  A state on a tripartite quantum system $\mathcal{H}_{A}\otimes
\mathcal{H}_{B}\otimes\mathcal{H}_{C} $ forms a Markov chain, i.e., quantum
conditional independence, if it can be reconstructed from its marginal on
$\mathcal{H}_{A}\otimes \mathcal{H}_{B}$ by a quantum operation from
$\mathcal{H}_{B}$ to $\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ via the famous
Petz map: a quantum Markov chain $\rho_{ABC}$ satisfies
$\rho_{ABC}=\rho_{BC}^{1/2}(\rho_B^{-1/2}\rho_{AB}\rho_B^{-1/2}\otimes
id_C)\rho_{BC}^{1/2}$.
  In this paper, we study the robustness of the Petz map for different metrics,
i.e., the closeness of marginals implies the closeness of the Petz map
outcomes. The robustness results are dimension-independent for infidelity
$\delta$ and trace distance $\epsilon$. The applications of robustness results
are
  The sample complexity of quantum Markov chain tomography, i.e., how many
copies of an unknown quantum Markov chain are necessary and sufficient to
determine the state, is $\tilde{\Theta}(\frac{(d_A^2+d_C^2)d_B^2}{\delta})$,
and $\tilde{\Theta}(\frac{(d_A^2+d_C^2)d_B^2}{\epsilon^2}) $.
  The sample complexity of quantum Markov Chain certification, i.e., to certify
whether a tripartite state equals a fixed given quantum Markov Chain
$\sigma_{ABC}$ or at least $\delta$-far from $\sigma_{ABC}$, is
${\Theta}(\frac{(d_A+d_C)d_B}{\delta})$, and
${\Theta}(\frac{(d_A+d_C)d_B}{\epsilon^2})$.
  $\tilde{O}(\frac{\min\{d_Ad_B^3d_C^3,d_A^3d_B^3d_C\}}{\epsilon^2})$ copies to
test whether $\rho_{ABC}$ is a quantum Markov Chain or $\epsilon$-far from its
Petz recovery state.
  We generalized the tomography results into multipartite quantum system by
showing $\tilde{O}(\frac{n^2\max_{i} \{d_i^2d_{i+1}^2\}}{\delta})$ copies for
infidelity $\delta$ are enough for $n$-partite quantum Markov chain tomography
with $d_i$ being the dimension of the $i$-th subsystem.
