  For an open quantum system to evolve under CPTP maps, assumptions are made on
the initial correlations between the system and the environment.
Hermitian-preserving trace-preserving (HPTP) maps are considered as the local
dynamic maps beyond CPTP. In this paper, we provide a succinct answer to the
question of what physical maps are in the HPTP realm by two approaches. The
first is by taking one step out of the CPTP set, which provides us with
Semi-Positivity (SP) TP maps. The second way is by examining the physicality of
HPTP maps, which leads to Semi-Nonnegative (SN) TP maps. Physical
interpretations and geometrical structures are studied for these maps. The
non-CP SPTP maps $\Psi$ correspond to the quantum non-Markovian process under
the CP-divisibility definition ($\Psi = \Xi \circ \Phi^{-1}$, where $\Xi$ and
$\Phi$ are CPTP). When removing the invertibility assumption on $\Phi$, we land
in the set of SNTP maps. A by-product of set relations is an answer to the
following question -- what kind of dynamics the system will go through when the
previous dynamic $\Phi$ is non-invertible. In this case, the only locally
well-defined maps are in $SN\backslash SP$, they live on the boundary of $SN$.
Otherwise, the non-local information will be irreplaceable in the system's
dynamic.
  With the understanding of physical maps beyond CPTP, we prove that the
current quantum error correction scheme is still sufficient to correct quantum
non-Markovian errors. In some special cases, lack of complete positivity could
provide us with more error correction methods with less overhead.
