= Purity
{synonym}

The purity of a <density operator> is $\gamma=\operatorname{Tr}(\rho^2)$. It equals one exactly for a <pure state>; in finite <dimension> $d$ it lies between $1/d$ and $1$. The minimum occurs exactly at the <maximally mixed state> $I/d$.

A worked <qubit> family interpolates between the <mixed state> $I/2$ and the <pure state> $\rho_+=|+\rangle\langle+|$, where $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$:
$$
\rho_p=p\rho_++(1-p)\frac I2
=\frac12\begin{pmatrix}1&p\\p&1\end{pmatrix},\qquad 0\leq p\leq1.
$$
The <eigenvectors> are $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$, with <eigenvalues> $\lambda_\pm=(1\pm p)/2$. Squaring and summing these <eigenvalues> gives
$$
\gamma(p)=\frac{(1+p)^2+(1-p)^2}{4}=\frac{1+p^2}{2},\qquad
P_X(+)=\operatorname{Tr}(\rho_p|+\rangle\langle+|)=\frac{1+p}{2}.
$$
Here $P_X(+)$ is the <Born rule> <probability> of the positive outcome in an $X$ <projective measurement>. The <Bloch vector> is $(p,0,0)$, since $\rho_p=(I+p\sigma_x)/2$ with $\sigma_x$ one of the <Pauli matrices>. At $p=0$ the purity is $1/2$; at $p=1/2$ it is $5/8$, with $P_X(+)=3/4$; at $p=1$ it is one. Within this family, $p=1$ is a <pure state> and every $p<1$ is a <mixed state>. Extending the formula to $p>1$ produces a negative <eigenvalue> $(1-p)/2$, so it ceases to be a <density operator> despite remaining <Hermitian> with <trace> one.

\Image[/density-matrix-purity-and-polarization.png]
{title=Purity and X-measurement probability as a qubit moves from the Bloch-ball centre to a pure state}

The <Bloch vector> length measures polarization: for any <qubit> <density operator>, $\gamma=(1+|\mathbf r|^2)/2$. This connects the interior of the <Bloch ball> with <mixed states> and its boundary, the <Bloch sphere>, with <pure states>. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, chapter 2, section 2.3.2, equation (2.62)]{ref}

For a short proof of the general bound, use the <spectral theorem>: the nonnegative <eigenvalues> $\lambda_j$ sum to one, so $\gamma=\sum_j\lambda_j^2$. Then
$$
\gamma-\frac1d=\sum_{j=1}^d\left(\lambda_j-\frac1d\right)^2\geq0,\qquad
1-\gamma=2\sum_{j<k}\lambda_j\lambda_k\geq0.
$$
The lower bound is attained only when all <eigenvalues> equal $1/d$; the upper bound is attained only when exactly one <eigenvalue> is nonzero, and hence equals one. Purity is therefore a simple quantitative test of mixedness, invariant under a <unitary operator>. It does not specify the entire <density operator>: even distinct <pure states> all have purity one. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, chapter 2, section 2.3.1, equation (2.60) and the following purity criterion]{ref}


 Purity of a density operator (source code)