= Solution
Strict trace-distance contraction means that for some $\eta<1$,
$$
\|\mathcal E(\rho)-\mathcal E(\sigma)\|_1
\leq\eta\|\rho-\sigma\|_1
$$
for all states. Equivalently, the induced trace norm on nonzero traceless Hermitian operators is strictly below one. This immediately gives a unique fixed state by the contraction mapping theorem.
The standard algebraic condition is that the channel be a <primitive quantum channel>: some finite products
$$
K_{\alpha_m}\cdots K_{\alpha_1}
$$
span the full matrix algebra, equivalently some power of the channel maps every nonzero positive operator to a positive-definite one. Spectrally, eigenvalue $1$ is then simple and every other eigenvalue has modulus less than one. This condition is necessary and sufficient for convergence to a unique full-rank fixed point. Irreducibility alone gives uniqueness but may leave periodic peripheral eigenvalues.
Solved by gpt-5.6-sol high.
Back to article page
