Sample optimal tomography of quantum Markov chains

作     者：Gao, Li Yu, Nengkun 

作者机构：Department of Mathematics University of Houston HoustonTX United States Centre for Quantum Software and Information Faculty of Engineering and Information Technology University of Technology SydneyNSW2007 Australia 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

主　　题：Tomography 

摘      要：A state on a tripartite quantum system HA ⊗ HB ⊗ HC forms a Markov chain, i.e., quantum conditional independence, if it can be reconstructed from its marginal on HA ⊗ HB by a quantum operation from HB to HB ⊗ HC via the famous Petz map: a quantum Markov chain ρABC satisfies ρABC = ρ1/2BC (ρ−B1/2ρABρ−B1/2 ⊗ idC)ρ1/2BC . 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 δ and trace distance ǫ. 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 (Formula presented), and (Formula presented) • The sample complexity of quantum Markov Chain certification, i.e., to certify whether a tripartite state equals a fixed given quantum Markov Chain σABC or at least δ-far from σABC, is (Formula presented), and (Formula presented) • (Formula presented) copies to test whether ρABC is a quantum Markov Chain or ǫ-far ǫ2 from its Petz recovery state. The bound is better than the standard tomography of general ρABC with dA dBdC. In other words, tomography is not always necessary for testing quantum conditional independence. We generalized the tomography results into multipartite quantum system ⊗ni=1Hi by showing (Formula presented) copies for infidelity δ are enough for n-partite quantum Markov chain δ tomography with di being the dimension of the i-th subsystem. We also prove the continuity of the Petz map for general quantum channels in 2 distance, which may be of independent interest. © 2022, CC0.