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

Souradeep Sasmal, Ritesh K. Singh, Prabuddha Roy, A. K. Pan

Abstract

The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension $m_*=2^{\lfloor n/2 \rfloor}$ (equivalently $\lfloor n/2 \rfloor$ copies of maximally entangled two-qubit pairs), together with $n$ numbers of anti-commuting observables on one side. To this end, we employ an $n$-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having $2^{n-1}$ and $n$ number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least $2^{\lfloor n/2 \rfloor}$ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with $n$ mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.

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

Abstract

The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension (equivalently copies of maximally entangled two-qubit pairs), together with numbers of anti-commuting observables on one side. To this end, we employ an -settings Bell inequality comprising two space-like separated observers, Alice and Bob, having and number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.
Paper Structure (20 sections, 7 theorems, 247 equations)

This paper contains 20 sections, 7 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bell scenario: Local, Quantum and Nosignalling correlations
  4. Bell functionals and inequalities
  5. Optimal quantum Bell violation, Quantum extremality and self-testing
  6. Results: Classical bound of the considered higher settings Bell functional
  7. Results: Optimal quantum bound of $\mathscr{G}_n$
  8. Results: Device-Independent Characterisation of the Optimal Quantum Strategy - Self-Testing from the Maximal Violation of $\mathscr{G}_n$
  9. Robust self-testing
  10. Summary and Discussion
  11. Acknowledgements
  12. Proof of theorem 1
  13. Evaluation of the optimal quantum bound of the Bell functional $\mathscr{G}_n$ via the SOS Method
  14. Evaluation of $\Tr[\qty{B_y,B_{y'}}\rho] \ \ \forall y \neq y'$
  15. Observable relations
  16. ...and 5 more sections

Key Result

Theorem 1

Any ontological model satisfying the condition of factorisability of the joint conditional probability must satisfy the following inequality For $n=2,3$, the above inequality given by Eq. (nbell) reduces to the well-known CHSH inequality CHSH1970 and elegant Bell inequality Gisin2007 respectively.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3: Blockwise self-testing of the Clifford realisation
  • proof
  • Lemma 1: Direct-sum composition of local isometries
  • proof
  • ...and 6 more