Finite channel families, semigroups, and CP-divisibility

Let $d\ge 1$ and let ${\mathrm{\Phi }}_t:M_d(\mathbb{C})\to M_d(\mathbb{C})$ be a family of finite quantum channels indexed by $t\ge 0$, where “finite quantum channel” has the meaning in Finite completely positive trace-preserving maps. The family obeys the semigroup law when ${\mathrm{\Phi }}_0={\mathrm{id}}_{M_d(\mathbb{C})}$ and ${\mathrm{\Phi }}_{t+s}={\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_s$ for every $s,t\ge 0$.

The family is CP-divisible when for every $0\le s\le t$ there exists a finite quantum channel $V_{t,s}:M_d(\mathbb{C})\to M_d(\mathbb{C})$ such that ${\mathrm{\Phi }}_t=V_{t,s}\circ {\mathrm{\Phi }}_s$. CP-divisibility is this specified factorization property; the definition does not assert that it is a universal meaning of “Markovian”.

Finite completely positive trace-preserving maps

The source literature distinguishes divisibility and time-homogeneous semigroup conditions and discusses several meanings of quantum Markovianity. This item fixes only the map properties used on this page.