Optimal Tomography of Quantum Markov Chains via Continuity of Petz Recovery States
Li Gao, Nengkun Yu · IEEE Transactions on Information Theory · 2025
In this work, we show that the Petz recovered state ρ1/2BC(ρ−1/2BρABρ−1/2B⊗IC)ρ1/2BCis continuous regarding its marginals ρABand ρBC. 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(ρAB, σAB) ≤ δ, 1 −F(ρBC, σBC) ≤ δ =⇒ 1 −F(ρABC, σABC) ≤ 18δ, ∥ ρAB− σAB∥1≤ ε, ∥ ρBC− σBC∥1≤ ε =⇒∥ ρABC− σABC∥≤ ε + 4ε1/2. As applications, we obtain the following applications in tomography of quantum Markov chains: • 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 ((d2A+d2C)d2B/δ), and ˜Θ((d2A+d2C)d2B/ϵ2), where δ denotes infidelity error and ϵ denotes trace distance. • The sample complexity of quantum Markov chain certification, i.e., to certify whether a tripartite state equals a given quantum Markov chain σABCor at least δ-far from σABC, is Θ((dA+dC)dB/δ), and Θ((dA+dC)dB/ϵ2). • Õ(mindAd3Bd3C,d3Ad3BdC/ϵ2) copies of sample are sufficient to certify whether ρABCis 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 ρABCis a quantum Markov chain (equals to its Petz recovered state) or not.