Scalable Self-Testing of Mutually Anticommuting Observables and Maximally Entangled Two-Qudits

1. Problem Statement

Device-independent (DI) quantum information protocols, such as randomness expansion and quantum key distribution, increasingly require the certification of multiple entangled states or high-dimensional entangled systems. While self-testing (certifying quantum states and measurements solely from input-output statistics) is well-established for single qubit pairs (e.g., via CHSH inequality), scaling this to higher dimensions or multiple copies presents significant challenges:

• Sequential vs. Parallel: Existing methods often rely on sequential self-testing, which assumes device stability over time and is vulnerable to temporal correlations. Parallel self-testing is preferred but often requires complex multi-outcome measurements or only certifies observables on one side.
• Dimension Independence: Many protocols presuppose the dimension of the Hilbert space, whereas a truly device-independent approach should not.
• Complexity: The structure of high-dimensional Hilbert spaces makes deriving optimal bounds and self-testing statements analytically difficult.

The authors aim to develop a scalable, dimension-independent framework to simultaneously self-test a maximally entangled two-qudit state and a set of mutually anticommuting observables using only dichotomic (two-outcome) measurements.

2. Methodology

The authors propose a novel $n$-setting Bell inequality and analyze it using the Sum-of-Squares (SOS) decomposition method.

• Bell Scenario:

  • Two observers, Alice and Bob.
  • Alice has $2^{n-1}$ measurement settings ($x$), and Bob has $n$ settings ($y$).
  • All measurements yield binary outcomes ($\pm 1$).
  • The Bell functional is defined as:
    $$G_n = \sum_{x=1}^{2^{n-1}} \sum_{y=1}^{n} (-1)^{z^x_y} A_x B_y$$
    where $z^x_y$ are bits derived from binary strings of length $n$.

• Derivation of Bounds:

  • Local Bound: The authors derive the classical (local ontic) bound $(G_n)_L$ analytically for arbitrary $n$.
  • Quantum Bound: Instead of assuming a specific dimension, they employ an SOS decomposition of the shifted Bell operator:
    $$G_n \mathbb{1} - G_n = \sum_{x} \frac{\omega_x}{2} M_x^\dagger M_x$$
    This allows them to determine the optimal quantum bound $(G_n)_{Q}^{opt}$ without presupposing the Hilbert space dimension.

• Self-Testing Logic:

  • They analyze the conditions required to saturate the optimal quantum bound.
  • They prove that any physical realization achieving this maximum must correspond to a specific reference strategy involving a maximally entangled state and observables forming an irreducible representation of the Clifford algebra.
  • They address the issue of complex conjugation: Since the Bell functional involves real coefficients and joint correlators, the optimal strategy is unique only up to local isometries and local complex conjugation (transposition).

• Robustness Analysis:

  • The authors derive quantitative bounds relating a deviation $\delta$ from the optimal quantum value to the fidelity between the realized and ideal strategies. This establishes the protocol's resilience to experimental noise.

3. Key Contributions

1. Scalable Self-Testing Framework: The paper introduces a single Bell functional that self-tests $\lfloor n/2 \rfloor$ copies of maximally entangled two-qubit pairs (equivalent to a local dimension $m^* = 2^{\lfloor n/2 \rfloor}$) and $n$ mutually anticommuting observables on one side.
2. Dimension-Independent Certification: The derivation of the quantum bound and the self-testing statement does not assume the dimension of the underlying quantum system. The dimension emerges naturally as the minimal dimension required to support $n$ anticommuting observables.
3. Clifford Algebra Structure: The work establishes that maximal violation of this specific Bell inequality uniquely identifies the local observables as forming a representation of the complex Clifford algebra $Cl_n(\mathbb{C})$.
4. Handling Complex Conjugation: The authors generalize the definition of self-testing to include equivalence under local complex conjugation (transposition), resolving ambiguities that arise in Bell functionals with real coefficients (extending previous results on the Elegant Bell Inequality).
5. Robustness Bounds: They provide explicit operator norm bounds showing that if the observed value deviates by $\delta$ from the maximum, the extracted state and measurements are within $O(\sqrt{\delta})$ of the ideal target, with constants scaling polynomially with $n$.

4. Key Results

• Optimal Quantum Bound: The maximum quantum value of the Bell functional is:
  $$(G_n)_{Q}^{opt} = 2^{n-1}\sqrt{n}$$
• Local Bound: The local ontic bound is given by:
  $$(G_n)_L = \left( \lfloor \frac{n}{2} \rfloor + 1 \right) \binom{n}{\lfloor \frac{n}{2} \rfloor + 1}$$
  (Note: For $n=2$, this reduces to the CHSH bound of 2; for $n=3$, it matches the Elegant Bell Inequality bound).
• State Structure: Any state achieving the maximal violation must be a direct sum of maximally entangled states supported on orthogonal local subspaces of dimension at least $m^* = 2^{\lfloor n/2 \rfloor}$.
• Observable Structure: The local observables on the support of the state must satisfy pairwise anticommutation relations:
  $$\{B_y, B_{y'}\} = 0 \quad \text{and} \quad \{A_x, A_{x'}\} \propto \delta_{xx'}$$
  These operators generate the Clifford algebra.
• Robustness: The distance between the physical strategy and the ideal reference strategy scales as $O(n^{1/4}\sqrt{\delta})$. This indicates that the protocol remains stable even as the number of settings $n$ increases.

5. Significance and Impact

• Scalability for DI Protocols: This work provides a practical route for certifying high-dimensional entanglement and complex measurement structures (Clifford gates) in a fully device-independent manner. This is crucial for scaling up DI Quantum Key Distribution (QKD) and Randomness Expansion, where higher dimensions or parallel copies can significantly boost key rates and randomness generation.
• Experimental Feasibility: By utilizing only dichotomic measurements (two outcomes), the protocol avoids the experimental difficulties associated with high-dimensional projective measurements (which often require many outcomes), making it more accessible for current quantum hardware.
• Theoretical Advancement: The paper bridges the gap between prepare-and-measure contextuality scenarios and Bell nonlocality, demonstrating that the algebraic structure of Clifford algebras can be self-tested via Bell inequalities. It also rigorously addresses the "complex conjugation" ambiguity, refining the theoretical definition of self-testing for real-coefficient Bell functionals.
• Future Applications: The framework enables the certification of quantum resources necessary for verifiable delegated quantum computing and blind quantum computing, where a client must trust a server's high-dimensional operations without knowing the internal workings of the server's device.

In summary, Sasmal et al. present a robust, scalable, and experimentally friendly method to certify high-dimensional quantum resources and Clifford measurements, overcoming previous limitations regarding dimension assumptions and measurement complexity.