Magic basis

A Bell basis with phases chosen so that local special-unitary transformations of the two qubits are represented by real four-dimensional rotations. For one standard ordering it is ${\Phi }_{+},i{\Phi }_{-},i{\Psi }_{+},{\Psi }_{-}$. Permuting these vectors changes generator conventions but preserves the invariant concurrence $|{\sum }_j{b}_j^2|$.^[ref]

The anti-Hermitian generators of local qubit rotations become real antisymmetric matrices in the magic basis. Their exponentials preserve $b^{\mathsf{T}}b$, and hence its modulus and pure-state concurrence. A common phase changes that bilinear quantity by a phase but not its modulus.