= Mixed state
= Mixed quantum state
{synonym}
A mixed state is a <density operator> that is not a rank-one <orthogonal projection>. In finite <dimension> it is characterized by $\operatorname{Tr}(\rho^2)<1$, so its <purity of a density operator> distinguishes it from a <pure state>.
Compare a <pure state> <quantum superposition> with an equal classical choice between two <computational-basis states> of a <qubit>:
$$
|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2},\qquad
\rho_+=|+\rangle\langle+|=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad
\rho_{\mathrm{mix}}=\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1|=\frac I2.
$$
A $Z$ <projective measurement> in the <computational basis> gives $P(0)=P(1)=1/2$ for both <density operators>. An $X$ <projective measurement> in the <Hadamard basis> $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2$ distinguishes them: the <Born rule> gives $P(+)=1$ for $\rho_+$ and $P(+)=1/2$ for $\rho_{\mathrm{mix}}$. These comparisons use separate copies of each preparation; measuring $Z$ first changes the state tested by a subsequent $X$ measurement.
\Image[/density-matrix-coherence-versus-mixture.png]
{title=Equal computational-basis populations can hide different coherence and X-measurement statistics}
The off-diagonal entries of $\rho_+$ encode <quantum coherence> in the <computational basis>. Such <quantum coherence> depends on the <basis>: $\rho_+$ is diagonal in the <Hadamard basis> but remains a <pure state>. Mixedness is independent of the <basis>, because a <unitary> change of <basis> preserves the <eigenvalues> and the <purity of a density operator>.
Even the <quantum state ensemble> is not unique:
$$
\frac I2=\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1|
=\frac12|+\rangle\langle+|+\frac12|-\rangle\langle-|.
$$
The off-diagonal entries of the two <Hadamard basis> projectors cancel. Thus the <density operator> retains precisely what measurements on the system can reveal, while an accessible preparation label carries additional information. A <mixed state> can also be the <reduced density matrix> of an <entangled state>, even when the whole system is in a <pure state>; classical ignorance about a selected component is not its only physical origin. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, chapter 2, section 2.3.1, equations (2.54)–(2.60)]{ref}
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