Purity

Formula

\gamma(\rho)=\operatorname{Tr}(\rho^2)=\sum_i\lambda_i^2.

For a $d$ -dimensional density operator,

\frac1d\le\gamma\le1,

with $\gamma=1$ exactly for a pure state and $\gamma=1/d$ exactly for $I_d/d$ . For a qubit,

\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma) \quad\Longrightarrow\quad \gamma=\frac{1+\lVert\mathbf r\rVert^2}{2}.

Other useful identities are

\gamma(\rho_A\otimes\rho_B)=\gamma(\rho_A)\gamma(\rho_B),

\rho_\beta=\frac{e^{-\beta H}}{Z(\beta)} \quad\Longrightarrow\quad \gamma(\rho_\beta)=\frac{Z(2\beta)}{Z(\beta)^2},

and, for a swap operator $S$ on two identical copies,

\operatorname{Tr}[S(\rho\otimes\rho)]=\gamma(\rho).

$\rho\ge0$ and $\operatorname{Tr}\rho=1$ .
The lower bound $1/d$ requires finite known dimension $d$ .
The normalized linear entropy convention may include a factor $d/(d-1)$ ; state the convention.
Swap estimation assumes independent, identically prepared copies.

Purity is basis independent but does not determine the whole spectrum.
A superposition of basis states can be pure; off-diagonal entries do not by themselves decide mixedness.
Unitary evolution preserves purity. General channels can increase or decrease it.
Reduced purity diagnoses entanglement only when the global bipartite state is known to be pure.
In infinite dimension, $0\lt\gamma\le1$ but there is no positive dimension-independent lower bound.

Proofs, preparation-versus-entanglement distinctions, thermal and two-copy examples, exercises, and references are at Pure vs Mixed States.