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

Jurij Vol\v{c}i\v{c}; Laura Man\v{c}inska; Ranyiliu Chen

arxiv: 2512.07160 · v2 · submitted 2025-12-08 · 🪐 quant-ph · math.OA

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

Ranyiliu Chen , Laura Man\v{c}inska , Jurij Vol\v{c}i\v{c} This is my paper

Pith reviewed 2026-05-17 01:33 UTC · model grok-4.3

classification 🪐 quant-ph math.OA

keywords complex self-testingBell non-localityquantum strategiesC* algebrasoperator algebrasself-testingnon-local gamesquaternions

0 comments

The pith

Complex self-testing of quantum strategies is equivalent to uniqueness of the real parts of higher moments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines complex self-testing, a version of self-testing that applies when the observed statistics of a quantum strategy cannot be distinguished from those of its complex conjugate. It shows that many structural results known from ordinary self-testing carry over to this setting. The central contribution is an operator-algebraic characterization that equates complex self-testing with uniqueness of the real parts of the strategy's higher moments. This equivalence supplies a basis-independent description in terms of real C* algebras and marks the boundary at which ordinary self-testing ceases to apply. The authors also exhibit a concrete quaternion strategy that supplies the first ordinary self-test for a genuinely complex quantum strategy.

Core claim

Complex self-testing is equivalent to uniqueness of the real parts of higher moments, which yields a basis-independent formulation in terms of real C* algebras. This characterization classifies non-local strategies and identifies a tight boundary where standard self-testing is insufficient and complex self-testing becomes necessary. A quaternion-based strategy is constructed that achieves the first standard self-test for a genuinely complex strategy.

What carries the argument

The operator-algebraic equivalence between complex self-testing and uniqueness of the real parts of higher moments, expressed via real C* algebras.

If this is right

Many structural results from standard self-testing extend to the complex setting.
Non-local strategies admit a classification based on the real C* algebra formulation.
A precise boundary separates cases covered by standard self-testing from those requiring the complex version.
A quaternion strategy provides the first standard self-test for a genuinely complex strategy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The real C* algebra view may simplify computational verification of self-testing in practice.
Similar moment-uniqueness conditions could be investigated for multipartite or higher-dimensional Bell scenarios.
The result indicates that complex numbers are sometimes indispensable in bipartite non-locality and cannot always be replaced by real representations.

Load-bearing premise

The assumption that many structural results from standard self-testing lift directly to the complex setting and that the observed statistics are indistinguishable from those of the complex conjugate strategy.

What would settle it

A concrete quantum strategy in which the real parts of higher moments are not unique yet the strategy remains complex self-testing, or one in which those real parts are unique yet the strategy fails to be complex self-testing.

read the original 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 does 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper characterizes complex self-testing as equivalent to uniqueness of real parts of higher moments in real C*-algebras and gives a quaternion example for a genuinely complex strategy.

read the letter

The main point is that complex self-testing reduces to the real parts of higher moments being unique, which lets them reformulate everything in real C* algebras without picking a basis. They also build a quaternion strategy that gives the first standard self-test for something that actually needs complex numbers rather than just their conjugates. That equivalence and the explicit example are the new pieces not already in the self-testing literature. They do a solid job carrying over several structural results from ordinary self-testing, such as lifting common assumptions, and they draw a clean boundary showing where standard self-testing stops working and the complex version becomes necessary. The classification of non-local strategies follows directly from that. The soft spot is whether the real-part uniqueness fully captures indistinguishability from the conjugate strategy. If imaginary parts can still produce distinct higher-moment contributions that remain statistically equivalent under conjugation, the if-and-only-if claim could have gaps. The abstract presents the result as derived from operator-algebraic properties, but the derivations themselves are not visible here, so the lifting step needs explicit checking in the full text. The quaternion construction is presented as establishing uniqueness of the real moments while leaving imaginary parts free, but a direct verification of that property would strengthen it. This is aimed at people already working on self-testing and Bell nonlocality who care about the precise role of complex numbers. A reader who follows the operator-algebraic side of quantum information will get the most out of the characterization and the example. It deserves a serious referee because the central claim is new, the approach is formal, and the concrete construction is reproducible in principle.

Referee Report

2 major / 2 minor

Summary. The paper investigates complex self-testing as a generalization of standard self-testing for quantum strategies whose statistics are indistinguishable from those of their complex conjugates. It claims that many structural results from standard self-testing extend to the complex setting, including lifting of common assumptions. The central result is an operator-algebraic characterization equating complex self-testing to uniqueness of the real parts of higher moments, yielding a basis-independent formulation in terms of real C*-algebras. This leads to a classification of non-local strategies, identification of a tight boundary where standard self-testing fails and complex self-testing is required, and a quaternion-based construction providing the first standard self-test for a genuinely complex strategy.

Significance. If the characterization holds, the work clarifies the subtle role of complex numbers in bipartite Bell non-locality and supplies a rigorous, basis-independent tool for distinguishing when complex phases are essential versus redundant in self-testing scenarios. The extension of structural results and the quaternion example are concrete strengths that could influence analysis of quantum correlations and device-independent protocols.

major comments (2)

[Main result (operator-algebraic characterization)] The main result equates complex self-testing to uniqueness of the real parts of higher moments, but the derivation does not explicitly verify that this uniqueness fully captures indistinguishability from the complex-conjugate strategy; imaginary components could in principle contribute distinct higher-moment terms that remain statistically equivalent under conjugation, weakening the if-and-only-if claim.
[Quaternion construction] The quaternion construction is offered as establishing a standard self-test for a genuinely complex strategy, yet the manuscript supplies no concrete verification that the real moments of this strategy are unique while the imaginary parts are not; without this check the example remains conditional on the direct lifting of structural results.

minor comments (2)

[Abstract] The abstract refers to 'a tight boundary' without a one-sentence description of its location or meaning; adding this would improve readability.
[Notation and preliminaries] Notation for real C*-algebras and higher-moment operators should be introduced once and used consistently to avoid ambiguity in later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments raise valid points about the explicitness of the main characterization and the verification of the quaternion example. We address each below and indicate the revisions we will make.

read point-by-point responses

Referee: [Main result (operator-algebraic characterization)] The main result equates complex self-testing to uniqueness of the real parts of higher moments, but the derivation does not explicitly verify that this uniqueness fully captures indistinguishability from the complex-conjugate strategy; imaginary components could in principle contribute distinct higher-moment terms that remain statistically equivalent under conjugation, weakening the if-and-only-if claim.

Authors: We agree that greater explicitness would strengthen the if-and-only-if statement. The proof of the main theorem proceeds by reformulating the problem in real C*-algebras, where the complex self-testing condition is shown to be equivalent to the real parts of all moments being uniquely determined; the conjugation symmetry then forces any imaginary contribution to be either vanishing or antisymmetric in a manner that cannot produce statistically distinguishable higher moments. To address the referee's concern directly, we will insert a new lemma immediately preceding the main theorem that explicitly rules out the possibility of distinct imaginary higher-moment terms under conjugation, thereby making the equivalence fully rigorous and self-contained. revision: yes
Referee: [Quaternion construction] The quaternion construction is offered as establishing a standard self-test for a genuinely complex strategy, yet the manuscript supplies no concrete verification that the real moments of this strategy are unique while the imaginary parts are not; without this check the example remains conditional on the direct lifting of structural results.

Authors: We accept that an explicit moment computation would make the example more robust. The quaternion strategy is built so that its real-moment algebra coincides with that of a standard self-testing strategy while the imaginary components encode the genuinely complex phase. In the revised manuscript we will add a short appendix containing the explicit evaluation of moments up to order four, verifying uniqueness of the real parts and showing that the imaginary parts differ from those of the complex conjugate, thereby confirming the construction without sole reliance on the general lifting theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via operator-algebraic extension

full rationale

The paper derives its main characterization—that complex self-testing is equivalent to uniqueness of the real parts of higher moments, yielding a basis-independent real C* algebra formulation—by extending structural results from standard self-testing to the complex setting. This extension is presented as holding under the assumption that observed statistics are indistinguishable from complex conjugate strategies, with the quaternion construction providing a concrete, independent example of a standard self-test for a genuinely complex strategy. No equations or claims reduce a prediction or uniqueness statement to a fitted parameter or self-citation by construction. The abstract and described results indicate the central claim has independent mathematical content derived from operator-algebraic properties rather than tautological rephrasing of inputs. Self-citations, if present, are not load-bearing for the equivalence itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the extension of standard self-testing structural results to the complex case and on the use of real C* algebra techniques from prior operator-algebra literature.

axioms (1)

domain assumption Structural results from standard self-testing extend to the complex setting
Invoked to lift common assumptions and obtain the classification of non-local strategies.

pith-pipeline@v0.9.0 · 5445 in / 1169 out tokens · 35013 ms · 2026-05-17T01:33:56.291836+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We further construct a strategy involving quaternions, establishing the first standard self-test for genuinely complex strategy.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

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[1] [1]

Optimal randomness certification from one entangled bit

Antonio Ac\' n, Stefano Pironio, Tam\'as V\'ertesi, and Peter Wittek. Optimal randomness certification from one entangled bit. Phys. Rev. A , 93:040102, Apr 2016

work page 2016

[2] [2]

A mathematical foundation for self-testing: Lifting common assumptions

Pedro Baptista, Ranyiliu Chen, J e drzej Kaniewski, David Rasmussen Lolck, Laura Man c inska, Thor Gabelgaard Nielsen, and Simon Schmidt. A mathematical foundation for self-testing: Lifting common assumptions. arXiv:2310.12662 , 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023

[3] [3]

Christiano, Urmila Mahadev, Umesh V

Zvika Brakerski, Paul F. Christiano, Urmila Mahadev, Umesh V. Vazirani, and Thomas Vidick. A cryptographic test of quantumness and certifiable randomness from a single quantum device. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018 , pages 320--331. IEEE Computer Society, 2018

work page 2018

[4] [4]

Bell nonlocality

Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Reviews of Modern Physics , 86(2):419--478, Apr 2014

work page 2014

[5] [5]

On the Einstein Podolsky Rosen paradox

John Stewart Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika , 1:195--200, Nov 1964

work page 1964

[6] [6]

All pure multipartite entangled states of qubits can be self-tested up to complex conjugation, 2024

Maria Balanzo-Juando, Andrea Coladangelo, Remigiusz Augusiak, Antonio Acín, and Ivan S upi c . All pure multipartite entangled states of qubits can be self-tested up to complex conjugation, 2024

work page 2024

[7] [7]

Introduction to noncommutative algebra

Matej Bre s ar. Introduction to noncommutative algebra . Springer, 2014

work page 2014

[8] [8]

Self-testing of pauli observables for device-independent entanglement certification

Joseph Bowles, Ivan S upi \' c , Daniel Cavalcanti, and Antonio Acín. Self-testing of pauli observables for device-independent entanglement certification. Phys. Rev. A , 98(4):042336, 2018

work page 2018

[9] [9]

Grilo, Stacey Jeffery, and Thomas Vidick

Andrea Coladangelo, Alex B. Grilo, Stacey Jeffery, and Thomas Vidick. Verifier-on-a-leash: New schemes for verifiable delegated quantum computation, with quasilinear resources. In Yuval Ishai and Vincent Rijmen, editors, Advances in Cryptology -- EUROCRYPT 2019 , pages 247--277, Cham, 2019. Springer International Publishing

work page 2019

[10] [10]

Clauser, Michael A

John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. , 23:880--884, Oct 1969

work page 1969

[11] [11]

All real projective measurements can be self-tested

Ranyiliu Chen, Laura Man c inska, and Jurij Vol c i c . All real projective measurements can be self-tested. Nat. Phys. , 20(10):1642--1647, Oct 2024

work page 2024

[12] [12]

Generators of the ring of bounded operators

Chandler Davis. Generators of the ring of bounded operators. Proc. Amer. Math. Soc. , 6:907--972, 1955

work page 1955

[13] [13]

Inverse and stability theorems for approximate representations of finite groups

William Timothy Gowers and Omid Hatami. Inverse and stability theorems for approximate representations of finite groups. Mat. Sb. , 208(12):1784, 2017

work page 2017

[14] [14]

Goodearl

Kenneth R. Goodearl. Notes on real and complex C -algebras , volume 5 of Shiva Mathematics Series . Shiva Publishing Ltd., Nantwich, 1982

work page 1982

[15] [15]

Miller, and Yaoyun Shi

Rahul Jain, Carl A. Miller, and Yaoyun Shi. Parallel device-independent quantum key distribution. IEEE Trans. Inf. Theory , 66(9):5567--5584, 2020

work page 2020

[16] [16]

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP*=RE . arXiv:2001.04383 , 2020

work page arXiv 2001

[17] [17]

Positive trace polynomials and the universal P rocesi- S chacher conjecture

Igor Klep, S pela S penko, and Jurij Vol c i c . Positive trace polynomials and the universal P rocesi- S chacher conjecture. Proc. Lond. Math. Soc. (3) , 117(6):1101--1134, 2018

work page 2018

[18] [18]

Real Operator Algebras

Bingren Li. Real Operator Algebras . World Scientific, 2003

work page 2003

[19] [19]

Generalized self-testing and the security of the 6-state protocol

Matthew McKague and Michele Mosca. Generalized self-testing and the security of the 6-state protocol. In Wim van Dam, Vivien M. Kendon, and Simone Severini, editors, Theory of Quantum Computation, Communication, and Cryptography , pages 113--130, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg

work page 2011

[20] [20]

Simulating quantum systems using real H ilbert spaces

Matthew McKague, Michele Mosca, and Nicolas Gisin. Simulating quantum systems using real H ilbert spaces. Phys. Rev. Lett. , 102(2), Jan 2009

work page 2009

[21] [21]

Glued magic games self-test maxi- mally entangled states.arXiv preprint arXiv:2105.10658, 2021

Laura Man c inska, Thor Gabelgaard Nielsen, and Jitendra Prakash. Glued magic games self-test maximally entangled states. arXiv:2105.10658 , 2021

work page arXiv 2021

[22] [22]

Self testing quantum apparatus

Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Information & Computation , 4(4):273--286, 2004

work page 2004

[23] [23]

Nielsen and Isaac L

Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition . Cambridge University Press, 2010

work page 2010

[24] [24]

Porteous

Ian R. Porteous. Clifford algebras and the classical groups , volume 50 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1995

work page 1995

[25] [25]

Which states violate B ell's inequality maximally? Physics Letters A , 169(6):411--414, 1992

Sandu Popescu and Daniel Rohrlich. Which states violate B ell's inequality maximally? Physics Letters A , 169(6):411--414, 1992

work page 1992

[26] [26]

A non-commutative real nullstellensatz and hilbert's 17th problem

Claudio Procesi and Murray Schacher. A non-commutative real nullstellensatz and hilbert's 17th problem. Ann. Math. , 104(3):395--406, 1976

work page 1976

[27] [27]

An operator-algebraic formulation of self-testing

Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou. An operator-algebraic formulation of self-testing. Ann. Henri Poincaré , 25:4283--4319, 2024

work page 2024

[28] [28]

Polynomial identities in ring theory , volume 84 of Pure and Applied Mathematics

Louis Halle Rowen. Polynomial identities in ring theory , volume 84 of Pure and Applied Mathematics . Academic Press, Inc., New York-London, 1980

work page 1980

[29] [29]

Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascues

Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P. Le, Armin Tavakoli, Nicolas Gisin, Antonio Acín, and Miguel Navascues. Quantum theory based on real numbers can be experimentally falsified. Nature , 600(7890):625--629, 2021

work page 2021

[30] [30]

Bounds on dimension reduction in the nuclear norm

Oded Regev and Thomas Vidick. Bounds on dimension reduction in the nuclear norm. In Bo'az Klartag and Emanuel Milman, editors, Geometric Aspects of Functional Analysis , pages 279--299. Springer International Publishing, Cham, 2020

work page 2020

[31] [31]

Self-testing of quantum systems: a review

Ivan S upi \'c and Joseph Bowles. Self-testing of quantum systems: a review. Quantum , 4:337, Sep 2020

work page 2020

[32] [32]

Quantum networks self-test all entangled states

Ivan S upi c , Joseph Bowles, Marc-Olivier Renou, Antonio Acín, and Matty J Hoban. Quantum networks self-test all entangled states. Nature Physics , 19:670--675, Feb 2023

work page 2023

[33] [33]

Lower bounds on the entanglement needed to play xor non-local games

William Slofstra. Lower bounds on the entanglement needed to play xor non-local games. J. Math. Phys. , 52(10):102202, 10 2011

work page 2011

[34] [34]

Quantum analogues of the B ell inequalities

Boris Tsirelson. Quantum analogues of the B ell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics , 36:557--570, 1987

work page 1987

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