Complex-qubit Jordan algebra

Definition

The complex-qubit Jordan algebra is

\mathfrak h_2(\mathbb C) =\{X\in M_2(\mathbb C):X^*=X\}, \qquad X\circ Y=\frac12(XY+YX).

It is the four-dimensional real Euclidean Jordan algebra of observables of a complex two-level quantum system.

Pauli and spin-factor descriptions

Every $X\in\mathfrak h_2(\mathbb C)$ is uniquely

X=\lambda I+u_1\sigma_1+u_2\sigma_2+u_3\sigma_3, \qquad (\lambda,u)\in\mathbb R\oplus\mathbb R^3,

where the $\sigma_i$ are the Pauli matrices. Their anticommutation relations give the spin-factor product, so

\mathfrak h_2(\mathbb C)\cong J(\mathbb R^3).

It is therefore a simple spin factor of rank $2$ .

Spectrum and states

The eigenvalues of $X=\lambda I+u\cdot\sigma$ are $\lambda\pm\|u\|$ . Thus $X$ is positive exactly when $\lambda\geq\|u\|$ . A density matrix has the Bloch representation

\rho=\frac12(I+r\cdot\sigma),\qquad \|r\|\leq1.

Embedding in a qutrit algebra

The upper-left $2\times2$ corner is a Jordan subalgebra of $\mathfrak h_3(\mathbb C)$ . This inclusion is not unital: it sends $I_2$ to $\operatorname{diag}(1,1,0)$ , rather than to $I_3$ .

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.