通过几何平均算子研究混合量子态的测地线
Geodesics for mixed quantum states via their geometric mean operator
AI总结：
本文利用纯化纤维丛、Bures 水平提升与几何平均算子构造任意维度混合态间测地线，并给出 Werner 态及量子比特 Bloch 球情形的解析结果。
摘要：
我们借助两个任意维度混合态的几何平均算子来考察它们之间的测地线。我们采用纤维丛方法：底空间 M 中两个混合态密度算子 ρ1 与 ρ2 之间的距离，由其纯化在（Hilbert–Schmidt）丛空间 E 中的最短距离给出。众所周知，后者等于沿 M 中 ρ1 与 ρ2 之间测地线在 E 中的水平提升所得到的 Bures 距离。水平提升是 E 中唯一一条曲线，它正交穿过 M 中该曲线上方的纤维 F⊂E，并投影回该曲线。我们简要回顾这一形式体系，并展示如何用它构造沿底空间测地线、在初始态 ρ1 与终态 ρ2 之间由仿射参数 s 参数化的中间混合量子态 ρ(s)。我们强调几何平均算子 M(s)=ρ1^{-1/2}√(ρ1^{1/2}ρ(s)ρ1^{1/2})ρ1^{-1/2} 所起的作用；其中 ρ1 与 ρ(s) 之间的 Uhlmann 根保真度为 √F(ρ1,ρ(s))=Tr[M(s)ρ1]=Tr[√(ρ1^{1/2}ρ(s)ρ1^{1/2})]，且 ρ(s)=M(s)ρ1M(s)。我们给出任意维度下最大混合态与纯态之间测地线的例子，以及维度 N=2^3 时 Werner 态 ρ(p)=(1-p)I/N+p|Ψ⟩⟨Ψ|（其中 |Ψ⟩={|GHZ⟩,|W⟩}）之间测地线的例子。对于后者，我们将 p→1 极限下的表达式与正交纯态 |GHZ⟩ 和 |W⟩ 之间无限多条可能测地线进行比较。最后，我们计算了维度 N=2 时 Bloch 球内连接两个任意端点量子比特密度矩阵的测地线上密度矩阵的解析形式。
英文摘要：
We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators $ρ_1$ and $ρ_2$ in the base space $M$ is given by the shortest distance in the (Hilbert Schmidt) bundle space $E$ of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in $E$ of the geodesic between the $ρ_1$ and $ρ_2$ in $M$. The horizontal lift is that unique curve in $E$ that orthogonally traverses the fibers $F\subset E$ above the curve in $M$, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states $ρ(s)$ along the base space geodesic parameterized by affine parameter $s$ between the initial $ρ_1$ and final $ρ_2$ states. We emphasize the role played by geometric mean operator $M(s) = ρ_1^{-1/2}\, \sqrt{ρ_1^{1/2}ρ(s)ρ_1^{1/2}}\,ρ_1^{-1/2}$, where the Uhlmann root fidelity between $ρ_1$ and $ρ(s)$ is given by $\sqrt{F}(ρ_1,ρ(s)) = Tr[M(s)\,ρ_1] = Tr[\sqrt{ρ_1^{1/2}ρ(s)ρ_1^{1/2}}]$, and $ρ(s) = M(s)\,ρ_1\,M(s)$. We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states $ρ(p) = (1-p) I/N + p\,|Ψ\rangle\langle Ψ|$ with $|Ψ\rangle = \{|GHZ\rangle, |W\rangle\}$ in dimension $N=2^3$. For the latter, we compare expressions in the limit $p\to1$ to the infinite number of possible geodesics between the orthogonal pure states $|GHZ\rangle$ and $|W\rangle$. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension $N=2$.
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