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

TL;DR

This work addresses how quantum operations generate entanglement by introducing entangling power $e_{\overline{p}_0}(\hat{U})$ and its deviation $\Delta_{\overline{p}_0}(\hat{U})$, capturing average strength and input-state dependence, respectively. It develops a general group-theoretic framework based on Haar averages and Schur–Weyl duality to derive closed-form expressions for both metrics across arbitrary bipartite dimensions, and applies them to two-qubit gates (CU, SWAP$^{\alpha}$, iSWAP) and generalized CX operations, revealing dimension- and parity-dependent effects invisible to EP alone. Key results include exact EP/EPD formulas, theorems linking zero EP to symmetry properties, and a universal SU(4) landscape exhibiting trade-offs between strength and uniformity, with no gate optimizing both simultaneously. The findings establish EPD as a fundamental, practical diagnostic that complements EP for gate design, benchmarking, and quantum circuit optimization, and they uncover fundamental constraints on uniform entanglement generation. The framework offers analytic benchmarks and a pathway to extend EP/EPD to higher-dimensional and multipartite settings, with implications for quantum algorithms and error mitigation.

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.

Figures (3)

• Figure 1: EP vs. EPD graphs for physically implementable two-qubit gates. EP and EPD graphs are depicted for $\mathrm{iSWAP}$, CU, and $\mathrm{SWAP}^{\alpha}$ gates with respect to the realistic Hamiltonian parameters. CU and $\mathrm{SWAP}^{\alpha}$ exhibit linear EP--EPD relations, but with distinct profiles: $\mathrm{SWAP}^{\alpha}$ shows symmetric variation, while CU increases monotonically. Their EP-to-EPD ratios and maxima differ. In contrast, $\mathrm{iSWAP}$ displays a nonlinear profile, yet reaches the same maximal EP and EPD as CU. See Tab. \ref{['tab:1']} for EP-to-EPD ratios.
• Figure 2: EP–EPD Landscape of Two-Qubit Gates. The shaded region shows the full range of EP and EPD achievable by general two-qubit unitaries. Specific gate families---CU (red), $\mathrm{SWAP}^\alpha$ (blue), and $\mathrm{iSWAP}$ (yellow)---follow characteristic EP--EPD trajectories due to their fixed internal structure. Some representative gates such as CNOT, $B$, $F_4$, and $\sqrt{\mathrm{SWAP}}$ are also indicated. The $B$-gate lies on the maximal EP boundary with moderate EPD, indicating strong yet consistent entanglement generation. The $F_4$-gate resides centrally, achieving moderate EP with low fluctuation, exemplifying a well-balanced entangler. CNOT shows maximal EP but moderate EPD, reflecting selective entangling behavior. In contrast, $\sqrt{\mathrm{SWAP}}$ displays high EPD despite a lower EP, indicating its entangling strength varies widely across inputs.
• Figure 3: The histogram of the generated entanglement $e=E(\hat{U}\left|\psi_1\right>\otimes\left|\psi_2\right>)$ for Haar-random product inputs ($N=2 \times 10^5$ samples per gate), shown for CNOT, $B$ (a perfect entangler with the same EP as CNOT), and $\sqrt{\mathrm{SWAP}}$. The linear entropy $e$ is bounded ($0 \le e \le 1/2$ for two-qubit systems), and the distributions are well-behaved yet exhibit distinct skewness and widths.

Theorems & Definitions (37)

• Definition 1
• Definition 2: Symmetric subspace
• Definition 3: Anti-symmetric subspace
• Definition 4: Partially symmetrized and anti-symmetrized projectors
• Proposition 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof