标题： 纠缠能力及其偏差：对输入状态依赖性和纠缠生成变异性的定量分析 显示英文标题

标题： Entangling Power and Its Deviation: A Quantitative Analysis on Input-State Dependence and Variability in Entanglement Generation

摘要： 量化量子操作的纠缠能力是量子信息科学中的基本任务。 传统上，这种能力通过纠缠功率（EP）来衡量，EP定义为当一个量子操作在所有可能的乘积态上均匀作用时所产生的平均纠缠。 然而，EP本身无法捕捉到纠缠生成中对输入态的复杂依赖性。 为了解决这个问题，我们定义了一个互补的度量——纠缠功率偏差（EPD），即所有乘积输入态上产生的纠缠的标准差，从而捕捉到纠缠行为的多方面性质。 我们开发了一个通用的群论框架，可以得出EP和EPD的闭式表达式。 我们的分析揭示了一种基本且此前未被探索的物理现象：增强纠缠能力不可避免地会增加对特定输入态的敏感性或偏差。 通过对典型的两量子比特门进行分析，我们发现具有相同EP的门可能表现出显著不同的EPD值，这说明根据物理实现的不同，纠缠生成的性质可能会有显著差异。 将我们的框架扩展到一类广义的受控单元操作，这些操作作用于任意维度的双子希尔伯特空间，我们（重新）确认了纠缠强度与均匀性之间的固有权衡。 此外，我们在纠缠生成中发现了一种微妙的维度奇偶性依赖行为，而EP本身无法检测到这一点。 这些发现表明，EPD是一个不可或缺的诊断工具——它与EP一起，提供了对纠缠结构更深入和更完整的表征。 显示英文摘要

摘要： Quantifying the entangling capability of quantum operations is a fundamental task in quantum information science. Traditionally, this capability is measured by the entangling power (EP), defined as the average entanglement generated when a quantum operation acts uniformly on all possible product states. However, EP alone cannot capture the intricate input-state-dependent nature of entanglement generation. To address this, we define a complementary metric -- entangling power deviation (EPD) -- as the standard deviation of entanglement generated over all product input states, thereby capturing the multifaceted nature of entangling behavior. We develop a general group-theoretical framework that yields closed-form expressions for both EP and EPD. Our analysis reveals a fundamental and previously unexplored physics: enhancing entangling capability inevitably increases sensitivity, or bias, toward specific input states. By analyzing representative two-qubit gates, we show that the gates with identical EP can exhibit markedly different EPD values, illustrating that the nature of entanglement generation can significantly differ depending on physical implementation. Extending our framework to a class of generalized controlled-unitary operations acting on bipartite Hilbert spaces of arbitrary dimensions, we (re)affirm the inherent trade-off between the entangling strength and uniformity. Moreover, we uncover a subtle dimension-parity-dependent behavior in entanglement generation, which EP alone fails to detect. These findings highlight EPD as an indispensable diagnostic tool -- one that, alongside EP, provides a deeper and more complete characterization of the entangling structure.