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An operator system approach to self-testing

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Pith's one-line read The paper proposes a general operator-system framework in which a correlation is self-tested exactly when it is maximal under a local dilation pre-order, and proves that in the general case such self-tests are abstract self-tests—unique…

desk verdict A serious operator-system framework for self-testing in the commuting model, with strong theorems; the abstract overstates the weak-to-abstract bridge, which is conditional on a centrally supported Haag model. read the letter →

arxiv 2506.17980 v1 pith:GCG5WHGH submitted 2025-06-22 quant-ph math.FAmath.OA

Jason Crann , Ivan G. Todorov , Lyudmila Turowska This is my paper · ORCID
classification quant-phmath.FAmath.OA MSC 46L0746L1081P68
keywords self-testingcommutingoperatormodelsystemsabstractself-testlocaldilationHaagquantumcorrelationsno-signalling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a general operator-system framework for self-testing in the commuting operator model, where bipartite systems are represented by bimodules over pairs of von Neumann algebras rather than tensor products of Hilbert spaces. It defines local isometries and an (approximate) local dilation pre-order on models of a correlation, and proposes that a correlation is self-tested when it has a maximal model in that pre-order. The central result is that, provided the model class contains a centrally supported Haag model of the correlation, a weak self-test is automatically an abstract self-test: the correlation's state extends uniquely to the maximal tensor product of the universal C*-covers. In the tensor-product setting with type I observable algebras the converse also holds, so self-testing and abstract self-testing coincide there. These results supply the first operational notion of self-testing in the commuting operator framework and apply to QNS correlations, the CHSH game, Clifford correlations, contextuality scenarios, quantum colourings, and Schur channels.

What carries the argument

The carrying objects are quantum commuting models $\mathcal{S} = (_A H_B, \varphi_A, \varphi_B, \xi)$: a Hilbert $A$-$B$-bimodule together with commuting unital completely positive maps from the two operator systems into $B(H)$ and a unit vector. A local isometry is built from a commuting diagram of $A$-local and $B$-local intertwiners, generalising the split tensor-product isometries $V_A \otimes V_B$; it induces the pre-orders $\preceq$ and $\preceq_a$ of (approximate) local dilation. A centrally supported Haag model is one where $B^o = A'$ (Haag duality) and the support projections of the state on $A$ and $B^o$ commute with the maps $\varphi_A, \varphi_B$; Lemma 3.11 shows that under approximate dilation the induced states on the maximal tensor product coincide, and Theorem 3.13 feeds this into the abstract self-test conclusion.

What would settle it

Find a correlation f and a class M of quantum commuting models over some operator systems, with M containing a centrally supported Haag model of f, such that f is a weak self-test for M but two different states on $C^*_u(S_A) \otimes_{\max} C^*_u(S_B)$ both extend f; that would refute Theorem 3.13. Alternatively, exhibit an application class in Section 5 whose intended models contain no centrally supported Haag model, which would show the main bridge does not reach that example.

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Extended reading notes

Core claim

The paper claims that for bipartite correlations described by states on the commuting tensor product $S_A \otimes_c S_B$ of operator systems, the operational condition of being a maximal element under the (approximate) local dilation pre-order—called a weak self-test—implies the algebraic condition of abstract self-testing, namely unique extension of the correlation state to $C^*_u(S_A) \otimes_{\max} C^*_u(S_B)$. The bridge is Theorem 3.13: if the model class $\mathcal{M}$ contains a centrally supported Haag model of $f$, then every weak self-test for $\mathcal{M}$ is an abstract self-test for the associated state set. Theorem 4.3 establishes the converse within the class $\mathcal{C}$ of tensor-product models whose observable algebras are type I: under extremality and unique extension to the minimal tensor product, an abstract self-test admits an irreducible ideal model and is a self-test. Thus the paper establishes that, in the commuting operator framework, maximality under entanglement-assisted local dilation forces uniqueness of the algebraic extension, with the caveat that the centrally supported Haag condition is needed for the general direction.

Load-bearing premise

The load-bearing premise is that the model class contains a centrally supported Haag model of the correlation f; without such a model the proof only shows weak self-tests preserve the correlation (Proposition 3.5), and for most of the Section 5 applications the paper does not verify this hypothesis.

Editorial extensions

If this is right

  • In any model class satisfying the Haag-model hypothesis, weak self-testing—maximality under approximate local dilation—forces unique state extension to $C^*_u(S_A)\otimes_{\max} C^*_u(S_B)$, so the operational and algebraic notions coincide in that generality.
  • For tensor-product models with type I observable algebras, abstract self-tests admit irreducible ideal models and are genuine self-tests, extending the known equivalence beyond the finite-dimensional Bell scenario.
  • The CHSH optimal correlation is an abstract self-test and a self-test among all quantum commuting models, so the standard self-test survives in the commuting operator framework.
  • The framework hosts self-tests for QNS correlations, Clifford correlations, contextuality scenarios, quantum colourings of complete graphs, and Schur channels, giving these areas a common definitional language.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to check whether the centrally supported Haag condition can be verified for the concrete application classes (QNS, Clifford, colourings); presently the paper only verifies the condition where explicitly noted, so the full reach of Theorem 3.13 in those classes remains open.
  • If robust self-testing in the sense of $\delta$-dilations implies weak self-testing under the same hypotheses, then device-independent certification would inherit the algebraic uniqueness conclusion, linking robustness to abstract self-tests.
  • Section 6 suggests an operator-system analogue of the unique extension property: states factoring through C*-envelopes rather than universal covers would characterise a finer, model-dependent form of self-testing.
  • The dilation of every stochastic operator matrix to a unistochastic one (Corollary 6.3) is a standalone statement that may find use outside self-testing, for instance in channel dilation problems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops an operator-system framework for self-testing in the commuting operator model. It defines a local isometry between bipartite quantum systems modeled by von Neumann bimodules, then introduces local dilation and approximate local dilation preorders on quantum commuting models, together with the associated notions of self-test, weak self-test, and abstract self-test. The central result, Theorem 3.13, shows under a hypothesis involving a centrally supported Haag model that weak self-testing implies abstract self-testing; Theorem 4.3 provides a converse for finite-dimensional type-I tensor-product models. The framework is applied to QNS correlations, POVM and PVM correlations, the CHSH game in the quantum commuting class, Clifford correlations, contextuality scenarios, quantum graph colourings, and Schur product channels. Section 6 relates several examples to unique state extension across tensor products of C*-envelopes.

Significance. If the main claims hold, the paper supplies a genuinely general operational definition of self-testing in the commuting operator model and connects it to unique state extension, thereby extending the operator-algebraic approach of Paddock--Slofstra--Zhao--Zhou. The detailed proofs of Theorem 2.9, Lemma 3.11, Theorem 4.3, and Theorem 5.19 are a notable strength, as is the breadth of examples treated. The paper is also commendably explicit about the status of some reductions, for instance in Question 7.2. However, the central bridge from weak self-testing to abstract self-testing is conditional on the existence of a centrally supported Haag model, and the applications in Section 5 do not verify this hypothesis for the genuinely non-spatial classes; the abstract's unconditional wording overstates the proved theorem.

major comments (3)
  1. [Abstract; Theorem 3.13; Corollary 3.14; Section 5] The abstract states that self-tests are 'in the general case always abstract self-tests', but Theorem 3.13 is explicitly conditional on the model class M containing a centrally supported Haag model of f. The proof of Lemma 3.11 uses central support of the ideal model at equation (13), and Proposition 3.12 transfers central support only from a centrally supported Haag model; Corollary 3.14 covers only classes closed under reduction that contain a Haag model. Section 5 does not verify this premise for the genuinely non-spatial quantum commuting classes used in its applications, and Question 7.2 explicitly leaves the reduction S ⪯ S_r open. Thus the 'always' in the abstract is not supported by the theorems; the bridge is proved only for classes with a Haag anchor.
  2. [Proposition 5.8] Proposition 5.8 asserts that if p is an extreme point in C_qc, then any quantum commuting model S for p admits a projective quantum commuting model ~S with S ⪯ ~S. The proof, however, constructs only projections P_{x,0} and Q_{y,0} from spectral decompositions and then states 'Then S ⪯ ~S' without exhibiting the local isometry, the auxiliary system, or the verification of the conditions in Definition 3.1. Since Corollary 5.9 relies on this proposition to conclude that the CHSH correlation is a self-test for all quantum commuting models, the missing construction is load-bearing.
  3. [Corollary 5.9; Remark 5.7] Corollary 5.9 asserts that p_{~S} is extreme in C_qc, citing the uniqueness of the optimal quantum commuting strategy from Remark 5.7. The direct abstract-self-test argument in Section 5.3.1 is written for states factoring through A_{X,A} ⊗_max A_{Y,B}, and the passage from that argument to uniqueness in the full class C_qc, which includes general POVM models rather than only PVM models, is not spelled out. The extremality step should be proved directly, or the reduction to PVM models should be exhibited; as written, Corollary 5.9 is an assertion rather than a complete proof.
minor comments (4)
  1. [Section 5.3.1] The sentence 'the C*-tensor product is unabiguiusly defined' contains a typo; it should read 'unambiguously'.
  2. [Proof of Theorem 5.16] The phrase 'By Cauchy-Schwartz' should be 'By Cauchy-Schwarz'.
  3. [Section 5.1, after equation (24)] The text 'We refer to Γ_S being an self-test' should read 'being a self-test'.
  4. [References] Reference [41] (McKague--Yang--Scarani) appears in the bibliography but is not cited in the body of the paper; please either add a citation in the introduction or remove the reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central conditional theorem is proved from operator-system machinery, and the applications are self-contained rather than restatements of their inputs.

full rationale

The paper's central implication (Theorem 3.13) is explicitly conditional: if M contains a centrally supported Haag model of f and f is a weak self-test for M, then f is an abstract self-test for the state set S. The proof is a genuine derivation: Proposition 3.12 transforms the Haag anchor into central support of the ideal model, and Lemma 3.11 then uses approximate local dilations and the commutation relation (13) to force equality of the extended states. This is not a definitional equivalence: the Haag-model hypothesis is an input condition, not a restatement of unique state extension, and the conclusion is derived rather than assumed. The abstract's wording ('in the general case always') overstates the theorem's scope, but overclaiming is a correctness concern, not circularity. In the applications, abstract self-tests are established by direct arguments (CHSH in Section 5.3, Clifford correlations in Theorem 5.12, colourings in Corollary 5.17, Schur channels in Theorem 5.19), or via Theorem 4.3 for type I tensor-product models, rather than by citing the target result. Self-citations to [55], [9,10], and [44] are used as published, parameter-free tools with proofs; they do not assume the theorem being proved. No parameters are fitted and no prediction is renamed from an input. Accordingly, there is no circular step to exhibit.

Assumptions & free parameters 0 free parameters · 10 assumptions · 5 invented entities

No free parameters are fitted; the paper is a mathematical framework. The axioms are standard operator-algebra results (Choi-Effros, Stinespring, commuting tensor product, direct integrals, NPA hierarchy, Clifford representations), plus domain assumptions about the operator systems T_{X,A}, R_{d^2,d}, and the existence of Haag models. Several of the domain assumptions come from the authors' own prior work, notably [55] and [9,10]; these are published with proofs, so they are not circular, but they are non-trivial inputs.

assumptions (10)
  • standard math The commuting tensor product S_A ⊗_c S_B embeds completely order isomorphically into C*_u(S_A) ⊗_max C*_u(S_B).
    Used throughout Section 3 to define f_S on S_A⊗_c S_B; follows from Kavruk-Paulsen-Todorov-Tomforde, [31, Theorem 6.4 and Corollary 6.5].
  • standard math Every pair of unital completely positive maps with commuting ranges determines a unique ucp map on the commuting tensor product.
    [31, Corollary 6.5]; used in equation (2) for f_S and throughout the model definitions.
  • domain assumption Haag models satisfy B^o = A', and the class M is assumed to contain a centrally supported Haag model of the correlation.
    Theorem 3.13 requires this to conclude that weak self-tests are abstract self-tests; Proposition 3.12 and Lemma 3.11 invoke Haag duality.
  • standard math Type I representations admit direct integral decompositions into irreducible representations (Dixmier [20]).
    Used in the proof of Theorem 4.3 to decompose π_A and π_B and to build the local isometry V_A⊗V_B.
  • standard math The classification of irreducible representations of C*(Z2 * Z2) (Ostrovskyi-Samoilenko [43]) and of Clifford algebras with even |X| (Tsirelson [56]).
    Used in Section 5.3 to identify the representation τ=π_{π/4}⊗π_{π/4} from anticommutation relations, and in Section 5.4 for Clifford correlations (Theorem 5.10).
  • domain assumption The NPA hierarchy characterizes quantum commuting correlations via positive completions of the moment matrix M(p).
    Used in Section 5.4 (Theorem 5.12(ii)) to define the class A_C of admissible positive semidefinite matrices.
  • domain assumption The universal operator systems T_{X,A} for QNS correlations have the stated universal properties with C*_u(T_{X,A}) = C_{X,A} ([55, Corollaries 5.3 and 5.4]).
    Foundational for Subsection 5.1; [55] is prior work by two of the present authors, but the results are published with proofs.
  • domain assumption Semi-classical SOMs correspond to ucp maps R_{X,A}→B(H) and the state space of R_{X,A}⊗_c R_{Y,B} is affinely isomorphic to CQNS correlations ([55, Theorems 7.5 and 7.7]).
    Used in Subsection 5.6 and Section 6.2; from the authors' own prior paper [55].
  • standard math The C*-envelope results of [6] give C*_e(R_{X,A}) = B_{X,A} and the inclusion R_{X,A}⊗_c R_{Y,B} ⊆ C*_e(R_{X,A})⊗_max C*_e(R_{Y,B}).
    Used in Section 6.2; [6] is by Bochniak, Kasprzak and Sołtan, independent of the present authors.
  • standard math Every finite dimensional C*-algebra decomposes as a direct sum of matrix algebras and unital *-homomorphisms M_d→M_m force d|m.
    Used in Remark 5.15 to justify that Hom(K_d^2,Q_d) representations land in ⊕_i M_{d n_i}; also used in the trace-preserving isomorphism in Theorem 5.16.
invented entities (5)
  • Local isometry between bipartite quantum systems in the commuting operator model independent evidence
    purpose: Provides the equivalence relation for self-testing in the commuting operator framework (Definition 2.3); replaces the tensor-split V_A⊗V_B.
    Its content is testable via the split case (Proposition 2.6), the pre-order property (Theorem 2.9), and the CHSH and colouring self-tests in Section 5 that use it.
  • Quantum commuting model (A H_B, φ_A, φ_B, ξ) over a pair of operator systems independent evidence
    purpose: The basic object whose correlations are self-tested (Definition 3.1); generalizes POVM and SOM models.
    Maps onto known objects: POVM qc-models, SOM qc-models, and QNS correlations via equation (24).
  • Approximate local dilation order ⪯_a and weak self-test independent evidence
    purpose: Allows a notion of robustness of self-tests through limits of local isometries (Definition 3.1(ii) and 3.6(ii)).
    Related to robust self-testing in [58]; the paper shows it preserves the correlation f_S (Proposition 3.5) and implies abstract self-testing under central support (Theorem 3.13).
  • Clifford correlations independent evidence
    purpose: A class of no-signalling correlations from representations of Clifford relations; shown to be abstract self-tests (Theorem 5.12).
    Defined via the quotient C_{X,Z2}/J_C = Clifford algebra; the self-testing statements are concrete corollaries.
  • Self-test for the quantum colouring game K_4→Q_2 independent evidence
    purpose: A self-test for a classical-to-quantum no-signalling correlation (Corollary 5.17 and Proposition 5.18).
    The self-testing statement is a concrete new result about the Pauli strategy Γ_{K4}.

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Pith. "Pith review of An operator system approach to self-testing." pith.science (2026). https://pith.science/paper/GCG5WHGH

@misc{pith2026250617980,
  author       = {Pith},
  title        = {Pith review of: An operator system approach to self-testing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCG5WHGH}},
  note         = {Machine review of arXiv:2506.17980}
}
read the original abstract

We develop a general framework for self-testing, in which bipartite correlations are described by states on the commuting tensor product of a pair of operator systems. We propose a definition of a local isometry between bipartite quantum systems in the commuting operator model, and define self-testing and abstract self-testing in the latter generality. We show that self-tests are in the general case always abstract self-tests and that, in some cases, the converse is also true. We apply our framework in a variety of instances, including to correlations with quantum inputs and outputs, quantum commuting correlations for the CHSH game, synchronous correlations, contextuality scenarios, quantum colourings and Schur quantum channels.

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Works this paper leans on

58 extracted references · 54 canonical work pages

  1. [1]

    Ac´ın, T

    A. Ac´ın, T. Fritz, A. Leverrier and A. B. Sainz , A combinatorial approach to nonlocality and contextuality, Comm. Math. Phys. 334 (2015), no. 2, 533-628

    work page 2015

  2. [2]

    Anoussis, A

    M. Anoussis, A. Chatzinikolaou and I. G. Todorov, Operator systems, contextu- ality and non-locality, preprint (2024), arXiv:2405.11152

    work page arXiv 2024

  3. [3]

    W. B. Arveson, Notes on the unique extension property , Unpublished note, 2003

    work page 2003

  4. [4]

    Bharti, M

    K. Bharti, M. Ray, A. Varvitsiotis, N. A. Warsi, A. Cabello and L.-C. Kwek, Robust self-testing of quantum systems via noncon- textuality inequalities , Phys. Rev. Letters 122 (2019), no. 25, 250403

    work page 2019

  5. [5]

    D. P. Blecher and C. Le Merdy, Operator Algebras and Their Modules: An oper- ator space approach, Oxford University Press, 2004

    work page 2004

  6. [6]

    Bochniak, P

    A. Bochniak, P. Kasprzak and P. M. So/suppress ltan, Quantum correlations on quantum spaces, Int. Math. Res. Not. IMRN 14 (2023), 12400-12440

    work page 2023

  7. [7]

    Bowles, I

    J. Bowles, I. Supi´c, D. Cavalcanti and A. Ac ´ın, Device-independent entangle- ment certification of all entangled states , Phys. Rev. Lett. 121 (2018), no. 18, 180503

    work page 2018

  8. [8]

    Bowles, I

    J. Bowles, I. ˇSupi´c, D. Cavalcanti and A. Ac ´ın, Self-testing of Pauli observ- ables for device-independent entanglement certification , Phys. Rev. A 98 2018, no. 4, 042336

    work page 2018

Show all 58 references
  1. [9]

    Brannan, S

    M. Brannan, S. J. Harris, I. G. Todorov and L. Turowska, Synchronicity for quantum non-local games , J. Funct. Anal. 284 (2023), no. 2, Paper No. 109738, 54 pp

    2023

  2. [10]

    Brannan, S

    M. Brannan, S. J. Harris, I. G. Todorov and L. Turowska , Quantum no- signalling bicorrelations, Adv. Math. 449 (2024), Paper No. 109732, 81 pp

    2024

  3. [11]

    Broadbent, A

    A. Broadbent, A. Mehta and Y. Zhao, Quantum delegation with an off-the-shelf device, Leibniz International Proceedings in Informatics (2023), arXiv:2304.03448

    2023 arXiv

  4. [12]

    L. G. Brown, Ext of certain free product C∗-algebras, J. Operator Theory 6 (1981), no. 1, 135-141

    1981

  5. [13]

    N. P. Brown and N. Ozawa, C∗-algebras and finite-dimensional approximations , American Mathematical Society, 2008

    2008

  6. [14]

    J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories , Phys. Rev. Letters 23 (1969), 880-884

    1969

  7. [15]

    Coladangelo, A

    A. Coladangelo, A. B. Grilo, S. Jeffery and T. Vidick , Verifier-on-a-leash: New schemes for verifiable delegated quantum computation, with quasilinear resources, Th. Comp. 20 (2024), no. 1, 1-87

    2024

  8. [16]

    Conlon, J

    A. Conlon, J. Crann, D. W. Kribs and R. H. Levene, Quantum teleportation in the commuting operator framework, Ann. Henri Poincar´ e 24 (2023), no. 5, 1779-1821

    2023

  9. [17]

    Crann, D

    J. Crann, D. W. Kribs, R. H. Levene and I. G. Todorov, State convertibility in the von Neumann algebra framework , Comm. Math. Phys. 378 (2020), no. 2, 1123- 1156

    2020

  10. [18]

    Crann, R

    J. Crann, R. H. Levene, I. G. Todorov and L. Turowska, Values of cooperative quantum games, preprint (2023), arXiv:2310.17735

    2023 arXiv

  11. [19]

    K. R. Davidson and M. Kennedy, The Choquet boundary of an operator system , Duke Math. J. 164 (2015), no. 15, 2989-3004

    2015

  12. [20]

    Dixmier, C*-algebras, North Holland, 2011

    J. Dixmier, C*-algebras, North Holland, 2011

    2011

  13. [21]

    Dixmier, Von Neumann algebras, Elsevier, 2011

    J. Dixmier, Von Neumann algebras, Elsevier, 2011

    2011

  14. [22]

    Duan and A

    R. Duan and A. Winter , No-signalling assisted zero-error capacity of quantum channels and an information theoretic interpretation of the Lov´ asz number , IEEE Trans. Inf. Theory 62 (2016), no. 2, 891-914

    2016

  15. [23]

    E. G. Effros and C. E. Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1-34

    1977

  16. [24]

    G. K. Eleftherakis, E. T. A. Kakariadis and I. G. Todorov, Symmetrisations of operator spaces, preprint (2025), arXiv:2503.15192. 64 J. CRANN, I. G. TODOROV, AND L. TUROWSKA

    2025 arXiv

  17. [25]

    L. Gao, S. Harris and M. Junge, Quantum teleportation and super-dense coding in operator algebras. Int. Math. Res. Not. IMRN (2021), no. 12, 9146-9179

    2021

  18. [26]

    Haagerup and M

    U. Haagerup and M. Musat, Factorization and dilation problems for completely positive maps on von Neumann algebras , Comm. Math. Phys. 303 (2011), no. 2, 555- 594

    2011

  19. [27]

    Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen , MIP*=RE, preprint (2020), arXiv:2001.04383

    2020 arXiv

  20. [28]

    V. F. R. Jones and V. S. Sunder, Introduction to subfactors, Cambridge University Press, 1997

    1997

  21. [29]

    R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras II, American Mathematical Society, 1997

    1997

  22. [30]

    P. N. Kar, Robust self-testing for synchronous correlations and games , preprint (2025), arXiv:2503.23500

    2025 arXiv

  23. [31]

    A. S. Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde, Tensor products of operator systems, J. Funct. Anal. 261 (2011), no. 2, 267-299

    2011

  24. [32]

    S.-J. Kim, V. I. Paulsen and C. Schafhauser, A synchronous game for binary con- straint systems, J. Math. Phys. 59 (2018), no. 3, 032201, 17 pp

    2018

  25. [33]

    Kitajima, Local operations and completely positive maps in algebraic quantum field theory, In: M

    Y. Kitajima, Local operations and completely positive maps in algebraic quantum field theory, In: M. Ozawa, J. Butterfield, H. Halvorson, M. R´ edei, Y. Kitajima, Y, F. Buscemi (eds) Reality and Measurement in Algebraic Quantum Theory, NWW 2015, Springer Proceedings in Mathema...

    2015

  26. [34]

    Klappenecker and M

    A. Klappenecker and M. R¨otteler, On the monomiality of nice error bases, IEEE Trans. Inf. Theory 51 (2005), no. 3, 1084-1089

    2005

  27. [35]

    L. J. Landau and R. F. Streater, On Birkoff’s theorem for doubly stochastic com- pletely positive maps of matrix algebras , Lin. Alg. App. 193, 107-127 (1993)

    1993

  28. [36]

    van Luijk, A

    L. van Luijk, A. Stottmeister, R. F. Werner and H. Wilming, Pure state en- tanglement and von Neumann algebras , preprint (2024), arXiv:2409.17739

    2024

  29. [37]

    Lupini, L

    M. Lupini, L. Man ˇcinska, V. I. Paulsen, D. E. Roberson, G. Scarpa, S. Severini, I. G. Todorov and A. Winter , Perfect strategies for non-local games, Math. Phys. Anal. Geom. 23 (2020), no. 1, Paper No. 7, 31 pp

    2020

  30. [38]

    Manˇcinska, J

    L. Manˇcinska, J. Prakash and C. Schafhauser , Constant-sized robust self-tests for states and measurements of unbounded dimension , Comm. Math. Phys. 405 (2024), no. 9, Paper No. 221, 36 pp

    2024

  31. [39]

    Mayers and A

    D. Mayers and A. Yao, Self testing quantum apparatus , Quantum Inf. Comp. 4 (2004), no. 4, 273-286

    2004

  32. [40]

    Mehta, C

    A. Mehta, C. Paddock and L. Wooltorton, Self-testing in the compiled setting via tilted-CHSH inequalities, preprint (2024), arXiv:2406.04986

    2024

  33. [41]

    McKague, T

    M. McKague, T. H. Yang and V. Scarani, Robust self-testing of the singlet , J. Phys. A 45 (2012), no. 45, 455304

    2012

  34. [42]

    Navascu´es, S

    M. Navascu´es, S. Pironio and A. Ac´ın, A convergent hierarchy of semdefinite pro- gramscharacterizing the set of quantum correlations , New J. Phys. 10 (2008), 073013

    2008

  35. [43]

    Ostrovskyi and Yu

    V. Ostrovskyi and Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators , Harwood Academic Publishers, 1999

    1999

  36. [44]

    Paddock, W

    C. Paddock, W. Slofstra, Y. Zhao and Y. Zhou, An operator-algebraic formula- tion of self-testing , Ann. Henri Poincare 25 (2024), no. 10, 4283-4319

    2024

  37. [45]

    V. I. Paulsen, Completely bounded maps and operator algebras , Cambridge Univer- sity Press, 2002

    2002

  38. [46]

    V. I. Paulsen, S. Severini, D. Stahlke, I. G. Todorov and A. Winter, Estimat- ing quantum chromatic numbers , J. Funct. Anal. 270 (2016), no. 6, 2188-2222

    2016

  39. [47]

    V. I. Paulsen and I. G. Todorov, Quantum chromatic numbers via operator sys- tems, Q. J. Math. 66 (2015), no. 2, 677-692. AN OPERATOR SYSTEM APPROACH TO SELF-TESTING 65

    2015

  40. [48]

    Pironio, A

    S. Pironio, A. Ac´ın, N. Brunner, N. Gisin, S. Massar and V. Scarani, Device- independent quantum key distribution secure against collective attacks , New J. Phys. 11 (2009), no. 4, 045021

    2009

  41. [49]

    B. W. Reichardt, F. Unger and U. Vazirani, Classical command of quantum sys- tems, Nature 496 (2013), 456-460

    2013

  42. [50]

    Seitz, Finite groups having only one irreducible representation of degree greater than one, Proc

    G. Seitz, Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc. 19 (1968), 459-461

    1968

  43. [51]

    Simon, Representations of finite and compact groups, American Mathematical So- ciety, 1996

    B. Simon, Representations of finite and compact groups, American Mathematical So- ciety, 1996

    1996

  44. [52]

    Sˇupic and J

    I. Sˇupic and J. Bowles, Self-testing of quantum systems: a review , Quantum 4, 337 (2020)

    2020

  45. [53]

    Takesaki, Theory of Operator Algebras I , Springer-Verlag, 2001

    M. Takesaki, Theory of Operator Algebras I , Springer-Verlag, 2001

    2001

  46. [54]

    Takesaki, Theory of Operator Algebras II , Springer-Verlag, 2003

    M. Takesaki, Theory of Operator Algebras II , Springer-Verlag, 2003

    2003

  47. [55]

    I. G. Todorov and L. Turowska, Quantum no-signalling correlations and non-local games, Comm. Math. Phys. 405 (2024), no. 6, Paper No. 141, 65 pp

    2024

  48. [56]

    B. S. Tsirelson, Quantum analogues of Bell’s inequalities. The case of two spatially divided domains, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 142 (1985), 174-194, 200

    1985

  49. [57]

    Verch and R

    R. Verch and R. F. Werner, Distillability and positivity of partial transposes in general quantum field systems , Rev. Math. Phys. 17 (2005), no. 5, 545–576

    2005

  50. [58]

    Zhao, Robust self-testing for nonlocal games with robust game algebras , preprint (2024), arXiv:2411.03259

    Y. Zhao, Robust self-testing for nonlocal games with robust game algebras , preprint (2024), arXiv:2411.03259. 1School of Mathematics & Statistics, Carleton University, Ottawa, ON, Canada H1S 5B6 Email address: jasoncrann@cunet.carleton.ca 2School of Mathematical Sciences, Uni...

    2024 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.