On the geometry of two qubits

Abstract

Two qubits are considered as a spinor in the four-dimensional complex Hilbert space that describes the state of a four-level quantum system. This system is basic for quantum computation and is described by the generalized Pauli equation, including the generalized Pauli matrices. The generalized Pauli matrices constitute the finite Pauli group \( {{\mathcal{P}}_2} \) for two qubits of order 26 and nilpotency class 2. It is proved that the commutation relation for the Pauli group \( {{\mathcal{P}}_2} \) and the incidence relation in an Hadamard 2-(15, 7, 3) design give rise to equivalent incidence matrices.