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Existence and uniqueness of Bloch coordinates

Statement

For every $r \in R^{3}$ , the matrix $ρ (r)$ of Bloch coordinate maps is Hermitian with trace one. Conversely, every Hermitian trace-one matrix $ρ \in M_{2} (C)$ has a unique vector $r \in R^{3}$ such that $ρ = ρ (r) = \frac{1}{2} (I + r_{x} X + r_{y} Y + r_{z} Z) .$ For a qubit density operator this vector is exactly $(tr (ρ X), tr (ρ Y), tr (ρ Z))$ , the coordinate map of Bloch coordinate maps. In particular all three trace coordinates of every density operator are real.

Facts & Assumptions

Given: The ordered basis $(∣ 0 ⟩, ∣ 1 ⟩)$ of $Q_{2}$ , a vector $r \in R^{3}$ , a Hermitian trace-one matrix $ρ$ , and the definitions in Bloch coordinate maps.

[F1]

Qubit spaces and states: $Q_{2} = C^{2}$ with ordered computational basis $(e_{0}, e_{1})$ .

[F2]

The computational basis is orthonormal: the computational basis $(e_{0}, e_{1})$ is orthonormal.

[F3]

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space: Hermitian means self-adjoint, so $ρ^{*} = ρ$ .

[F4]

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: in an orthonormal basis the matrix of the adjoint is the conjugate transpose.

[F5]

Density operator and Non-negative and positive operators: a density operator has trace one and is non-negative; non-negative means self-adjoint with nonnegative quadratic form.

[F6]

The Pauli matrices: $X, Y, Z$ have the displayed $2 \times 2$ matrices in the ordered computational basis.

[F7]

Pauli matrix algebra and trace identities: $tr (σ_{j} σ_{k}) = 2 δ_{j k}$ and $tr (σ_{j}) = 0$ .

[F8]

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

[F9]

The trace $tr (A)$ as the sum of the diagonal entries: trace is the sum of the diagonal entries.

Proof

1.1givenF1F6F8F9algebra

For arbitrary real $r$ , the matrices in [F6] give $ρ (r) = \frac{1}{2} (\begin{array}{cc} 1 + r_{z} & r_{x} - i r_{y} \\ r_{x} + i r_{y} & 1 - r_{z} \end{array})$ . Its diagonal entries are real, its lower-left entry is the conjugate of the upper-right, and its trace is $1$ , so it is Hermitian trace one.

1.2givenF1F2F3F4F6F8algebra

Write the arbitrary Hermitian trace-one matrix $ρ = (\begin{array}{cc} a & z \\ \overline{z} & 1 - a \end{array})$ with $a \in R$ : [F2] makes the computational basis orthonormal, and [F3]–[F4] give the conjugate-transpose matrix criterion for $ρ^{*} = ρ$ , so its diagonal entries are real and its lower-left entry is the conjugate of the upper-right; trace one fixes the second diagonal entry. Put $r_{x} = 2 Re z$ , $r_{y} = - 2 Im z$ , and $r_{z} = 2 a - 1$ , which are real. The matrices in [F6] and formula in [F8] give $ρ (r) = \frac{1}{2} (\begin{array}{cc} 1 + r_{z} & r_{x} - i r_{y} \\ r_{x} + i r_{y} & 1 - r_{z} \end{array}) = (\begin{array}{cc} a & z \\ \overline{z} & 1 - a \end{array}) = ρ$ . This proves existence.

2.1F5F6F7F8F9step 1.2algebra∎

For the entries $a, z$ of step 1.2, direct multiplication using [F6] and summing diagonal entries by [F9] gives $tr (ρ X) = z + \overline{z} = 2 Re z = r_{x}$ , $tr (ρ Y) = i (z - \overline{z}) = - 2 Im z = r_{y}$ , and $tr (ρ Z) = a - (1 - a) = 2 a - 1 = r_{z}$ . If $ρ = ρ (s)$ for another real vector $s$ , [F7] applied to [F8] gives $tr (ρ σ_{j}) = s_{j}$ for each $j$ , so $s = r$ and uniqueness follows. When $ρ$ is a density operator, [F5] makes it Hermitian and trace one, and [F8] identifies these same real trace coordinates with the map $r (ρ)$ .

Depends on

Used by

Bloch coordinates of |0> and H|0> Example
Purity is the boundary of the Bloch ball Theorem
The Bloch ball is exactly the qubit density-state space Theorem

Cited to discharge well-definedness by Bloch coordinate maps.

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27) (standard reference, not scraped)
Aram Harrow, 6.443/8.371/18.436 Quantum Information Science II, Lecture 1: Quantum states and operations (2018) (standard reference, not scraped)