pith. sign in

arxiv: 2606.19033 · v1 · pith:BHTPAIL7new · submitted 2026-06-17 · 🪐 quant-ph

Contextuality as a Diagnostic of Translation-Symmetry Breaking in Translation-Invariant 1D Hamiltonians

Xiao Zeng , Kaiyan Yang , Lingxia Zhang , Zizhu Wang This is my paper

Pith reviewed 2026-06-26 20:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords contextualitytranslation symmetry breaking1D quantum chainsground statesperiodic boundary conditionswitness inequalitiessymmetry breaking
0
0 comments X

The pith

In translation-invariant 1D chains, maximal contextuality violation coincides with spontaneous breaking of one-site translation symmetry to yield p-periodic ground states.

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

The paper shows that contextuality witnesses in infinite one-dimensional translation-invariant quantum chains reach their strongest quantum violation precisely when the ground state spontaneously breaks one-site translation symmetry. This produces ground states that are strictly periodic with period p greater than 1. The alignment appears along continuous interpolations connecting classical bounds to quantum-optimal Hamiltonians, where the classical bound corresponds to the point at which symmetry breaking selects one unique period over competing classical possibilities. The witnesses permit exact reduction of the infinite-chain problem to finite p-site rings that preserve both classical and quantum bounds, turning the certification into local energy measurements on small periodic systems.

Core claim

Within the witness families studied, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, producing strictly p-periodic ground states with p>1. Along natural continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking point where competing classical periodicities are lifted in favor of a unique quantum-selected period. At the quantum optimum, the studied families admit exact finite-size reductions: a translation-invariant contextuality witness induces a p-site periodic-boundary-condition inequality with identical classical and quantum bounds.

What carries the argument

A translation-invariant contextuality witness that reduces the infinite-chain problem to a p-site periodic-boundary-condition inequality while preserving identical classical and quantum bounds.

If this is right

  • The classical bound of the witness marks the location of the symmetry-breaking transition along the interpolation path.
  • The reduction produces compact finite-ring inequalities that can be tested with local energy measurements and are tight in several cases.
  • Contextuality acquires a thermodynamic interpretation as a diagnostic of which periodicity is selected in the ground state.
  • The mechanism is established analytically for representative two- and three-body models and supported numerically for broader families.

Where Pith is reading between the lines

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

  • The same diagnostic relation might be testable in other symmetry-breaking scenarios such as discrete time-translation or spatial reflection.
  • If the reduction technique extends, contextuality witnesses could become practical probes for periodicity in experimental many-body platforms without requiring full tomography.
  • Competing periodicities at the classical bound suggest that contextuality strength could quantify the selection pressure exerted by quantum fluctuations.

Load-bearing premise

The coincidence between maximal violation and symmetry breaking holds only for the specific families of contextuality witnesses considered along the continuous interpolations examined.

What would settle it

Finding, within one of the studied witness families, a translation-invariant Hamiltonian whose ground state breaks one-site translation symmetry yet fails to achieve the maximal quantum violation, or achieves the maximal violation without symmetry breaking.

Figures

Figures reproduced from arXiv: 2606.19033 by Kaiyan Yang, Lingxia Zhang, Xiao Zeng, Zizhu Wang.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic classical-to-quantum symmetry [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The diagram representation of the correlation [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A diagram representation of uMPS. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Structure of the ground state [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: provides a direct visualization of the finite￾size scaling under OBC. The finite-size energy densities converge to the TI energy density as n goes to infinity. Examining the two-body energies (nearest neighbor) of the ground state reveals a clear pattern: when n = 3k or n = 3k + 2, the two-body energies exhibit a periodic structure composed of {−2 p 2,−2,−2}, except for the last one or two terms. In contra… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: An example of a domino loop. There are three [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we show that, in infinite one-dimensional translation-invariant chains, contextuality can acquire a genuinely thermodynamic meaning: within the witness families studied, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, producing strictly $p$-periodic ground states with $p>1$. Along natural continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking point where competing classical periodicities are lifted in favor of a unique quantum-selected period. At the quantum optimum, the studied families admit exact finite-size reductions: a translation-invariant contextuality witness induces a $p$-site periodic-boundary-condition inequality with identical classical and quantum bounds (hence no loss under reduction), and in several cases the resulting finite inequalities are tight. This reduction turns an infinite-chain contextuality certification into a compact, hardware-testable benchmark on a small ring, requiring only local energy measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using a translation-invariant adaptation of semidefinite-program hierarchies together with variational matrix-product-state algorithms.

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

0 major / 2 minor

Summary. The paper claims that, within specific families of contextuality witnesses for infinite translation-invariant 1D chains, the maximal quantum violation coincides with spontaneous breaking of one-site translation symmetry, yielding strictly p-periodic ground states (p>1). Along continuous interpolations between classical-bound and quantum-optimal Hamiltonians, the classical bound marks a symmetry-breaking transition; the witnesses admit exact finite-size reductions to p-site periodic-boundary inequalities that preserve both classical and quantum bounds exactly (hence no loss), with several cases being tight. The mechanism is established analytically for representative two- and three-body models and corroborated via translation-invariant SDP hierarchies and MPS variational methods, turning the certification into a compact, locally measurable benchmark on small rings.

Significance. If the central claim holds, the work supplies a thermodynamic interpretation of contextuality in many-body systems by linking maximal violation directly to symmetry breaking, with the exact finite-size reductions providing a concrete, hardware-testable diagnostic. Analytic proofs for representative models, exact bound preservation under reduction, and the combination of SDP hierarchies with MPS constitute clear strengths that make the result falsifiable and practically relevant within the stated scope of witness families and continuous interpolations.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement (perhaps as a numbered remark) that the coincidence is scoped to the witness families studied and does not extend to arbitrary contextuality witnesses; this is already implicit but would prevent misreading.
  2. Notation for the p-periodic reductions (e.g., the mapping from infinite-chain witnesses to finite-ring inequalities) could be introduced with a small diagram or table summarizing the preserved bounds for the two- and three-body cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We are pleased that the central claims, analytic proofs, exact finite-size reductions, and practical diagnostic aspects were viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent external methods

full rationale

The central claim—that maximal quantum violation of the studied contextuality witnesses coincides with spontaneous one-site translation symmetry breaking—is established via analytic proofs on representative two- and three-body models together with corroboration from translation-invariant SDP hierarchies and MPS variational algorithms. These are external, independently verifiable computational tools whose outputs are not defined in terms of the symmetry-breaking point itself. The result is explicitly scoped to the witness families and continuous interpolations considered, with finite-size reductions shown to preserve bounds exactly without introducing self-referential fits or load-bearing self-citations. No step reduces the prediction to a fitted input or prior author result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper rests on standard quantum mechanics and translation invariance of the Hamiltonian; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Standard quantum mechanics and the validity of contextuality inequalities as probes of many-body correlations
    Invoked throughout the abstract as the framework for witnesses and violations.
  • domain assumption The Hamiltonians are translation-invariant and the ground states can be analyzed via symmetry breaking
    Central to the claim about infinite chains and p-periodic states.

pith-pipeline@v0.9.1-grok · 5765 in / 1429 out tokens · 32375 ms · 2026-06-26T20:57:42.412921+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

63 extracted references · 9 canonical work pages · 1 internal anchor

  1. [1]

    ,an,0,a n,1,a n,2) =1,(19) where(a 2k+1,0,a 2k+1,1,a 2k+1,2) = (0, 0, 0), and (a2k,0,a 2k,1,a 2k,2) = (0, 1, 0), fork=0,

    Then-partite in- equality with open boundary conditions (OBC) given by 1 n−1 n−1X i=1 〈σ(i) 1 σ(i+1) 0 +σ (i) 1 σ(i+1) 1 σ(i) 2 σ(i+1) 0 +σ (i) 2 σ(i+1) 1 〉 ≥ −2 (18) is also a valid inequality for anyn>2, since the classi- cal bound can always be saturated by a LHV model (see details in Section IVA) given by Q(a1,0,a 1,1,a 1,2, . . . ,an,0,a n,1,a n,2) =...

  2. [2]

    ForQ(010101) =1, the ground state is|222〉⊗∞

  3. [3]

    ForQ(110101) =Q(010111) =Q(011101) =1/3, the ground state is|122〉⊗∞

  4. [4]

    ForQ(001101) =Q(110101) =Q(010100) = Q(010011) =1/4, the ground state is|0122〉⊗∞

  5. [5]

    ForQ(001101) =Q(110101) =Q(010111) = Q(011100) =Q(110011) =1/5, the ground state is|01221〉 ⊗∞

  6. [6]

    In fact, the TI ground sector ofHcan be characterized using the domino-loop construction

    ForQ(000011) =Q(001101) =Q(110101) = Q(010111) =Q(011100) =Q(110000) =1/6, the ground state is|001221〉⊗∞. In fact, the TI ground sector ofHcan be characterized using the domino-loop construction. By the TI ground sector, we mean the set of energy-density-minimizing states within the class of TI orp-periodic states; sym- metrizingap-periodicstateproducesaT...

  7. [7]

    energy fingerprint

    Therefore,L p ≤L TI, and henceLp =L TI. By the domino-loop construction, we find that there is a vertex solution with periodpfor each model in Ta- ble I, and the results are listed in Table II. Consequently, the finite PBC inequalities all share the same classical bound with the TI inequalities. The classical bound of the 232-type inequality can be attain...

  8. [8]

    J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz.1, 195 (1964)

    1964

  9. [9]

    Brunner, D

    N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S.Wehner,Bellnonlocality,ReviewsofModernPhysics86, 419 (2014)

    2014

  10. [10]

    Kochen and E

    S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, inErnst Specker Selecta (Birkhäuser Basel, 1990) pp. 235–263

    1990

  11. [11]

    Budroni, A

    C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-Å. Larsson, Kochen-Specker contextuality, Reviews of Mod- ern Physics94, 045007 (2022)

    2022

  12. [12]

    Cabello, Bell non-locality and Kochen–Specker contex- tuality: How are they connected?, Foundations of Physics 51, 61 (2021), arXiv:1904.05306 [quant-ph]

    A. Cabello, Bell non-locality and Kochen–Specker contex- tuality: How are they connected?, Foundations of Physics 51, 61 (2021), arXiv:1904.05306 [quant-ph]

    work page arXiv 2021

  13. [13]

    Howard, J

    M. Howard, J. Wallman, V. Veitch, and J. Emerson, Con- textuality supplies the ‘magic’ for quantum computation, Nature510, 351 (2014)

    2014

  14. [14]

    Contextuality as a resource for models of quantum computation on qubits

    J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, and R. Raussendorf, Contextuality as a resource for models of quantum computation with qubits, Physical Review Let- ters119, 120505 (2017), arXiv:1610.08529 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Bravyi, D

    S. Bravyi, D. Gosset, and R. König, Quantum advantage with shallow circuits, Science362, 308 (2018)

    2018

  16. [16]

    Bravyi, D

    S. Bravyi, D. Gosset, R. König, and M. Tomamichel, Quan- tum advantage with noisy shallow circuits, Nature Physics 16, 1040 (2020)

    2020

  17. [17]

    Zurel, C

    M. Zurel, C. Okay, and R. Raussendorf, Hidden variable model for universal quantum computation with magic states on qubits, Physical Review Letters125, 260404 (2020), arXiv:2004.01992 [quant-ph]

    work page arXiv 2020

  18. [18]

    Gupta, D

    S. Gupta, D. Saha, Z.-P. Xu, A. Cabello, and A. S. Majum- dar,Quantumcontextualityprovidescommunicationcom- plexity advantage, Physical Review Letters130, 080802 (2023), arXiv:2205.03308 [quant-ph]

    work page arXiv 2023

  19. [19]

    Acín, Detecting nonlocality in many-body quantum states, Science344, 1256 (2014)

    J.Tura,R.Augusiak,A.B.Sainz,T.Vértesi,M.Lewenstein, and A. Acín, Detecting nonlocality in many-body quantum states, Science344, 1256 (2014)

    2014

  20. [20]

    J. Tura, A. B Sainz, T. Vértesi, A. Acín, M. Lewenstein, and R. Augusiak, Translationally invariant multipartite Bell in- equalities involving only two-body correlators, Journal of Physics A: Mathematical and Theoretical47, 424024 (2014)

    2014

  21. [21]

    M. Hu, E. Vallée, T. Seynnaeve, P. Emonts, F. Moham- madi, and J. Tura, Characterizing translation-invariant Bell inequalities using tropical algebra and graph poly- topes, Phys. Rev. A113, 032421 (2026)

    2026

  22. [22]

    M.HuandJ.Tura,Tropicalcontractionoftensornetworks as a Bell inequality optimization toolset, Phys. Rev. Lett. 136, 100202 (2026)

    2026

  23. [23]

    J. Tura, G. De las Cuevas, R. Augusiak, M. Lewenstein, A. Acín, and J. I. Cirac, Energy as a detector of nonlocality of many-body spin systems, Physical Review X7, 021005 (2017)

    2017

  24. [24]

    Z. Wang, S. Singh, and M. Navascués, Entanglement and nonlocality in infinite 1D systems, Physical Review Letters 118, 230401 (2017)

    2017

  25. [25]

    Fine, Hidden variables, joint probability, and the Bell inequalities, Physical Review Letters48, 291 (1982)

    A. Fine, Hidden variables, joint probability, and the Bell inequalities, Physical Review Letters48, 291 (1982)

    1982

  26. [26]

    K. Yang, X. Zeng, Y. Luo, G. Yang, L. Shu, M. Navascués, and Z. Wang, Contextuality in infinite one-dimensional translation-invariant local Hamiltonians, npj Quantum In- formation8, 89 (2022)

    2022

  27. [27]

    C. K. Majumdar and D. K. Ghosh, On next-nearest- neighbor interaction in linear chain. I, Journal of Math- ematical Physics10, 1388 (1969)

    1969

  28. [28]

    C. K. Majumdar and D. K. Ghosh, On next-nearest- neighbor interaction in linear chain. II, Journal of Mathe- matical Physics10, 1399 (1969)

    1969

  29. [29]

    F. D. M. Haldane, Spontaneous dimerization in theS=1 2 Heisenbergantiferromagneticchainwithcompetinginter- actions, Physical Review B25, 4925 (1982)

    1982

  30. [30]

    Okamoto and K

    K. Okamoto and K. Nomura, Fluid-dimer critical point in S = 1/2 antiferromagnetic Heisenberg chain with next nearest neighbor interactions, Physics Letters A169, 433 (1992)

    1992

  31. [31]

    M. E. Fisher and W. Selke, Infinitely many commensurate phasesinasimpleIsingmodel,PhysicalReviewLetters44, 1502 (1980)

    1980

  32. [32]

    Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Reports on Progress in Physics 45, 587 (1982)

    P. Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Reports on Progress in Physics 45, 587 (1982)

    1982

  33. [33]

    W.Selke,TheANNNImodel—theoreticalanalysisandex- perimental application, Physics Reports170, 213 (1988)

    1988

  34. [34]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics16, 407 (1961)

    1961

  35. [35]

    Haldanegap

    M. Oshikawa, M. Yamanaka, and I. Affleck, Magnetiza- tionplateausinspinchains: “Haldanegap”forhalf-integer 17 spins, Physical Review Letters78, 1984 (1997)

    1984

  36. [36]

    Oshikawa, Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice, Physical Review Letters84, 1535 (2000)

    M. Oshikawa, Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice, Physical Review Letters84, 1535 (2000)

    2000

  37. [37]

    M. B. Hastings, Lieb-Schultz-Mattis in higher dimensions, Physical Review B69, 104431 (2004)

    2004

  38. [38]

    Zauner-Stauber, L

    V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman, Variational optimization algorithms for uniform matrix product states, Physical Re- view B97, 045145 (2018)

    2018

  39. [39]

    Haegeman, J

    J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Ver- schelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Physical Review Letters 107, 070601 (2011)

    2011

  40. [40]

    D.Perez-Garcia,F.Verstraete,M.Wolf,andJ.Cirac,Matrix product state representations, Quantum Information and Computation7, 401 (2007)

    2007

  41. [41]

    Navascués, S

    M. Navascués, S. Pironio, and A. Acín, Bounding the set of quantumcorrelations,PhysicalReviewLetters98,010401 (2007)

    2007

  42. [42]

    Navascués, S

    M. Navascués, S. Pironio, and A. Acín, A convergent hier- archy of semidefinite programs characterizing the set of quantum correlations, New Journal of Physics10, 073013 (2008)

    2008

  43. [43]

    Collins and N

    D. Collins and N. Gisin, A relevant two qubit Bell in- equality inequivalent to the CHSH inequality, Journal of Physics A: Mathematical and General37, 1775 (2004), arXiv:quant-ph/0306129

    work page arXiv 2004

  44. [44]

    Scarani,Bell Nonlocality(Oxford University Press, 2019)

    V. Scarani,Bell Nonlocality(Oxford University Press, 2019)

    2019

  45. [45]

    Blakaj and M

    V. Blakaj and M. M. Wolf, On the set of reduced states of translation invariant, infinite quantum systems, Letters in Mathematical Physics114, 28 (2024)

    2024

  46. [46]

    K.Yang, Y.Zhu, X.Zeng, Z.Zou, M.-H.Yung,andZ.Wang, Cost of locally approximating high-dimensional ground states of contextual quantum models, Communications Physics8, 204 (2025)

    2025

  47. [47]

    Ringbauer, M

    M. Ringbauer, M. Meth, L. Postler, R. Stricker, R. Blatt, P. Schindler, and T. Monz, A universal qudit quantum pro- cessor with trapped ions, Nature Physics18, 1053 (2022)

    2022

  48. [48]

    Blatt, P

    P.Hrmo, B.Wilhelm, L.Gerster, M.W.vanMourik, M.Hu- ber, R. Blatt, P. Schindler, T. Monz, and M. Ringbauer, Na- tive qudit entanglement in a trapped ion quantum proces- sor, Nature Communications14, 2242 (2023)

    2023

  49. [49]

    Henriet, L

    L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O.Reymond,andC.Jurczak,Quantumcomputingwith neutral atoms, Quantum4, 327 (2020)

    2020

  50. [50]

    Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Frontiers in Physics8, 589504 (2020)

    2020

  51. [51]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum pro- cessor, Nature Communications5, 4213 (2014)

    2014

  52. [52]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S.Endo, K.Fujii, J.R.McClean, K.Mitarai, X.Yuan, L.Cin- cio, and P. J. Coles, Variational quantum algorithms, Na- ture Reviews Physics3, 625 (2021)

    2021

  53. [53]

    Calajó, G

    G. Calajó, G. Magnifico, C. Edmunds, M. Ringbauer, S. Montangero, and P. Silvi, Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits, PRX Quantum5, 040309 (2024), arXiv:2402.07987 [quant- ph]

    work page arXiv 2024

  54. [54]

    T. V. Zache, D. González-Cuadra, and P. Zoller, Fermion- qudit quantum processors for simulating lattice gauge theories with matter, Quantum7, 1140 (2023), arXiv:2303.08683 [quant-ph]

    work page arXiv 2023

  55. [55]

    X. Gao, P. Appel, N. Friis, M. Ringbauer, and M. Huber, On the role of entanglement in qudit-based circuit compres- sion, Quantum7, 1141 (2023), arXiv:2209.14584 [quant- ph]

    work page arXiv 2023

  56. [56]

    Wang and M

    Z. Wang and M. Navascués, Two-dimensional translation- invariant probability distributions: approximations, char- acterizations and no-go theorems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences474, 20170822 (2018)

    2018

  57. [57]

    Lagae, J

    A. Lagae, J. Kari, and P. Dutré, Aperiodic sets of square tiles with colored corners, Report CW460(2006)

    2006

  58. [58]

    Gottesman and S

    D. Gottesman and S. Irani, The quantum and classical complexity of translationally invariant tiling and Hamil- tonian problems, in2009 50th Annual IEEE Symposium on Foundations of Computer Science(IEEE, 2009) pp. 95–104

    2009

  59. [59]

    T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf, Undecidabil- ity of the spectral gap, Nature528, 207 (2015)

    2015

  60. [60]

    Ioannou and D

    M. Ioannou and D. Rosset, Noncommutative polynomial optimization under symmetry (2021), arXiv:2112.10803 [quant-ph]

    work page arXiv 2021

  61. [61]

    thesis, Université de Genéve (2015)

    Rosset, Denis,Characterization of correlations in quantum networks, Ph.D. thesis, Université de Genéve (2015). 18 Appendix A: The analytical solution of the 232-type contextuality witness

    2015

  62. [62]

    (A1) Using MPS-based optimization and NPA, we found that the maximum quantum violation can be saturated when d=5,D=4

    The optimal observables and state The 232-type contextuality witness is in the form of 〈E〉TI =〈σ (1) 1 σ(2) 0 +σ (1) 1 σ(2) 1 −σ (1) 2 σ(2) 0 +σ (1) 2 σ(2) 1 〉TI ≥ −2. (A1) Using MPS-based optimization and NPA, we found that the maximum quantum violation can be saturated when d=5,D=4. ThesimplestoptimalobservableswefoundforEq.(A1) are σ0 =   1 0 0 0 ...

  63. [63]

    Incorporating LTI and symmetries in the NPA hierarchy The goal in this subsection is to derive the tight lower bound of Eq. (A1). To compute the tight lower bound, we use NPA, which is a convergent hierarchy of SDPs charac- terizing the set of quantum correlations. The tricky part of Eq. (A1) is that the operators{σ(1) x σ(2) y }are evaluated on the two-b...