Hitchin functionals are related to measures of entanglement

Abstract

According to the Black Hole/Qubit Correspondence (BHQC) certain black hole entropy formulas in supergravity can be related to multipartite entanglement measures of quantum information.Here we show that the origin of this correspondence is a connection between Hitchin functionals used as action functionals for form theories of gravity related to topological strings, and entanglement measures for systems with a small number of constituents. The basic idea acting as a unifying agent in these seemingly unrelated fields is stability connected to the mathematical notion of special prehomogeneous vector spaces associated to Freudenthal systems coming from simple Jordan algebras. It is shown that the nonlinear function featuring these functionals and defining Calabi-Yau and generalized Calabi-Yau structures is the Freudenthal dual a concept introduced recently in connection with the BHQC. We propose to use the Hitchin invariant for three-forms in 7 dimensions as an entanglement measure playing a basic role in classifying three-fermion systems with seven modes. The representative of the class of maximal tripartite entanglement is the threeform used as a calibration for compactification on manifolds with G 2 holonomy. The idea that entanglement measures are related to action functionals from which the usual correspondence of the BHQC follows at the tree level suggests that one can use the BHQC in a more general context.

Black holes and general Freudenthal transformations

“…x can = U x = U x, (9.63) 34 Generically, this is actually a larger class of charge configurations, but in the special cases of the T 3 , ST 2 , STU models it is precisely the D0D4 subsector. For N = 8, imposing A is diagonal restricts to the D0D4 subsector.…”

“…Freudenthal duality in the context of entanglement and Hitchin functionals can actually be related back to its application to black holes via the black-hole/qubit correspondence[34,[36][37][38][39] …”

Magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy

“…It is related to the quantum theory of topological strings [52]. An alternative form of this functional is given by writing Hitchin's invariant in the following form [54]. Recall our labelling convention (1, 2, 3, 4, 5, 6) = (1, 2, 3, 1, 2, 3).…”

“…For example, in type IIB theory one can consider wrapped D3-branes [55]. The wrapping configurations then can be reinterpreted either as three qubits [56], or more generally, as three-fermion states [54] related to our three-form P , or our observable Π of Eq. (80).…”

“…For more details on the connection between the critical points of the generalized Hitchin functional and black hole entropy we orient the reader to the paper of Pestun [53]. Interestingly, this entropy structure inherently connected to the generalized Hitchin functional has an alternative interpretation in terms of the 32 amplitudes of 4 real, unnormalized threequbit states built up from six qubits [15,54]. This structure is coming from the "tripartite entanglement of seven qubits" interpretation of the (111) quartic invariant of Refs.…”