The Veldkamp space of multiple qubits

TL;DR

This paper introduces a geometric framework called the Veldkamp space to analyze the commutation relations of multiple qubits' Pauli operators, revealing structures related to quantum entanglement and black hole entropy.

Contribution

It develops a novel geometric approach to encode qubit operator relations and identifies hyperplanes linked to entanglement measures and black hole entropy formulas.

Findings

Identifies geometric hyperplanes corresponding to self-dual operators.

Reveals connections between hyperplanes and entanglement structures like Mermin squares.

Links the geometry of three-qubit hyperplanes to black hole entropy formulas.

Abstract

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.