: The $2\times 2\times 2$ matrix corresponding to twistor picture of a 3-cubit in Levay. The combination of 4 upper corners forms a 4-vector $Z$. All lower corners are used to form a 4-vector $W$.
• Figure 4: The $2\times 2\times 2$ matrix corresponding to supergravity black holes Behrndt:1996hu in the hatted basis, $\hat{p}$ and $\hat{q}$. One has to slice this cube vertically so that the back side is cut from the front side. In this way we will separate the 4 corners in the front forming a $\hat{p}$ vector and the 4 corners in the back forming a $\hat{q}$ vector.
• Figure 5: The $2\times 2\times 2$ matrix with all entries vanishing except one, e.g. $q_0$. We show it by a corner with a circle. This corresponds to a non-entangled completely separable state describing a black hole with just one charge, $q_0$, with vanishing area of the horizon.