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Certifying Majorana Fermions with Elegant-Like Bell Inequalities and a New Self-Testing Equivalence

Patryk Michalski, Arturo Konderak, Wojciech Bruzda, Remigiusz Augusiak

TL;DR

This work introduces ROCN (row-orthogonal, column-normalized) matrices as a general method to construct correlation Bell inequalities with analytically computable quantum bounds. The authors show these ROCN Bell inequalities are maximally violated by Clifford observables on maximally entangled states, enabling device-independent certification of Majorana fermions, and they identify a new self-testing equivalence arising from partial transposition. A key result is the exact quantum bound $β_Q^h = n$ and the classical bound characterization, with self-testing guaranteed when a rank condition on an induced matrix $M$ is satisfied; for odd numbers of generators, an additional Clifford-related equivalence arises. The framework unifies and extends classic Bell inequalities (e.g., CHSH, Gisin’s elegant inequality) and connects to Hadamard and Platonic families, providing analytic insight into quantum-classical gaps and potential applications in Majorana physics and quantum certification. The work also discusses Hadamard-based realizations, Hadamard excess, and implications for quantum networks via potential extensions of ROCN to nonlocal or bilocal scenarios, highlighting both practical and foundational significance for device-independent quantum information.

Abstract

Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable through quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. Our framework generalizes both the Clauser-Horne-Shimony-Holt and Gisin's elegant inequalities, yielding Bell expressions maximally violated by any number of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, our inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into the definition of self-testing beyond invariance under local isometries and transposition. This equivalence arises from partial transposition applied to the shared state and to the measurements, which in specific cases leaves all observed correlations unchanged.

Certifying Majorana Fermions with Elegant-Like Bell Inequalities and a New Self-Testing Equivalence

TL;DR

This work introduces ROCN (row-orthogonal, column-normalized) matrices as a general method to construct correlation Bell inequalities with analytically computable quantum bounds. The authors show these ROCN Bell inequalities are maximally violated by Clifford observables on maximally entangled states, enabling device-independent certification of Majorana fermions, and they identify a new self-testing equivalence arising from partial transposition. A key result is the exact quantum bound and the classical bound characterization, with self-testing guaranteed when a rank condition on an induced matrix is satisfied; for odd numbers of generators, an additional Clifford-related equivalence arises. The framework unifies and extends classic Bell inequalities (e.g., CHSH, Gisin’s elegant inequality) and connects to Hadamard and Platonic families, providing analytic insight into quantum-classical gaps and potential applications in Majorana physics and quantum certification. The work also discusses Hadamard-based realizations, Hadamard excess, and implications for quantum networks via potential extensions of ROCN to nonlocal or bilocal scenarios, highlighting both practical and foundational significance for device-independent quantum information.

Abstract

Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable through quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. Our framework generalizes both the Clauser-Horne-Shimony-Holt and Gisin's elegant inequalities, yielding Bell expressions maximally violated by any number of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, our inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into the definition of self-testing beyond invariance under local isometries and transposition. This equivalence arises from partial transposition applied to the shared state and to the measurements, which in specific cases leaves all observed correlations unchanged.

Paper Structure

This paper contains 19 sections, 6 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bell nonlocality & Bell inequalities
  4. Self-testing
  5. Clifford algebra
  6. Bell inequalities from ROCN matrices
  7. Quantum bound
  8. Classical bound
  9. Self-testing from ROCN Bell inequalities
  10. Applications
  11. Comparison with other families of Bell inequalities
  12. A note on Hadamard matrices
  13. Classical value and excess of a Hadamard matrix
  14. Conclusions
  15. Acknowledgments
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\{A_i\}_{i=1}^m$ be a set of observables acting on a finite-dimensional Hilbert space $\mathcal{H}$ and satisfying the canonical anticommutation relations eq:anticomm_th1. Let $m=2r+\varepsilon$, with $\varepsilon\in\{0,1\}$ and $r=\lfloor m/2\rfloor$. Then, up to a unitary transformation, the With respect to this decomposition, the first $2r$ operators $A_i$ can be expressed in terms of Pau

Figures (1)

  • Figure 1: The three classes of Bell inequalities are compared. The Hadamard truncated (HT) class, introduced in Subsection \ref{['sec:hadamard_matrices']}, generalizes both the CHSH and Gisin's elegant Bell inequality (EBI). The ROCN family, discussed in this work, and the Platonic Bell inequalities, introduced in Ref. Pal2022, both contain the Hadamard truncated Bell inequalities.

Theorems & Definitions (19)

  • Definition 1: Self-testing
  • Theorem 1: Jordan--Wigner Representation Theorem
  • Definition 2: ROCN matrix
  • Proposition 1: Quantum bound
  • Definition 3
  • proof : Proof of Proposition \ref{['prop:quantum_bound']}.
  • Remark 1
  • Theorem 2: Self-testing from ROCN Bell inequalities
  • Remark 2
  • Remark 3
  • ...and 9 more