论文标题
量子马尔可夫链的最佳层析成像
Sample optimal tomography of quantum Markov chains
论文作者
论文摘要
三方量子系统上的状态$ \ MATHCAL { $ \ MATHCAL {H} _ {A} \ otimes \ Mathcal {H} _ {B} $通过$ \ Mathcal {h} _ {h} _ {b} $ to $ \ \ \ \ \ \ \ m natercal {h} _ { Markov链$ρ_{abc} $满足$ρ_{abc} =ρ_{bc}^{1/2}（ρ_b^{ - 1/1/2}ρ_{ab}ρ_B}ρ_b^{ - 1/2} { - 1/2} \ otimes ID_C） 在本文中，我们研究了Petz图对不同指标的鲁棒性，即边际的亲密性意味着Petz Map结果的亲密关系。鲁棒性结果与不忠$δ$和跟踪距离$ε$无关。鲁棒性结果的应用是 Quantum Markov链层析成像的样本复杂性，即，需要$ \tildeθ（\ frac {（d_a^2+d_c^2）d_b^2}δ）$ \tildeθ（\ frac {（\ frac {（\ frac） $ \tildeθ（\ frac {（d_a^2+d_c^2）d_b^2} {ε^2}）$。 量子马尔可夫链认证的样本复杂性，即证明三方状态是否等于固定的给定量子马尔可夫链$σ_{abc} $还是至少与$σ_{abc} $相等的$δ$ -FAR $θ（\ frac {（d_a+d_c）d_b} {ε^2}）$。 $ \ tilde {o}（\ frac {\ min \ {d_ad_b^3d_c^3，d_a^3d_b^3d_b^3d_c \}} {ε^2} {ε^2}）$ copies $ copies $ copies以测试$ρ_{abc} $是量子马可分子链链还是$ - $ - $ - - - - ε-ε-ε-ε$ε-ε-am uscessive。 我们通过显示$ \ tilde {o}（\ frac {n^2 \ max_ {i} \ {d_i^2d_ {i+1}^2}^2 \}}δ）$ copies $ n $ nimum dimummakov caplite $ nimummarkov $ i $ -th子系统。
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 $ρ_{ABC}$ satisfies $ρ_{ABC}=ρ_{BC}^{1/2}(ρ_B^{-1/2}ρ_{AB}ρ_B^{-1/2}\otimes id_C)ρ_{BC}^{1/2}$. 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 $δ$ and trace distance $ε$. 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 $\tildeΘ(\frac{(d_A^2+d_C^2)d_B^2}δ)$, and $\tildeΘ(\frac{(d_A^2+d_C^2)d_B^2}{ε^2}) $. The sample complexity of quantum Markov Chain certification, i.e., to certify whether a tripartite state equals a fixed given quantum Markov Chain $σ_{ABC}$ or at least $δ$-far from $σ_{ABC}$, is $Θ(\frac{(d_A+d_C)d_B}δ)$, and $Θ(\frac{(d_A+d_C)d_B}{ε^2})$. $\tilde{O}(\frac{\min\{d_Ad_B^3d_C^3,d_A^3d_B^3d_C\}}{ε^2})$ copies to test whether $ρ_{ABC}$ is a quantum Markov Chain or $ε$-far from its Petz recovery state. We generalized the tomography results into multipartite quantum system by showing $\tilde{O}(\frac{n^2\max_{i} \{d_i^2d_{i+1}^2\}}δ)$ copies for infidelity $δ$ are enough for $n$-partite quantum Markov chain tomography with $d_i$ being the dimension of the $i$-th subsystem.
