发表机构
Northwestern University; Argonne National Laboratory; University of Chicago; University of Arizona（西北大学; 阿贡国家实验室; 芝加哥大学; 亚利桑那大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明幺正算子纠缠能力在相对相位为0或π的角点处平稳，并给出其用七个局域幺正不变量表示的公式，适用于满足U²∝I的门，如克利福德门，并通过例子展示极值类型。
AI 中文摘要
作用于二分希尔伯特空间上的幺正算子的纠缠能力，衡量了它从乘积态产生的纠缠，并对输入进行平均。有限维幺正算子具有谱分解 $U=\sum_{a=1}^{n}e^{i\theta_a}P_a$，其中 $e^{i\theta_a}$ 是特征值，$P_a$ 是对应的特征投影算子，$n$ 是不同特征值的个数。在去除整体相位后，纠缠能力是固定谱投影算子下相对特征相位 $(n-1)$-环面上的函数。我们证明该函数在环面上所有 $2^{n-1}$ 个点处都是平稳的，这些点处每个相对相位为 $0$ 或 $\pi$，我们将其定义为“角点”。在整体相位允许下，每个角点处的 $U$ 是一个广义反射 $R=\mathbb{I}-2Q$，满足 $R^2=\mathbb{I}$，其中 $Q$ 是相对相位为 $\pi$ 的谱投影算子之和。在角点处，纠缠能力用 $Q$ 的七个局域幺正不变量表示。一个幺正门 $U$ 能被实现为某个投影算子族的角点，当且仅当 $U^2\propto\mathbb{I}$，这一条件被许多克利福德和非克利福德门满足。我们用双量子比特门、$SU(N)$ 信道分解和双位点自旋链来说明该定理，获得了极小值、极大值和鞍点的例子。此外，在全相位环面上是鞍点的角点，可以沿不同时间演化轨迹表现为局部极大值或极小值。
英文摘要
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition $U=\sum_{a=1}^{n}e^{iθ_a}P_a$, where $e^{iθ_a}$ are the eigenvalues, $P_a$ the corresponding eigen-projectors, and $n$ is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the $(n-1)$-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all $2^{n-1}$ points on the torus where every relative phase is $0$ or $π$, which we define as \textit{corners}. Up to an overall phase, $U$ at each corner is a generalized reflection $R=\mathbb{I}-2Q$ satisfying $R^2=\mathbb{I}$, where $Q$ is the sum of spectral projectors whose relative phase is $π$. At the corner the entangling power is expressed in terms of seven local-unitary invariants of $Q$. A unitary gate $U$ can be realized as a corner of some projector family if and only if $U^2\propto\mathbb{I}$, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, $SU(N)$ channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
Comments10 pages, 2 figures
