Optimal Tomography of Quantum Markov Chains via Continuity of Petz Recovery States

In this work, we show that the Petz recovered state $\rho _{BC}^{1/2}(\rho _{B}^{-1/2}\rho _{AB}\rho _{B}^{-1/2}\otimes I_{C})\rho _{BC}^{1/2}$ is continuous regarding its marginals $\rho _{AB}$ and $\rho _{BC}$ . In terms of infidelity $1-F(\rho,\sigma)=1-\mathop {\mathrm {tr}}\nolimits |\sqrt {\rho }\sqrt {\sigma }|$ and trace norm $\parallel \! \rho -\sigma \! \parallel _{1}=\mathop {\mathrm {tr}}\nolimits (|\rho -\sigma |)$ , we obtain the following dimension-independent estimate: $1-F(\rho _{AB},\sigma _{AB}) \le \delta \hspace {.1cm}, 1-F(\rho _{BC},\sigma _{BC}) \le \delta \hspace {.1cm}\Longrightarrow 1-F(\rho _{ABC},\sigma _{ABC})\le 18\delta, \parallel \! \rho _{AB}-\sigma _{AB} \! \parallel _{1} \le \varepsilon \hspace {.1cm}, \parallel \! \rho _{BC}-\sigma _{BC} \! \parallel _{1} \le \varepsilon \hspace {.1cm}\Longrightarrow \parallel \! \rho _{ABC}-\sigma _{ABC} \! \parallel _{}\le \varepsilon +4\varepsilon ^{\frac {1}{2}}$ . As applications, we obtain the following applications in tomography of quantum Markov chains: 1) 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 $\tilde {\Theta }\left ({{\frac {(d_{A}^{2}+d_{C}^{2})d_{B}^{2}}{\delta }}}\right)$ , and $\tilde {\Theta }\left ({{\frac {(d_{A}^{2}+d_{C}^{2})d_{B}^{2}}{\epsilon ^{2}}}}\right)$ , where $\delta $ denotes infidelity error and $\epsilon $ denotes trace distance. 2) The sample complexity of quantum Markov chain certification, i.e., to certify whether a tripartite state equals a given quantum Markov chain $\sigma _{ABC}$ or at least $\delta $ -far from $\sigma _{ABC}$ , is ${\Theta }\left ({{\frac {(d_{A}+d_{C})d_{B}}{\delta }}}\right)$ , and ${\Theta }\left ({{\frac {(d_{A}+d_{C})d_{B}}{\epsilon ^{2}}}}\right)$ . 3) $\tilde {O}\left ({{\frac {\min \{d_{A}d_{B}^{3}d_{C}^{3},d_{A}^{3}d_{B}^{3}d_{C}\}}{\epsilon ^{2}}}}\right)$ copies of sample are sufficient to certify whether $\rho _{ABC}$ is a quantum Markov chain or $\epsilon $ -far from its Petz recovered state in trace distance. This implies that full state tomography is not always necessary for testing whether $\rho _{ABC}$ is a quantum Markov chain (equals to its Petz recovered state) or not.