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Qubits, Bloch Sphere, Pauli Algebra, and One-Qubit Gates — Examples

1 · Prerequisites

2 · Summary

The companion is a worked finite-matrix check. It computes all three Pauli trace coordinates for $∣ 0 ⟩$ and $H ∣ 0 ⟩ = ∣ + ⟩$ , verifies that both projectors are density operators, checks their radii and purity, and uses only the A-page definitions and proved results. This page is a logical leaf and supplies no prerequisite to an A page.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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5 · Examples, counterexamples and false statements

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Bloch coordinates of |0> and H|0>

Example

Let $ρ_{0} = ∣ 0 ⟩ ⟨ 0 ∣$ . Apply $H$ to obtain $∣ + ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) / \sqrt{2}$ and $ρ_{+} = ∣ + ⟩ ⟨ + ∣$ . The Bloch vectors are $(0, 0, 1)$ and $(1, 0, 0)$ , respectively, and both states lie on the Bloch sphere.

Facts & Assumptions

Given: The ordered qubit basis $(∣ 0 ⟩, ∣ 1 ⟩)$ and the matrices $X, Y, Z, H$ of the preceding definitions.

[F1]

Qubit spaces and states: $∣ 0 ⟩ = e_{0}$ and $∣ 1 ⟩ = e_{1}$ are the ordered orthonormal basis vectors of $Q_{2}$ .

[F2]

Pure state and ray: for a unit representative $u$ , the pure-state projector is $∣ u ⟩ ⟨ u ∣$ .

[F3]

Dirac Notation Conversion: $∣ u ⟩ ⟨ v ∣$ sends $z$ to $⟨ z, v ⟩_{r o o t} u$ ; in the standard coordinate basis its matrix is $[u] [v]^{†}$ .

[F4]

The Pauli matrices: $X, Y, Z$ are the displayed $2 \times 2$ matrices.

[F5]

Exact one-qubit gates and rotations: $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ .

[F6]

Bloch coordinate maps: $r_{j} (ρ) = tr (ρ σ_{j})$ and $ρ (r) = \frac{1}{2} (I + \sum_{j} r_{j} σ_{j})$ .

[F7]

Existence and uniqueness of Bloch coordinates: every Hermitian trace-one $2 \times 2$ matrix has a unique Bloch vector.

[F8]

Purity is the boundary of the Bloch ball: a qubit density operator is pure exactly when its Bloch vector has Euclidean norm one.

[F9]

The computational basis is orthonormal: $∣ 0 ⟩, ∣ 1 ⟩$ are orthonormal.

[F10]

Ray-to-projector is well-defined and bijective: every unit-vector ray projector is non-negative with trace one, hence is a density operator.

Verification

1.1F1F2F3F4F6F9F10algebra

By [F1] and [F9], $∣ 0 ⟩ = (1, 0)^{T}$ is a unit vector; [F2]–[F3] give the rank-one projector convention, so direct coordinates give $ρ_{0} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ . Multiplication by [F4] gives $tr (ρ_{0} X) = 0$ , $tr (ρ_{0} Y) = 0$ , and $tr (ρ_{0} Z) = 1$ , so [F6] gives $r (ρ_{0}) = (0, 0, 1)$ . The matrix satisfies $ρ_{0}^{2} = ρ_{0}$ and $tr (ρ_{0}) = 1$ ; by [F2] it is the projector of the pure ray $[∣ 0 ⟩]$ , and [F10] makes it a density operator. Thus also $tr (ρ_{0}^{2}) = 1$ .

1.2F1F2F3F4F5F6F9F10algebra

By [F1] and [F5], $H ∣ 0 ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) / \sqrt{2} = ∣ + ⟩$ . From [F9], $∥ + ∥^{2} = \frac{1}{2} (⟨ 0, 0 ⟩_{r o o t} + ⟨ 0, 1 ⟩_{r o o t} + ⟨ 1, 0 ⟩_{r o o t} + ⟨ 1, 1 ⟩_{r o o t}) = 1$ , so [F2]–[F3] give $ρ_{+} = ∣ + ⟩ ⟨ + ∣ = \frac{1}{2} (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ . Multiplying by [F4] gives $tr (ρ_{+} X) = 1$ , $tr (ρ_{+} Y) = 0$ , and $tr (ρ_{+} Z) = 0$ , so [F6] gives $r (ρ_{+}) = (1, 0, 0)$ . Direct multiplication gives $ρ_{+}^{2} = ρ_{+}$ and $tr (ρ_{+}) = 1$ ; by [F2], $ρ_{+}$ is a pure-ray projector, so [F10] makes it a density operator and $tr (ρ_{+}^{2}) = 1$ .

2.1F7F8F10step 1.1step 1.2algebra∎

The computed vectors have norms $∥ (0, 0, 1) ∥_{2} = \sqrt{0^{2} + 0^{2} + 1^{2}} = 1$ and $∥ (1, 0, 0) ∥_{2} = \sqrt{1^{2} + 0^{2} + 0^{2}} = 1$ . Steps 1.1 and 1.2 prove both matrices are density operators. By [F7] these are the unique Bloch vectors, and [F8] confirms both states are pure and lie on the Bloch sphere.