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arxiv: 2606.21626 · v1 · pith:CLV24JRPnew · submitted 2026-06-19 · 🪐 quant-ph

Systematic derivation of Tsirelson bounds in arbitrary dimensions

Lorenzo Coccia , Matteo Padovan , Giuseppe Vallone This is my paper

Pith reviewed 2026-06-26 13:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Tsirelson boundsBell nonlocalitysum-of-squares decompositionsquantum correlationsqubitsquditsbipartite systems
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The pith

Sum-of-squares decompositions enable systematic derivation of Tsirelson bounds in arbitrary dimensions.

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

The paper introduces a method to derive tight bounds on quantum correlations, known as Tsirelson bounds, for systems of any dimension using sum-of-squares decompositions. This approach recovers known results for qubits and qudits while identifying new bounds. A reader would care because determining these bounds for high-dimensional systems has been challenging, limiting understanding of quantum nonlocality beyond simple cases. The method aims to characterize both quantum and local bounds through algebraic decompositions without dimension-specific adjustments.

Core claim

We propose a systematic derivation of bipartite Tsirelson and local bounds written in terms of sum-of-squares decompositions. Using this method, we discover novel bounds and recover established results for maximally entangled states of qubits and qudits.

What carries the argument

Sum-of-squares decompositions that express correlation expressions to derive bounds on the quantum set in bipartite Bell scenarios.

If this is right

  • Novel Tsirelson bounds are obtained for high-dimensional systems where previous analytic methods were unavailable.
  • Known Tsirelson bounds for maximally entangled qubit and qudit states are recovered as special cases.
  • Both quantum (Tsirelson) and classical (local) bounds are derived uniformly through the same decomposition technique.
  • The approach extends the characterization of the quantum set to arbitrary local dimensions without case-by-case analysis.

Where Pith is reading between the lines

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

  • The decompositions might be turned into an algorithmic procedure that automates bound derivation for user-specified Bell expressions and dimensions.
  • Similar sum-of-squares techniques could be tested on non-maximally entangled states or on scenarios with more than two parties to check for broader applicability.
  • The method may intersect with semidefinite programming hierarchies, potentially offering tighter relaxations or exact solutions for certain correlation polytopes.

Load-bearing premise

The assumption that sum-of-squares decompositions provide a complete and tight characterization of the quantum set for arbitrary dimensions without requiring additional constraints or numerical verification specific to each dimension.

What would settle it

An explicit quantum strategy consisting of a maximally entangled state and projective measurements in dimension three that achieves a correlation value strictly larger than the Tsirelson bound derived via the sum-of-squares method for the same Bell expression.

Figures

Figures reproduced from arXiv: 2606.21626 by Giuseppe Vallone, Lorenzo Coccia, Matteo Padovan.

Figure 1
Figure 1. Figure 1: Comparison of the quantum-to-local bound ratios for the Bell operator (105) from [31] and the novel Bell operator (120) derived in this work. convenient framework in Eqs. (140) and (141), the generalized version of the SATWAP operator for two inputs is derived in Eq. (150). Comment on the notation: In this last section we will not need to use different subscripts for normal operators and for projectors. We… view at source ↗
Figure 2
Figure 2. Figure 2: Ratio between quantum bound and local bound of the generalized SATWAP operator (152) for different values of β and d. 5 Conclusions and outlook In this work, we addressed the problem of determining Tsirelson and local bounds for arbi￾trarily high-dimensional systems, in the case of two separated parties. We presented a system￾atic derivation of Bell operators whose Tsirelson bounds are saturated by biparti… view at source ↗
read the original abstract

The study of Bell nonlocality and the bounds of quantum correlations, the so-called Tsirelson bounds, is fundamental to quantum information science and the exploration of the limits of quantum theory. While quantum bounds for qubit systems have been extensively characterized, determining tight quantum bounds for correlations attainable with high-dimensional quantum states and measurements remains a significant challenge. In this work, we propose a systematic derivation of bipartite Tsirelson and local bounds written in terms of sum-of-squares decompositions. Using this method, we discover novel bounds and recover established results for maximally entangled states of qubits and qu$d$its.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a systematic method for deriving bipartite Tsirelson bounds (and local bounds) for Bell expressions in arbitrary dimensions by rewriting them as sum-of-squares decompositions. The approach is applied to recover known tight bounds for maximally entangled qubit and qudit states and to obtain novel candidate bounds.

Significance. A reliable systematic derivation of tight Tsirelson bounds for high-dimensional systems would be valuable, as current methods are often dimension-specific or numerical. The SOS framework is a standard tool in quantum information, and recovering known results for qubits/qudits provides a useful sanity check; however, the significance hinges on whether the novel bounds are provably tight rather than relaxations.

major comments (2)
  1. [Method and novel bounds sections] The central claim that the SOS decompositions yield the actual Tsirelson bounds (rather than upper bounds) for novel cases in arbitrary d requires an explicit achievability argument or duality certificate showing the bound is attained by some quantum state and measurements. The abstract and method description indicate that SOS identities alone are used; without a matching lower bound or explicit realization for the new bounds, the identification with the quantum set remains unproven (see skeptic note on tightness).
  2. [Results for qudits] For the recovered results on maximally entangled states, the paper correctly matches literature values, but the extension to arbitrary dimensions and novel bounds relies on the assumption that the chosen SOS is complete without additional constraints; this needs explicit verification or a general proof that no gap exists.
minor comments (2)
  1. [Section 2] Clarify the notation for the Bell expression and the precise form of the SOS decomposition in the main text or an appendix to make the systematic aspect reproducible.
  2. [Numerical checks] Include a table or explicit comparison of the new bounds against known numerical SDP relaxations for small d to support the claims.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for the careful review and valuable comments on the tightness of the derived bounds. We address each major point below and will revise the manuscript accordingly to avoid overstating the results for novel cases.

read point-by-point responses

  1. Referee: [Method and novel bounds sections] The central claim that the SOS decompositions yield the actual Tsirelson bounds (rather than upper bounds) for novel cases in arbitrary d requires an explicit achievability argument or duality certificate showing the bound is attained by some quantum state and measurements. The abstract and method description indicate that SOS identities alone are used; without a matching lower bound or explicit realization for the new bounds, the identification with the quantum set remains unproven (see skeptic note on tightness).

    Authors: We agree that SOS decompositions provide rigorous upper bounds on the quantum value. The manuscript recovers known tight Tsirelson bounds for qubits and qudits by matching literature values, which serve as a sanity check. For the novel bounds in arbitrary dimensions, no explicit quantum realizations or duality certificates are provided. We will revise the abstract, method description, and conclusions to present these explicitly as candidate upper bounds obtained via the SOS method, rather than claiming they are the exact Tsirelson bounds, and note that achievability remains open. revision: yes

  2. Referee: [Results for qudits] For the recovered results on maximally entangled states, the paper correctly matches literature values, but the extension to arbitrary dimensions and novel bounds relies on the assumption that the chosen SOS is complete without additional constraints; this needs explicit verification or a general proof that no gap exists.

    Authors: The results for maximally entangled qudits are verified by exact matching to established tight bounds in the literature. For arbitrary d, the SOS polynomials are selected according to the monomial structure of the Bell expression, but the manuscript does not contain a general proof of completeness or absence of gaps in the relaxation. We will add a dedicated paragraph in the results section acknowledging this assumption, providing any low-dimensional numerical checks where feasible, and clarifying that the method yields systematic upper bounds whose tightness is not guaranteed for all novel cases. revision: partial

standing simulated objections not resolved
  • Explicit achievability arguments or quantum realizations attaining the novel bounds for arbitrary d
  • A general proof that the chosen SOS decompositions are complete with no relaxation gap for arbitrary dimensions

Circularity Check

0 steps flagged

No circularity; derivation is method-based upper bounds with known-case recovery

full rationale

The abstract and description present a systematic SOS-based rewriting to obtain candidate bounds, recovering established tight results for maximally entangled qubit/qudit states (where achievability is external) and proposing novel ones. No quoted equations or steps reduce the claimed Tsirelson identification to a self-definition, fitted input renamed as prediction, or self-citation chain. The method generates upper bounds via SOS; any gap to actual quantum maximum for novel cases is a completeness question, not a circular reduction by construction. The derivation chain is self-contained against external benchmarks for the recovered cases.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information from abstract only to identify free parameters, axioms, or invented entities.

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discussion (0)

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Reference graph

Works this paper leans on

41 extracted references · 31 canonical work pages · 2 internal anchors

  1. [1]

    Physics Physique Fizika1, 195–200 (1964) https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

    J. S. Bell. “On the Einstein-Podolsky-Rosen paradox”. In:Physics Physique Fizika1 (1964), pp. 195–200.doi:10.1103/PhysicsPhysiqueFizika.1.195

    work page doi:10.1103/physicsphysiquefizika.1.195 1964

  2. [2]

    Bell nonlocality

    Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. “Bell nonlocality”. In:Rev. Mod. Phys.86 (2 Apr. 2014), pp. 419–478.doi: 10.1103/RevModPhys.86.419

    work page doi:10.1103/revmodphys.86.419 2014

  3. [3]

    Valerio Scarani.The device-independent outlook on quantum physics (lecture notes on the power of Bell’s theorem). 2015. arXiv:1303.3081 [quant-ph]

    Pith/arXiv arXiv 2015

  4. [4]

    doi:10.22331/q-2020-09-30-337 , url =

    Ivan Šupić and Joseph Bowles. “Self-testing of quantum systems: a review”. In:Quantum 4 (Sept. 2020), p. 337.issn: 2521-327X.doi:10.22331/q-2020-09-30-337

    work page doi:10.22331/q-2020-09-30-337 2020

  5. [5]

    Advances in device-independent quantum key distribution

    Víctor Zapatero, Tim van Leent, Rotem Arnon-Friedman, Wen-Zhao Liu, Qiang Zhang, Harald Weinfurter, and Marcos Curty. “Advances in device-independent quantum key distribution”. In:npj Quantum Information9.10 (2023).doi:10 . 1038 / s41534 - 023 - 00684-x

    2023

  6. [6]

    Securityofdevice-independentquantumkeydistribution protocols: a review

    Ignatius W. Primaatmaja, Koon Tong Goh, Ernest Y.-Z. Tan, John T.-F. Khoo, Shouvik Ghorai,andCharlesC.-W.Lim.“Securityofdevice-independentquantumkeydistribution protocols: a review”. In:Quantum7 (Mar. 2023), p. 932.issn: 2521-327X.doi:10.22331/ q-2023-03-02-932

    2023

  7. [7]

    Hidden Variables, Joint Probability, and the Bell Inequalities

    Arthur Fine. “Hidden Variables, Joint Probability, and the Bell Inequalities”. In:Phys. Rev. Lett.48 (5 Feb. 1982), pp. 291–295.doi:10.1103/PhysRevLett.48.291

    work page doi:10.1103/physrevlett.48.291 1982

  8. [8]

    Quantum generalitazions of Bell’s inequality

    B. S. Cirelson. “Quantum generalitazions of Bell’s inequality”. In:Lett. Math. Phys.4 (1980), pp. 93–100.doi:10.1007/BF00417500

    work page doi:10.1007/bf00417500 1980

  9. [9]

    Device-independent secure cor- relations in sequential quantum scenarios

    Matteo Padovan, Alessandro Rezzi, and Lorenzo Coccia. “Device-independent secure cor- relations in sequential quantum scenarios”. In:Quantum10 (June 2026), p. 2131.issn: 2521-327X.doi:10.22331/q-2026-06-11-2131

    work page doi:10.22331/q-2026-06-11-2131 2026

  10. [10]

    Quantum bounds and device-independent security with rank-one qubit measurements

    Lorenzo Coccia, Matteo Padovan, Andrea Pompermaier, Mattia Sabatini, Marco Avesani, Davide G. Marangon, Paolo Villoresi, and Giuseppe Vallone. “Quantum bounds and device-independent security with rank-one qubit measurements”. In:npj Quantum In- formation12.1 (Jan. 2026), p. 29.issn: 2056-6387.doi:10.1038/s41534-025-01175-x

    work page doi:10.1038/s41534-025-01175-x 2026

  11. [11]

    Secure and robust randomness with sequential quantum measure- ments

    Matteo Padovan, Giulio Foletto, Lorenzo Coccia, Marco Avesani, Paolo Villoresi, and Giuseppe Vallone. “Secure and robust randomness with sequential quantum measure- ments”. In:npj Quantum Information10.1 (Sept. 2024), p. 94.issn: 2056-6387.doi: 10.1038/s41534-024-00879-w

    work page doi:10.1038/s41534-024-00879-w 2024

  12. [12]

    Experimental test of sequential weak measurements for certified quan- tum randomness extraction

    Giulio Foletto, Matteo Padovan, Marco Avesani, Hamid Tebyanian, Paolo Villoresi, and Giuseppe Vallone. “Experimental test of sequential weak measurements for certified quan- tum randomness extraction”. In:Phys. Rev. A103 (6 June 2021), p. 062206.doi:10. 1103/PhysRevA.103.062206

    2021

  13. [13]

    Maximal device-independent randomness in every dimension

    Máté Farkas, Jurij Volčič, Sigurd A. L. Storgaard, Ranyiliu Chen, and Laura Mančinska. “Maximal device-independent randomness in every dimension”. In:Nature Physics22.2 (Feb. 2026), pp. 319–324.issn: 1745-2481.doi:10.1038/s41567-025-03141-y. 5 Conclusions and outlook 37

    work page doi:10.1038/s41567-025-03141-y 2026

  14. [14]

    Optimal randomness certification from one entangled bit

    Antonio Acín, Stefano Pironio, Tamás Vértesi, and Peter Wittek. “Optimal randomness certification from one entangled bit”. In:Phys. Rev. A93 (4 Apr. 2016), p. 040102.doi: 10.1103/PhysRevA.93.040102

    work page doi:10.1103/physreva.93.040102 2016

  15. [15]

    2017.url:https://oqp.iqoqi.oeaw.ac.at/ (visited on 04/08/2026)

    IQOQI Vienna.Open Quantum Problems. 2017.url:https://oqp.iqoqi.oeaw.ac.at/ (visited on 04/08/2026)

    2017

  16. [16]

    QuantumanaloguesoftheBellinequalities.Thecaseoftwospatiallysep- arated domains

    B.S.Tsirel’son.“QuantumanaloguesoftheBellinequalities.Thecaseoftwospatiallysep- arated domains”. In:J. Sov. Math.36.4 (1987), pp. 557–570.doi:10.1007/BF01663472

    work page doi:10.1007/bf01663472 1987

  17. [17]

    Masanes.Necessary and sufficient condition for quantum-generated correlations

    Ll. Masanes.Necessary and sufficient condition for quantum-generated correlations. 2003. arXiv:quant-ph/0309137 [quant-ph]

    Pith/arXiv arXiv 2003

  18. [18]

    Quantum Correlations in the Minimal Scenario

    Thinh P. Le, Chiara Meroni, Bernd Sturmfels, Reinhard F. Werner, and Timo Ziegler. “Quantum Correlations in the Minimal Scenario”. In:Quantum7 (2023), p. 947.doi: 10.22331/q-2023-03-16-947. arXiv:2111.06270 [quant-ph]

    work page doi:10.22331/q-2023-03-16-947 2023

  19. [19]

    Quantum statistics in the minimal Bell sce- nario

    Victor Barizien and Jean-Daniel Bancal. “Quantum statistics in the minimal Bell sce- nario”. In:Nature Physics21.4 (Apr. 2025), pp. 577–582.issn: 1745-2481.doi:10.1038/ s41567-025-02782-3

    2025

  20. [20]

    Extremal Tsirelson Inequalities

    Victor Barizien and Jean-Daniel Bancal. “Extremal Tsirelson Inequalities”. In:Phys. Rev. Lett.133.1 (2024), p. 010201.doi:10.1103/PhysRevLett.133.010201. arXiv:2401. 12791 [quant-ph]

    work page doi:10.1103/physrevlett.133.010201 2024

  21. [21]

    Custom Bell inequalities from formal sums of squares

    Victor Barizien, Pavel Sekatski, and Jean-Daniel Bancal. “Custom Bell inequalities from formal sums of squares”. In:Quantum8 (May 2024), p. 1333.issn: 2521-327X.doi: 10.22331/q-2024-05-02-1333

    work page doi:10.22331/q-2024-05-02-1333 2024

  22. [22]

    Bounding the Set of Quantum Correlations

    Miguel Navascués, Stefano Pironio, and Antonio Acín. “Bounding the Set of Quantum Correlations”. In:Physical Review Letters98.1 (Jan. 2007), p. 010401.doi:10 . 1103 / PhysRevLett.98.010401

    2007

  23. [23]

    A convergent hierarchy of semidef- inite programs characterizing the set of quantum correlations

    Miguel Navascués, Stefano Pironio, and Antonio Acín. “A convergent hierarchy of semidef- inite programs characterizing the set of quantum correlations”. In:New Journal of Physics 10.7 (July 2008), p. 073013.issn: 1367-2630.doi:10.1088/1367-2630/10/7/073013

    work page doi:10.1088/1367-2630/10/7/073013 2008

  24. [24]

    Bounding large-scale Bell inequalities

    Luke Mortimer. “Bounding large-scale Bell inequalities”. In:Phys. Rev. A111.5 (2025), p. 052442.doi:10.1103/PhysRevA.111.052442. arXiv:2412.08532 [quant-ph]

    work page doi:10.1103/physreva.111.052442 2025

  25. [25]

    Noisy qudit vs multiple qubits: conditions on gate efficiency for enhancing fidelity

    Nicolas Gigena, Ekta Panwar, Giovanni Scala, Mateus Araújo, Máté Farkas, and Anubhav Chaturvedi. “Self-testing tilted strategies for maximal loophole-free nonlocality”. In:npj Quantum Information11.1 (May 2025), p. 82.issn: 2056-6387.doi:10.1038/s41534- 025-01029-6

    work page doi:10.1038/s41534- 2025

  26. [26]

    Marco Fanizza, Larissa Kroell, Arthur Mehta, Connor Paddock, Denis Rochette, William Slofstra, and Yuming Zhao.The NPA hierarchy does not always attain the commuting operator value. 2025. arXiv:2510.04943 [quant-ph]

    arXiv 2025

  27. [27]

    Self- testing quantum systems of arbitrary local dimension with minimal number of mea- surements

    Shubhayan Sarkar, Debashis Saha, Jedrzej Kaniewski, and Remigiusz Augusiak. “Self- testing quantum systems of arbitrary local dimension with minimal number of mea- surements”. In:npj Quantum Information7.1 (Oct. 2021), p. 151.issn: 2056-6387.doi: 10.1038/s41534-021-00490-3. 5 Conclusions and outlook 38

    work page doi:10.1038/s41534-021-00490-3 2021

  28. [28]

    Exploring the Limits of Quantum Nonlocality with Entangled Photons

    Bradley G. Christensen, Yeong-Cherng Liang, Nicolas Brunner, Nicolas Gisin, and Paul G. Kwiat. “Exploring the Limits of Quantum Nonlocality with Entangled Photons”. In: Phys. Rev. X5 (4 Dec. 2015), p. 041052.doi:10.1103/PhysRevX.5.041052

    work page doi:10.1103/physrevx.5.041052 2015

  29. [29]

    Tight Analytic Bound on the Trade- Off between Device-Independent Randomness and Nonlocality

    Lewis Wooltorton, Peter Brown, and Roger Colbeck. “Tight Analytic Bound on the Trade- Off between Device-Independent Randomness and Nonlocality”. In:Phys. Rev. Lett.129 (15 Oct. 2022), p. 150403.doi:10.1103/PhysRevLett.129.150403

    work page doi:10.1103/physrevlett.129.150403 2022

  30. [30]

    Bell inequalities: many questions, a few answers

    Nicolas Gisin. “Bell inequalities: many questions, a few answers”. In: May 2007. arXiv: quant-ph/0702021

    Pith/arXiv arXiv 2007

  31. [31]

    Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments

    Armin Tavakoli, Máté Farkas, Denis Rosset, Jean-Daniel Bancal, and Jedrzej Kaniewski. “Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments”. In:Sci. Adv.7.7 (2021), abc3847.doi:10.1126/sciadv.abc3847. arXiv: 1912.03225 [quant-ph]

    work page doi:10.1126/sciadv.abc3847 2021

  32. [32]

    Bell Inequalities Tailored to Maximally Entangled States

    Alexia Salavrakos, Remigiusz Augusiak, Jordi Tura, Peter Wittek, Antonio Acín, and Stefano Pironio. “Bell Inequalities Tailored to Maximally Entangled States”. In:Phys. Rev. Lett.119 (4 July 2017), p. 040402.doi:10.1103/PhysRevLett.119.040402

    work page doi:10.1103/physrevlett.119.040402 2017

  33. [33]

    An Operator- Algebraic Formulation of Self-testing

    Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou. “An Operator- Algebraic Formulation of Self-testing”. In:Annales Henri Poincare25.10 (2024), pp. 4283– 4319.doi:10.1007/s00023-023-01378-y. arXiv:2301.11291 [quant-ph]

    work page doi:10.1007/s00023-023-01378-y 2024

  34. [34]

    Sum-of-squaresdecompositionsforafamilyofClauser- Horne-Shimony-Holt-like inequalities and their application to self-testing

    CédricBampsandStefanoPironio.“Sum-of-squaresdecompositionsforafamilyofClauser- Horne-Shimony-Holt-like inequalities and their application to self-testing”. In:Phys. Rev. A91 (5 May 2015), p. 052111.doi:10.1103/PhysRevA.91.052111

    work page doi:10.1103/physreva.91.052111 2015

  35. [35]

    Self-testing properties of Gisin’s elegant Bell inequality

    Ole Andersson, Piotr Badziąg, Ingemar Bengtsson, Irina Dumitru, and Adán Cabello. “Self-testing properties of Gisin’s elegant Bell inequality”. In:Phys. Rev. A96 (3 Sept. 2017), p. 032119.doi:10.1103/PhysRevA.96.032119

    work page doi:10.1103/physreva.96.032119 2017

  36. [36]

    Bloch vectors for qudits

    Reinhold A. Bertlmann and Philipp Krammer. “Bloch vectors for qudits”. In:J. Phys. A 41.23 (2008), p. 235303.doi:10.1088/1751- 8113/41/23/235303. arXiv:0806.1174 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1751- 2008

  37. [37]

    & Schuch, N

    Daniel McNulty and Stefan Weigert. “Mutually Unbiased Bases in Composite Dimensions – A Review”. In:Quantum10 (Apr. 2026), p. 2051.issn: 2521-327X.doi:10.22331/q- 2026-04-01-2051

    work page doi:10.22331/q- 2026

  38. [38]

    Bell inequalities for arbitrarily high dimensional systems

    Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. “Bell Inequalities for Arbitrarily High-Dimensional Systems”. In:Phys. Rev. Lett.88.4 (2002), p. 040404.doi:10.1103/PhysRevLett.88.040404. arXiv:quant-ph/0106024

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.88.040404 2002

  39. [39]

    Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing

    F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín. “Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing”. In:Phys. Rev. Lett.124.2 (2020), p. 020402. doi:10.1103/PhysRevLett.124.020402. arXiv:1812.10428 [quant-ph]

    work page doi:10.1103/physrevlett.124.020402 2020

  40. [40]

    Topological fault-tolerance in cluster state quantum computation

    Rafael Santos, Debashis Saha, Flavio Baccari, and Remigiusz Augusiak. “Scalable Bell inequalities for graph states of arbitrary prime local dimension and self-testing”. In:New J. Phys.25.6 (2023), p. 063018.doi:10.1088/1367- 2630/acd9e3. arXiv:2212.07133 [quant-ph]. A Linearly dependent qubit strategies39

    work page doi:10.1088/1367- 2023

  41. [41]

    Self-testing maximally-dimensional gen- uinely entangled subspaces within the stabilizer formalism

    Owidiusz Makuta and Remigiusz Augusiak. “Self-testing maximally-dimensional gen- uinely entangled subspaces within the stabilizer formalism”. In:New J. Phys.23 (2021), p. 043042.doi:10.1088/1367-2630/abee40. arXiv:2012.01164 [quant-ph]. A Linearly dependent qubit strategies In this appendix, we investigate the effect of linear dependencies among Alice’s o...

    work page doi:10.1088/1367-2630/abee40 2021