Pseudo-pure state ($\rho =(1-p)I/d+p|\psi ⟩⟨\psi |$)

A mixture of the maximally mixed state and a chosen pure state. Unitary conjugation preserves its identity part; every traceless observable has expectation $p⟨\psi |O|\psi ⟩$. The coefficient $p$ is effective polarization rather than literal purity: $\mathrm{tr}{\rho }^2=1/d+(1-1/d)p^2$. Preserving the largest population of $n$ weakly polarized equal spins and averaging the rest gives $p\simeq nϵ/[2(2^n-1)]$, displaying the signal penalty of this high-temperature ensemble preparation.