= Trace distance between qubit states
{title2=$D(\rho,\sigma)=|\mathbf r-\mathbf s|/2$}
= Qubit trace distance
{synonym}
For <qubit> <density operators>, the <trace distance> has a particularly simple geometry. Write their <Bloch vectors> as $\mathbf r,\mathbf s$ in the <Bloch ball>:
$$
\rho=\frac{I+\mathbf r\cdot\boldsymbol\sigma}{2},\qquad
\sigma=\frac{I+\mathbf s\cdot\boldsymbol\sigma}{2},\qquad
\Delta=\rho-\sigma=\frac{(\mathbf r-\mathbf s)\cdot\boldsymbol\sigma}{2},
$$
where $I$ is the <identity operator> and $\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z)$ denotes the <Pauli matrices>. This <Bloch vector> representation includes both <pure states> on the <Bloch sphere> and <mixed states> in the interior of the <Bloch ball>. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, §2.3.2, equations (2.62)–(2.69)]{ref}
The <Pauli matrix multiplication law> gives $[(\mathbf r-\mathbf s)\cdot\boldsymbol\sigma]^2=|\mathbf r-\mathbf s|^2I$. Hence the two <eigenvalues> of $\Delta$ are $\pm|\mathbf r-\mathbf s|/2$, and their absolute sum gives
$$
D(\rho,\sigma)=\frac{|\mathbf r-\mathbf s|}{2}.
$$
The <trace distance> is half the straight-line <Euclidean distance> between the <Bloch vectors>, not the angular or surface distance along the <Bloch sphere>. Opposite <Bloch vectors> have <trace distance> one; the centre $I/2$ is at <trace distance> $1/2$ from every <pure state>.
For the <pure states> $|0\rangle$ and $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$, the <Bloch vectors> are $\mathbf r=(0,0,1)$ and $\mathbf s=(1,0,0)$. Their straight-line separation is $\sqrt2$, so
$$
D(|0\rangle\langle0|,|+\rangle\langle+|)=\frac1{\sqrt2}.
$$
These <Bloch vectors> are perpendicular, but the <pure states> are not orthogonal: $|\langle0|+\rangle|=1/\sqrt2$. The <Helstrom measurement for the zero and plus states> measures along the difference direction $\mathbf n=(\mathbf r-\mathbf s)/\sqrt2=(-1,0,1)/\sqrt2$, rather than along either input <Bloch vector>.
\Image[/trace-distance-bloch-geometry.png]
{title=Trace distance is half a Bloch-vector chord, and the optimal zero-versus-plus measurement follows the difference direction}
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