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

In this work, we show that the Petz recovered state ρ_BC^1/2 (ρ_B^-1/2 ρ_ABρ_B^-1/2⊗IC) ρ_BC^1/2 is continuous regarding its marginals ρ_AB and ρ_BC. In terms of infidelity 1 - F(ρ, σ) = 1 - tr | √ρ √σ and trace norm || ρ-σ ||₁= tr(|ρ-σ|), we obtain the following dimension-independent estimate: 1 - F(ρ_AB, σ_AB) ≤ δ ⇒1 - F(ρ_BC, σ_BC) ≤ δ =) 1 - F(ρ_ABC; σ;_ABC) ≤ 18δ, ||k ρ_AB - σ;_AB ||₁ ≤ ϵ, ||ρ_BC-σ;_BC ||₁≤ ϵ ⇒ || ρ_ABC-σ_ABC ||≤ ϵ+4ϵ 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 Θ (d_A² A+_cd² )d_B²/δ , and d_A² A+_cd² )d_B²/ϵ² , where δ denotes infidelity error and ϵ 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 σ_ABC or at least δ-far from σ_ABC, is Θ (d_AA+_cd )d_B/δ , and d_A A+_cd )d_B/ϵ². 3) O (min{d_Ad³d_B³d_C³d_BB³ d_C}/ϵ²) copies of sample are sufficient to certify whether σ_ABC is a quantum Markov chain or ϵ-far from its Petz recovered state in trace distance. This implies that full state tomography is not always necessary for testing whether ρ_ABC is a quantum Markov chain (equals to its Petz recovered state) or not.