Symmetries and Geometries of Qubits, and their Uses

TL;DR

This paper explores the symmetries and geometries of multiple qubits, connecting Lie algebraic, geometric, and algebraic approaches to deepen understanding of quantum information structures.

Contribution

It unifies Lie algebraic and geometric perspectives on qubit symmetries, linking SU(4) generators to finite geometries and combinatorial designs, and extends these concepts to multi-qubit systems.

Findings

SU(4) generators correspond to finite projective geometries

Connections established between Lie groups and combinatorial designs

Framework extended to multi-qubit and higher-dimensional systems

Abstract

The symmetry SU(2) and its geometric Bloch Sphere rendering are familiar for a qubit (spin-1/2) but extension of symmetries and geometries have been investigated far less for multiple qubits, even just a pair of them, that are central to quantum information. In the last two decades, two different approaches with independent starting points and motivations have come together for this purpose. One was to develop the unitary time evolution of two or more qubits for studying quantum correlations, exploiting the relevant Lie algebras and especially sub-algebras of the Hamiltonians involved, and arriving at connections to finite projective geometries and combinatorial designs. Independently, geometers studying projective ring lines and associated finite geometries have come to parallel conclusions. This review brings together both the Lie algebraic and group representation perspective of quantum physics and the geometric algebraic one, along with connections to complex quaternions. Together, all this may be seen as further development of Felix Klein's Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two-qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives may provide further insight into problems in quantum information. Extensions are considered for multiple qubits and higher spin or higher dimensional qudits.