Non-stabilizerness and violations of CHSH inequalities

Technical Summary: Non-stabilizerness and Violations of CHSH Inequalities

Problem Statement
The violation of Clauser-Horne-Shimony-Holt (CHSH) inequalities is a hallmark of quantum non-locality, traditionally attributed to entanglement. However, recent theoretical developments indicate that entanglement alone is insufficient; non-stabilizer resources (often termed "magic") are also necessary to achieve quantum advantage and violate these inequalities. While the necessity of these resources is established, the specific structural requirements regarding how entanglement and non-stabilizer resources must interplay to enable a CHSH violation remain quantitatively uncharacterized. Specifically, it is unclear whether non-stabilizer resources must be local or non-local, symmetric or asymmetric, and how these properties affect the probability of violation under imperfect control.

Methodology
The authors adopt a resource-theoretic framework where the initial state preparation and measurement are defined as "resource-free." They consider a standard resource-free preparation $\omega_0 = |00\rangle\langle 00|$ and a resource-free CHSH operator $B_0$ composed of Pauli measurements. All resources required for a violation are encoded in a unitary evolution $U$ such that the final state is $\omega_U = U\omega_0 U^\dagger$.

To quantify non-stabilizer resources, the paper employs Stabilizer Entropy (SE), specifically the 2-SE ($M_2$), defined as the shifted 2-Rényi entropy of the probability distribution over Pauli strings. The authors distinguish between:

1. Total Non-stabilizerness ($M_2$): The magic of the state.
2. Non-local Non-stabilizerness ($M_{NL}$): The amount of magic that cannot be removed by local unitaries, defined as the minimum $M_2$ achievable via local unitary operations.
3. Local Non-stabilizerness ($M_{LOC}$): The difference $M_2 - M_{NL}$.

The study utilizes three primary analytical tools:

• Theoretical Proofs: Deriving necessary and sufficient conditions for CHSH violations using properties of the Clifford group and stabilizer states.
• Haar Averaging: Computing the probability of violation for random unitaries and conditional probabilities given fixed resource values (entanglement entropy, $M_{NL}$, $M_{LOC}$).
• Isospectral Twirling: A technique from representation theory to construct ensembles of unitaries with fixed spectra (controlled by a "core" unitary $U_c$) but randomized eigenvectors (averaged over a subgroup $G$, such as the Clifford group or full Unitary group). This models scenarios with limited control over state preparation.

Key Contributions and Results

1. Necessity of Non-Stabilizer Resources:
   The paper proves (Theorem 1) that neither pure nor mixed stabilizer states (where $M_2=0$) can violate CHSH inequalities, regardless of the measurement settings. Furthermore, if the preparation unitary is a Clifford operator, the resulting state remains a stabilizer state, and no violation occurs.

2. Structural Requirements for Violation:

   • Asymmetry: Theorem 2 demonstrates that if the unitary preparation $U$ is symmetric with respect to the swap of qubits (commuting with the swap operator), no CHSH violation is possible. The resources must be asymmetric between Alice and Bob.
   • Locality of Magic: Counterintuitively, the paper proves that non-local non-stabilizerness is detrimental to CHSH violations. Theorem 3 establishes an upper bound on the maximal CHSH expectation value $C(\theta)$ as a function of non-local magic $M_{NL}$:
     $$C(\theta) \leq 2\sqrt{2} - \frac{\ln(2)}{\sqrt{2}} M_{NL}(\theta)$$
     Consequently, any non-zero $M_{NL}$ strictly prevents the state from reaching the Tsirelson bound ($2\sqrt{2}$). To maximize violation, the unitary must inject local non-stabilizerness ($M_{LOC}$). Theorem 4 confirms that if $M_{LOC} = 0$, no violation is possible.

3. Probabilistic Analysis via Haar Measure:
   Numerical simulations on $5 \times 10^7$ Haar-random two-qubit states reveal:

   • The probability of a random unitary violating CHSH is approximately 2.51%.
   • While states with high $M_{NL}$ are more likely to violate the inequality than those with low $M_{NL}$, the magnitude of the violation is suppressed by the presence of $M_{NL}$.
   • The probability of violation is non-monotonic with respect to local magic ($M_{LOC}$); very high $M_{LOC}$ often correlates with high $M_{NL}$ in random states, which hinders the violation.

4. Isospectral Twirling and Ensemble Construction:
   The authors utilize isospectral twirling to systematically construct ensembles of unitaries that maximize the probability of violation under limited control.

   • Asymmetric Twirling: Averaging over the Clifford group or full Unitary group on only one qubit (asymmetrically) yields significantly higher probabilities of violation compared to symmetric twirling over both qubits.
   • Efficiency of Clifford Groups: Surprisingly, performing isospectral twirling over the Clifford group on a single qubit yields violation probabilities comparable to those obtained by averaging over the full Unitary group, provided the core unitary possesses the necessary non-stabilizer resources.
   • Robustness: The probability of violation is found to be more dependent on the asymmetry of the twirling than on the precise amount of non-stabilizerness, provided the core unitary is non-Clifford.

Significance
The paper provides a quantitative characterization of the "magic" required for non-locality. It shifts the understanding of CHSH violations from a purely entanglement-centric view to a resource-theoretic perspective where the structure of non-stabilizer resources is paramount. The key finding—that non-stabilizer resources must be local and asymmetric to facilitate CHSH violations—challenges the naive intuition that non-local magic is the primary driver of non-locality.

By employing isospectral twirling, the work offers a systematic method for designing quantum control protocols (ensembles of unitaries) that are robust against noise and limited control, specifically targeting higher probabilities of observing non-local correlations. This has implications for experimental setups where perfect control over unitary evolution is unattainable, such as in satellite-based quantum experiments or systems with imperfect gate fidelities. The results suggest that even with restricted control (e.g., using Clifford groups), one can engineer high-probability violations if the asymmetry and locality of the resources are correctly managed.