数据处理不等式的恢复图稳定性
Recovery map stability for the Data Processing Inequality
摘要：
数据处理不等式（DPI）指出，Umegaki相对熵$S(ρ||σ) := {\rm Tr}[ρ(\log ρ- \log σ)]$在完全正迹保持（CPTP）映射的作用下是不增加的。设${\mathcal M}$是有限维von Neumann代数，${\mathcal N}$是von Neumann子代数，如果它是，则${\mathcal E}_τ$是从${\mathcal M}$到${\mathcal N}$的跟踪条件期望.对于${\mathcal N}$中的密度矩阵$ρ$和$σ$，设$ρ_{\mathcal N} := {\mathcal E}_τρ$和$σ_{\mathcal N} := {\mathcal E}_τσ$。由于${\mathcal E}_τ$是CPTP，DPI说是$S(ρ||σ) \geq S(ρ_{\mathcal N}||σ_{\mathcal N})$，一般情况很容易从这一点推导出来。Petz的一个定理说，存在等式当且仅当$σ= {\mathcal R}_ρ(σ_{\mathcal N} )$，其中${\mathcal R}_ρ$是Petz恢复映射，它与从${\mathcal M} $到${\mathcal N} $的Accardi-Cecchini粗粒化算子${\mathcal A}_ρ$对偶。在最简单的形式中，我们的界是$$S(ρ||σ) - S(ρ_{\mathcal N} ||σ_{\mathcal N} ) \geq \left(\frac{1}{8π}\right)^{4} \|Δ_{σ,ρ}\|^{-2} \| {\mathcal R}_{ρ_{\mathcal N}} -σ\|_1^4 $$，其中$Δ_{σ,ρ}$是相对模运算符。我们还证明了各种拟相对熵的相关结果。对Petz方程$σ= {\mathcal R}_ρ(σ_{\mathcal N} )$的解集的精确描述，相当于确定了Accardi-Cecchini粗粒化映射的不动点集。建立在以前的工作，我们提供了一个彻底详细的描述的一组解决方案的Petz方程，并获得我们所有的结果在一个简单的自我，包含的方式。
英文摘要：
The Data Processing Inequality (DPI) says that the Umegaki relative entropy $S(ρ||σ) := {\rm Tr}[ρ(\log ρ- \log σ)]$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let ${\mathcal M}$ be a finite dimensional von Neumann algebra and ${\mathcal N}$ a von Neumann subalgebra if it. Let ${\mathcal E}_τ$ be the tracial conditional expectation from ${\mathcal M}$ onto ${\mathcal N}$. For density matrices $ρ$ and $σ$ in ${\mathcal N}$, let $ρ_{\mathcal N} := {\mathcal E}_τρ$ and $σ_{\mathcal N} := {\mathcal E}_τσ$. Since ${\mathcal E}_τ$ is CPTP, the DPI says that $S(ρ||σ) \geq S(ρ_{\mathcal N}||σ_{\mathcal N})$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if $σ= {\mathcal R}_ρ(σ_{\mathcal N} )$, where ${\mathcal R}_ρ$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator ${\mathcal A}_ρ$ from ${\mathcal M} $ to ${\mathcal N} $. In it simplest form, our bound is $$S(ρ||σ) - S(ρ_{\mathcal N} ||σ_{\mathcal N} ) \geq \left(\frac{1}{8π}\right)^{4} \|Δ_{σ,ρ}\|^{-2} \| {\mathcal R}_{ρ_{\mathcal N}} -σ\|_1^4 $$ where $Δ_{σ,ρ}$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation $σ= {\mathcal R}_ρ(σ_{\mathcal N} )$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.
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