Average Pauli-Entangling Power (APEP)

• APEP is a measure that quantifies the mean entanglement generated by quantum operations on Pauli states, reflecting a gate’s intrinsic, basis-invariant capacity to entangle.
• Leveraging group-theoretic methods and permutation operators, APEP provides closed-form benchmarks that correlate with operator magic and information scrambling in noisy circuits.
• APEP serves as a practical diagnostic tool in quantum computing for assessing resource conversion, nonlocal entanglement, and sample-complexity in benchmarking tasks.

Average Pauli-Entangling Power (APEP) quantifies the mean entanglement generated by a quantum operation when averaged over a set of input Pauli product states. In contrast to state-dependent metrics, it captures a gate’s or channel’s intrinsic, basis-agnostic capacity to entangle, and provides a robust operational benchmark for gate design, benchmarking, information scrambling, and resource generation. APEP is directly connected to operator entanglement, information scrambling, and resource-theoretic concepts such as magic monotones, and is calculable through group-theoretic methods, especially using permutation operator formalism and representation theory for structured ensembles such as the @@@@1@@@@ or the Clifford group. The following sections develop the formal definitions, mathematical structure, operational meaning, group-theoretic computation, and key physical applications of APEP.

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1. Definition and Mathematical Foundations

APEP is defined as the average entanglement generated by a quantum operation (typically a unitary or a quantum channel) when acting on a basis set of Pauli product states. For an operator $U$ acting on $\mathcal{H}_A \otimes \mathcal{H}_B$, and a Pauli group $\tilde{\mathcal{P}}_N$, the APEP is formally

$$P_E(U) = \mathbb{E}_{P \in \tilde{\mathcal{P}}_N} \left[ E_{\text{lin}}\left( U^\dagger P U \right) \right]$$

where $E_{\text{lin}}$ denotes the operator’s linear entanglement (1 minus the sum of squared Schmidt coefficients over a bipartition) (Dallas et al., 24 Sep 2025). In the context of bipartite gates, the linear entanglement for an operator $$O/\sqrt{d} = \sum_i \sqrt{\lambda_i}\, V_i \otimes W_i$$ is

$E_{\text{lin}}(O) = 1 - \sum_i \lambda_i^2$

This measurement averages the “Pauli entangling” capability over the entire abelianized Pauli group, isolating the gate’s nonlocal ability to generate operator entanglement from product inputs.

Further, APEP is a first-moment quantity. It provides a scalar, basis-invariant summary of how a unitary or noisy channel maps local (Pauli) operators to entangled operators, which forms the foundation for assessing nonlocal magic, information scrambling, and benchmarking tasks (Cho et al., 1 Aug 2025, Dallas et al., 24 Sep 2025).

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2. Connection to Operator Entanglement and Magic

Pauli operators are mapped to Pauli operators under Clifford circuits; therefore, ideal Clifford circuits have zero APEP and cannot generate operator entanglement over a bipartition. The presence of noise or non-Clifford (“magic”) gates can convert product Paulis into non-factorizable operators, resulting in nonzero APEP (Dallas et al., 24 Sep 2025). The magnitude of APEP thus serves as a proxy for the nonlocal “magic” produced in a circuit—a crucial resource for universal quantum computation.

Given an ensemble-averaged calculation, APEP can be written in terms of four-copy permutation operators (Dallas et al., 24 Sep 2025): $$P_E(U) = 1 - \frac{1}{d^2} \mathrm{Tr}\left( T^{A}_{(12)(34)}\, U^{\otimes 4}\, Q\, (U^{\dagger})^{\otimes 4} \right)$$ where

$$Q = \frac{1}{d^2} \sum_{P \in \tilde{\mathcal{P}}_N} P^{\otimes 4}$$

and $T^{A}_{(12)(34)}$ is a specific permutation acting on four copies of subsystem A. This approach leverages the representation theory of the Clifford group and symmetries of the Pauli group to yield closed-form expressions in the macroscopic limit.

In noisy Clifford circuits, APEP quantifies the degree to which local magic generation is converted via scrambling into nonlocal entanglement (operator-level magic). Analytical expressions for APEP, particularly in the regime of large circuits ($L \to \infty$), demonstrate a “butterfly effect”: local noise can trigger macroscopic nonlocal magic creation (Dallas et al., 24 Sep 2025). The relationship between the magic capacity of the error channel and the resulting APEP is highly nonlinear and depends both on the magnitude of single-site magic and on the number of affected sites.

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3. Group-Theoretic and Permutation Operator Approach

The evaluation of APEP and related first- and second-moment quantities (such as entangling power deviation, EPD (Cho et al., 1 Aug 2025)) relies on group-theoretic techniques. For averaging over the Pauli basis, Haar-uniform distribution, or the Clifford group, Schur-Weyl duality allows the reduction of ensemble averages to linear combinations of permutation operators (Cho et al., 1 Aug 2025, Dallas et al., 24 Sep 2025).

Typical expressions are of the form: $$e_p(\hat{U}) = 2\, \mathrm{Tr}\left[\hat{U}^{\otimes 2} \hat{\Omega}^{(2)} \hat{U}^{\dagger\otimes 2} \hat{P}_{13}^{-}\right]$$ for EP (or APEP when the ensemble is Pauli), and

$$\Delta_p(\hat{U})^2 = 4\, \mathrm{Tr}\left[\hat{U}^{\otimes 4} \hat{\Omega}^{(4)} \hat{U}^{\dagger\otimes 4} \hat{P}_{13}^{-}\hat{P}_{57}^{-}\right] - e_p(\hat{U})^2$$

for the associated EPD (Cho et al., 1 Aug 2025). Here, $\hat{\Omega}^{(2)}$, $\hat{\Omega}^{(4)}$, $\hat{P}_{13}^{-}$, and $\hat{P}_{57}^{-}$ are permutation and projection operators encapsulating the full permutation symmetry of the Pauli or Haar ensemble.

Recent work applies representation theory of the Clifford group and Weingarten calculus to analytically compute such averages even for large-scale circuits, which is essential for studying macroscopic information dynamics (Dallas et al., 24 Sep 2025).

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4. Operational Interpretations: Benchmarking, Magic Generation, and Scrambling

APEP and related operator entanglement measures have several operational interpretations:

• Nonlocal Magic Generation: In noisy Clifford circuits, APEP quantifies the conversion of local “magic” (stabilizer non-Pauli character) into nonlocal operator entanglement—a necessary step for fault tolerance and quantum speedup (Dallas et al., 24 Sep 2025).
• Information Scrambling: APEP serves as a diagnostic for how much initially local information is nonlocally delocalized (scrambled) by the dynamics. A direct connection exists between APEP and the algebraic out-of-time-ordered correlator (A-OTOC) in these contexts.
• Sample-Complexity and Learning: In learning and benchmarking Pauli channels, APEP captures how much entanglement can be created per application, which is directly linked to exponential quantum advantages for parameter estimation protocols (Chen et al., 2021, Chen et al., 2023).
• Resource Quantification: APEP aligns with resource-theoretic monotones like magic capacity and provides insight into gate sets’ ability to generate useful nonclassicality beyond mere average entanglement production.

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5. Statistical Structure and Trade-offs: Uniformity and Deviation

While APEP (or EP) captures the first moment (the mean) of entanglement generated over a Pauli or Haar ensemble, the distribution’s width (captured by EPD) measures the uniformity of entanglement creation (Cho et al., 1 Aug 2025). A key result is the discovery of a universal trade-off: gates with higher APEP/EP are necessarily more sensitive to specific input states, reflected in a higher EPD.

For different physical implementations or classes of quantum gates, the same APEP can correspond to varied EPDs, revealing different “biases” in gate entanglement structure. In large systems, both APEP and EPD become diagnostic tools for the efficacy and uniformity of entanglement and magic distribution, crucial for random circuit design and quantum computing applications (Cho et al., 1 Aug 2025).

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6. Extensions: Probabilistic Gates, Mixed States, and Structured Systems

APEP can be generalized to various scenarios:

• Probabilistic Quantum Gates: For linear-optical or otherwise non-deterministic devices, APEP takes the form of an average over successful outcomes weighted by output probabilities and entanglement measures (e.g., negativity) (Lemr et al., 2012). For a probabilistic operation $\mathcal{A}$ acting on input $\rho_i$, APEP generalizes as

$$\mathrm{APEP} = \frac{1}{N} \sum_{i=1}^N p_s(\rho_i) \mathcal{E}(\mathcal{A}[\rho_i])$$

where $p_s(\rho_i)$ is the success probability.

• Mixed-State Inputs: For settings where inputs are mixed (e.g., noisy experimental realizations), APEP may be defined as the averaged output entanglement over product mixed states of fixed purity, connecting to the maximally entangled mixed state (MEMS) structure for two-qubit gates (Guan et al., 2013).
• Symmetric and Restricted Systems: In symmetric multiqubit systems, the Pauli basis is replaced by the spin-coherent or Majorana constellation basis, with group-invariant (SU(2)) averaging and geometric structure determining APEP (Serrano-Ensástiga et al., 4 Oct 2024, Morachis et al., 2021).

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7. Summary Table: APEP Formulations Across Contexts

Physical Context	Input Ensemble	Output Metric	Analytical Approach/Formula
Noisy Clifford circuits	Pauli basis over $\mathcal{H}$	Linear operator entanglement	$$P_E(U) = 1-\frac{1}{d^2}\mathrm{Tr}(T^{A}_{(12)(34)} U^{\otimes 4} Q (U^\dagger)^{\otimes 4})$$ (Dallas et al., 24 Sep 2025)
Haar-random gates	Haar measure over product states	State linear entropy (or other)	Permutation operator formalism, Schur-Weyl duality (Cho et al., 1 Aug 2025)
Probabilistic quantum gates	Pauli or other (weighted by $p_s$)	Negativity or related measure	$$\mathrm{APEP} = \tfrac{1}{N} \sum_i p_s(\rho_i) \mathcal{E}(\mathcal{A}[\rho_i])$$ (Lemr et al., 2012)
Symmetric/Bosonic systems	Spin-coherent states	Reduced state entropy	Inner product of SU(2)-invariant vectors (Serrano-Ensástiga et al., 4 Oct 2024, Morachis et al., 2021)

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8. Physical Significance and Implications

APEP provides a universal language for comparing quantum gates, noise channels, and quantum circuits with respect to their intrinsic capacity to generate entanglement and nonclassical resources. Its rigorous connection to operator entanglement, magic monotones, and information scrambling allows it to serve as:

• A tool for certifying non-Clifford (or “magic”) resource conversion by noisy circuits (Dallas et al., 24 Sep 2025)
• An analytical benchmark for gate benchmarking, randomized compiling, and quantum process learning (Chen et al., 2021, Chen et al., 2023)
• A diagnostic for the optimality and robustness of random or structured gates in high-dimensional, fault-tolerant, or error-prone environments (Chen et al., 2012, Cho et al., 1 Aug 2025)
• An operational parameter controlling the butterfly effect in information scrambling, and the transition to universal quantum computation (Dallas et al., 24 Sep 2025)

The measure is robust to input structure and scalable to macroscopic limits, with closed-form evaluation possible via group-theoretic or representation-theoretic tools. Detailed knowledge of both APEP and its fluctuations (EPD) is essential for precision design and analysis of quantum information processes.