Qubit - Knowlpedia

Definition

A qubit is a quantum system whose state Hilbert space is two-dimensional over $\mathbb C$ , hence isomorphic after choosing an orthonormal basis to

H\cong\mathbb C^2.

A pure state is a ray in $H$ , equivalently a rank-one density operator, and a general state is a positive trace-one operator $\rho\in M_2(\mathbb C)$ .

States and observables

After choosing $\{|0\rangle,|1\rangle\}$ , a normalized state vector is

|\psi\rangle=\alpha|0\rangle+\beta|1\rangle, \qquad |\alpha|^2+|\beta|^2=1,

with vectors differing by global phase representing the same pure state. Mixed states are density operators.

The observables are the self-adjoint matrices

\mathfrak h_2(\mathbb C)=\{A\in M_2(\mathbb C):A^*=A\}.

With $A\circ B=\tfrac12(AB+BA)$ , they form the complex-qubit Jordan algebra.

Bloch-ball picture

Every density operator is uniquely

\rho=\tfrac12(I+\mathbf r\cdot\boldsymbol\sigma), \qquad \mathbf r\in\mathbb R^3,\quad \|\mathbf r\|\le1.

Pure states form the boundary sphere; mixed states fill the ball.

Relation to the

F_4

construction

In the $F_4$ stabilizer characterization of the Standard Model group, a Jordan subalgebra $X\cong\mathfrak h_2(\mathbb C)$ is interpreted as the observable algebra of a qubit inside a qutrit observable algebra $B\cong\mathfrak h_3(\mathbb C)$ .

References

Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, tenth-anniversary edition, Cambridge University Press, 2010. DOI record. Relevant: §§1.2 and 2.2.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv record. Relevant: §1.