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Beyond real: Investigating the role of complex numbers in self-testing

Ranyiliu Chen, Laura Mančinska, Jurij Volčič

TL;DR

This work extends self-testing to complex quantum strategies, introducing complex self-testing and an operator-algebraic characterization that ties it to the uniqueness of the real parts of higher moments across all reproducing strategies. It employs real $C^*$-algebras to provide a basis-independent, algebraic formulation and identifies a boundary where standard self-testing is insufficient, necessitating complex self-testing; a quaternion-based construction demonstrates a nontrivial self-test in this intermediate regime. A key technical advance is showing that complex self-testing corresponds to a unique real-state on a universal real POVM algebra, linking moment structures to operator-algebraic realizations. Finally, the paper presents a quaternion-based self-test and analyzes minimal projections needed to generate quaternionic matrix algebras, revealing fundamental limits and new avenues for nonlocality certification in the complex domain.

Abstract

We investigate complex self-testing, a generalization of standard self-testing that accounts for quantum strategies whose statistics is indistinguishable from their complex conjugate's. We show that many structural results from standard self-testing extend to the complex setting, including lifting of common assumptions. Our main result is an operator-algebraic characterization: complex self-testing is equivalent to uniqueness of the real parts of higher moments, leading to a basis-independent formulation in terms of real C* algebras. This leads to a classification of non-local strategies, and a tight boundary where standard self-testing do not apply and complex self-testing is necessary. We further construct a strategy involving quaternions, establishing the first standard self-test for genuinely complex strategy. Our work clarifies the structure of complex self-testing and highlights the subtle role of complex numbers in bipartite Bell non-locality.

Beyond real: Investigating the role of complex numbers in self-testing

TL;DR

This work extends self-testing to complex quantum strategies, introducing complex self-testing and an operator-algebraic characterization that ties it to the uniqueness of the real parts of higher moments across all reproducing strategies. It employs real -algebras to provide a basis-independent, algebraic formulation and identifies a boundary where standard self-testing is insufficient, necessitating complex self-testing; a quaternion-based construction demonstrates a nontrivial self-test in this intermediate regime. A key technical advance is showing that complex self-testing corresponds to a unique real-state on a universal real POVM algebra, linking moment structures to operator-algebraic realizations. Finally, the paper presents a quaternion-based self-test and analyzes minimal projections needed to generate quaternionic matrix algebras, revealing fundamental limits and new avenues for nonlocality certification in the complex domain.

Abstract

We investigate complex self-testing, a generalization of standard self-testing that accounts for quantum strategies whose statistics is indistinguishable from their complex conjugate's. We show that many structural results from standard self-testing extend to the complex setting, including lifting of common assumptions. Our main result is an operator-algebraic characterization: complex self-testing is equivalent to uniqueness of the real parts of higher moments, leading to a basis-independent formulation in terms of real C* algebras. This leads to a classification of non-local strategies, and a tight boundary where standard self-testing do not apply and complex self-testing is necessary. We further construct a strategy involving quaternions, establishing the first standard self-test for genuinely complex strategy. Our work clarifies the structure of complex self-testing and highlights the subtle role of complex numbers in bipartite Bell non-locality.

Paper Structure

This paper contains 13 sections, 24 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notion
  3. Bell scenarios
  4. Complex dilation and complex self-testing
  5. Basic properties, and lifting assumptions in complex self-testing
  6. An operator-algebraic characterization
  7. Complex self-testing implies a unique real part of moments
  8. Unique real part of moments implies complex self-testing
  9. The operator-algebraic formulation
  10. Realness of quantum strategies
  11. A quaternion middle ground possibility
  12. A self-test involving quaternions
  13. Generating quaternions by few projections

Key Result

Lemma 2.2

A strategy $(\ket{\psi},\{E_{xa}\},\{F_{yb}\})$ is support-preserving if and only if there exist operators $\hat{E}_{xa}$, $\hat{F}_{yb}$ such that $E_{xa}\otimes \mathsf{Id}\ket{\psi}=\mathsf{Id}\otimes\hat{E}_{xa}\ket{\psi}$ and $\mathsf{Id}\otimes F_{yb}\ket{\psi}=\hat{F}_{yb}\otimes \mathsf{Id}\

Theorems & Definitions (55)

  • Definition 2.1
  • Lemma 2.2: Lemmas 4.3 & 4.4 of paddock2023operatoralgebraic; see also Lemma 3.3 of baptista2023mathematicalfoundationselftestinglifting for the approximate version
  • Definition 2.3: Local dilation
  • Definition 2.4: Complex local dilation
  • Definition 2.5: complex local dilation, alternative
  • Lemma 2.6
  • proof
  • Definition 2.7: complex self-testing
  • Proposition 3.1: Counterpart of Proposition 3.4 in baptista2023mathematicalfoundationselftestinglifting
  • proof
  • ...and 45 more