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

Abstract

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 shows that any nontrivial entangling operation necessarily exhibits input-state dependence: nonzero EP implies a nonzero EPD. 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 the gate functionality. Extending our framework to a class of generalized controlled-unitary operations acting on bipartite Hilbert spaces of arbitrary dimensions, we further analyze the interplay between the entangling strength and uniformity, as quantified by EP and EPD. 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.