Pith. sign in

REVIEW 4 major objections 1 minor 69 references

The multi-word-representation number of a Cartesian or rooted product equals the larger factor's number; lexicographic powers give a sublinear bound on the extremal function and graphs of order 2593 or more need not bipartition into two wor

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 15:36 UTC pith:NTP3EQ7K

load-bearing objection Abstract-only package: product formulas for μ, a sublinear τ(n) bound, and a concrete WB threshold look like real niche advances, but the supplied full text is the wrong paper so nothing is checkable. the 4 major comments →

arxiv 2603.29629 v5 pith:NTP3EQ7K submitted 2026-03-31 math.CO cs.DM

On graph products and multi-word-representability

Benny George Kenkireth , Gopalan Sajith , Sreyas Sasidharan This is my paper
classification math.CO cs.DM MSC 05C7605C6268R10
keywords word-representable graphsmulti-word-representation numbergraph productslexicographic productcomparability graphsextremal function τ(n)Word-representable Bipartition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The multi-word-representation number μ(G) measures how many word-representable graphs are needed to cover every edge of G. The paper determines how this number behaves under the six classical graph products. For Cartesian and rooted products it is exactly the maximum of the two factors; for the corona product it lies between that maximum and the maximum plus one, with explicit conditions that force equality with the lower bound; for the lexicographic product it lies between the maximum and the sum, again with a covering condition that forces the lower bound. Logarithmic upper bounds are given for the tensor and strong products. The same machinery shows that the lexicographic power G^{[k]} is word-representable if and only if G itself is a comparability graph, supplies two simple upper bounds on μ of such powers, and yields the sublinear estimate τ(n) ≤ n^{log_8 6 + ε} for the extremal function that counts the worst-case μ. Finally the paper settles the Word-representable Bipartition problem in the negative: every order n ≥ 2593 admits a graph that cannot be split into two induced word-representable subgraphs.

Core claim

Under the Cartesian and rooted products μ is multiplicative in the strong sense μ(H) = max{μ(G1), μ(G2)}; under the corona and lexicographic products the same lower bound holds and is tight under natural covering hypotheses, while the upper bounds are at most one larger or the sum, respectively; lexicographic powers characterise comparability graphs and produce the first sublinear bound on the extremal function τ(n); and for every n ≥ 2593 there exist n-vertex graphs with no bipartition into two word-representable induced subgraphs.

What carries the argument

The multi-word-representation number μ(G) together with the auxiliary covering number cov_comp(G) (the least number of comparability graphs needed to cover G). These two parameters control the product formulae, the power characterisation, the bound on τ(n) obtained from lexicographic powers, and the explicit constructions that refute the Word-representable Bipartition claim for n ≥ 2593.

Load-bearing premise

The concrete numerical claims (the exponent log_8 6 + ε and the threshold n ≥ 2593) rest on explicit constructions and counting arguments whose correctness cannot be verified from the abstract alone.

What would settle it

Either exhibit a single Cartesian or rooted product whose multi-word-representation number strictly exceeds the maximum of the factors, or produce a graph of order at least 2593 that does bipartition into two word-representable induced subgraphs, or improve the exponent in the bound for τ(n) below log_8 6.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 1 minor

Summary. The abstract claims a systematic study of the multi-word-representation number μ(G) under six standard graph products. It asserts exact equalities μ(H)=max{μ(G1),μ(G2)} for Cartesian and rooted products; sandwich bounds for corona and lexicographic products with stated tightness conditions involving cov_comp; logarithmic bounds for tensor and strong products; the characterization that the lexicographic power G^{[k]} is word-representable iff G is a comparability graph, with related upper bounds on μ(G^{[k]}); a sublinear extremal bound τ(n)≤n^{log_8 6+ε} obtained via lexicographic powers; and a negative resolution of the Word-representable Bipartition problem for all n≥2593 via explicit n-vertex counterexamples.

Significance. If the stated theorems hold, the paper would give a coherent product calculus for multi-word-representability, a clean comparability characterization of lexicographic powers, a concrete sublinear bound on the extremal function τ(n), and a definitive negative answer to WB above a fixed order. Those are standard, checkable combinatorial contributions of clear interest to the word-representable-graphs community. The abstract is precise and free of free parameters. However, none of the proofs, constructions, or counting arguments appear in the supplied manuscript body, so the significance remains conditional on verification that is currently impossible.

major comments (4)
  1. The CACHEABLE full-manuscript text is not the paper under review. It is an unrelated quant-ph manuscript (Entanglement in prepare-and-measure scenarios…, arXiv:2603.29625). Consequently every load-bearing claim of 2603.29629—Cartesian/rooted equalities, corona and lexicographic sandwiches and their cov_comp tightness conditions, the G^{[k]} characterization, the τ(n) exponent, and the n≥2593 WB counterexamples—cannot be inspected, let alone verified. This is not a local gap; it is the absence of the argument itself.
  2. Abstract claim “μ(H)=max{μ(G1),μ(G2)} for Cartesian and rooted products”: without the product definitions, the covering constructions, and the lower-bound arguments that would appear in the corresponding sections, the equality cannot be checked for correctness or for edge cases (e.g., empty factors, disconnected graphs).
  3. Abstract claim “τ(n)≤n^{log_8 6+ε} via lexicographic powers” and “negative WB answer for every n≥2593”: both rest on explicit constructions and counting that are not present. Any error in the missing base graphs, the power iteration, or the order threshold would invalidate those two strongest quantitative statements. They cannot be stress-tested from the abstract alone.
  4. Abstract claim “G^{[k]} is word-representable iff G is a comparability graph”: the “only if” direction is nontrivial and typically requires an alternating-cycle or semi-transitive-orientation obstruction. That argument is missing from the supplied text, so the characterization remains unconfirmed.
minor comments (1)
  1. The abstract is well-written and notation (μ, cov_comp, τ(n), the six products) is standard; no presentation issues can be assessed beyond the abstract because the body is the wrong paper.

Circularity Check

0 steps flagged

No circularity: abstract defines μ and products then states independent combinatorial bounds; full text is mismatched unrelated paper so no load-bearing reductions can be exhibited.

full rationale

The supplied abstract for 2603.29629 introduces μ(G) as the min number of word-representable graphs covering G, then claims equalities and sandwich bounds for six standard graph products, a characterization of word-representability of lexicographic powers via comparability graphs, a sublinear extremal bound on τ(n) obtained from those powers, and an existence result for WB-counterexamples of order n≥2593. These are ordinary definitional setup followed by claimed theorems; nothing is defined in terms of the target quantity, no free parameters are fitted to data and then re-predicted, and no uniqueness or ansatz is imported via self-citation. The FULL MANUSCRIPT TEXT that follows is an entirely different quant-ph paper (prepare-and-measure entanglement, arXiv:2603.29625) whose equations and protocols have no bearing on the graph-theoretic claims, so no concrete reduction of any derivation step can be quoted. Under the hard rule that circularity may be asserted only when a specific reduction is exhibited from the paper’s own text, the only admissible finding is absence of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

Abstract-only review of a pure math combinatorics paper. No free parameters fitted to data. Background notions (word-representable graphs, multi-word-representation number μ, comparability graphs, cov_comp, standard graph products, extremal function τ(n), WB problem) are domain assumptions imported from the literature. No invented physical entities. Concrete numerical claims (n≥2593, exponent log_8 6+ε) depend on constructions not visible in the abstract.

axioms (4)
  • domain assumption Word-representable graphs and the multi-word-representation number μ(G) are well-defined as in the prior literature the paper builds on.
    Central object of study; abstract treats μ as established.
  • domain assumption Standard definitions of lexicographic, Cartesian, rooted, corona, tensor, and strong products.
    All product theorems rest on these definitions.
  • domain assumption Comparability graphs and the covering number cov_comp(G) behave as used in the tightness conditions and power characterization.
    Used for G^{[k]} characterization and lexicographic tightness.
  • domain assumption The extremal function τ(n) is the quantity bounded by n^{log_8 6+ε} via lexicographic powers.
    Abstract invokes τ(n) without restating its definition.

pith-pipeline@v1.1.0-grok45 · 23643 in / 2578 out tokens · 23125 ms · 2026-07-13T15:36:53.413642+00:00 · methodology

0 comments
read the original abstract

The multi-word-representation number $\mu(G)$ is the minimum number of word-representable graphs whose union is $G$. We investigate $\mu(H)$ for graph products $H$ obtained from $G_1$ and $G_2$ via six fundamental products: lexicographic, Cartesian, rooted, corona, tensor, and strong. We prove $\mu(H) = \max\{\mu(G_1), \mu(G_2)\}$ for Cartesian and rooted products. For the corona product, we show $\max\{\mu(G_1), \mu(G_2)\} \le \mu(H) \le \max\{\mu(G_1), \mu(G_2)\} + 1$, and show that the lower bound is tight when $\mu(G_1) > \mu(G_2)$ or $G_2$ admits a covering by $\mu(G_2)$ word-representable graphs, one of which is a comparability graph. For the lexicographic product, we show $\max\{\mu(G_1), \mu(G_2)\} \le \mu(H) \le \mu(G_1) + \mu(G_2)$, and show that the lower bound is tight when $\mathrm{cov}_{\mathrm{comp}}(G_2) \le \max\{\mu(G_1), \mu(G_2)\}$. We provide logarithmic bounds for tensor and strong products. We prove $G^{[k]}$ is word-representable if and only if $G$ is a comparability graph. We establish bounds $\mu(G^{[k]}) \le \mathrm{cov}_{\mathrm{comp}}(G)$ and $\mu(G^{[k]}) \le k$ for non-comparability word-representable graphs. Using lexicographic powers, we obtain the sublinear bound $\tau(n) \le n^{\log_8 6+\epsilon}$ for the extremal function $\tau(n)$. Finally, we address the Word-representable Bipartition (WB) problem, proving a negative answer for $n \geq 2593$: showing that for every such $n$, there exists a graph of order $n$ that cannot be vertex-partitioned into two word-representable induced subgraphs.

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

69 extracted references · 2 canonical work pages

  1. [1]

    Cleve and H

    R. Cleve and H. Buhrman, Substituting quantum entanglement for communication, Phys. Rev. A56, 1201 (1997)

    1997

  2. [2]

    Brassard, Quantum communication complexity, , 1593 (2001), quant-ph/0101005

    G. Brassard, Quantum communication complexity, , 1593 (2001), quant-ph/0101005. 6

    Pith/arXiv arXiv 2001

  3. [3]

    Brukner, M

    ˇC. Brukner, M. ˙Zukowski, J.-W. Pan, and A. Zeilinger, Bell’s inequalities and quantum communication complexity, Phys. Rev. Lett.92, 127901 (2004)

    2004

  4. [4]

    Buhrman, Łukasz Czekaj, A

    H. Buhrman, Łukasz Czekaj, A. Grudka, M. Horodecki, P. Horodecki, M. Markiewicz, F. Speelman, and S. Strelchuk, Quantum communication complexity advantage im- plies violation of a bell inequality, Proceedings of the National Academy of Sciences113, 3191 (2016), https://www.pnas.org/doi/pdf/10.1073/pnas.1507647113

    work page doi:10.1073/pnas.1507647113 2016

  5. [5]

    Tavakoli, M

    A. Tavakoli, M. ˙Zukowski, and ˇC. Brukner, Does violation of a Bell inequality always imply quantum advantage in a commu- nication complexity problem?, Quantum4, 316 (2020)

    2020

  6. [6]

    Buhrman, R

    H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Nonlocal- ity and communication complexity, Reviews of Modern Physics 82, 665–698 (2010)

    2010

  7. [7]

    Pauwels, S

    J. Pauwels, S. Pironio, E. Z. Cruzeiro, and A. Tavakoli, Adaptive advantage in entanglement-assisted communications, Physical Review Letters129, 10.1103/physrevlett.129.120504 (2022)

    doi:10.1103/physrevlett.129.120504 2022

  8. [8]

    J. B. Brask, N. Brunner, J. Pauwels, D. Rusca, and A. Tavakoli, Quantum correlations in prepare-and-measure sce- narios and their semi-device-independent applications (2026), arXiv:2603.23604 [quant-ph]

    arXiv 2026

  9. [9]

    P. E. Frenkel and M. Weiner, Classical information storage in an n-level quantum system, Communications in Mathematical Physics340, 563–574 (2015)

    2015

  10. [10]

    Massar and S

    S. Massar and S. Pironio, Violation of local realism ver- sus detection efficiency, Physical Review A68, 10.1103/phys- reva.68.062109 (2003)

    doi:10.1103/phys- 2003

  11. [11]

    P. E. Frenkel and M. Weiner, On entanglement assistance to a noiseless classical channel, Quantum6, 662 (2022)

    2022

  12. [12]

    Vieira, C

    C. Vieira, C. de Gois, L. Pollyceno, and R. Rabelo, Interplays between classical and quantum entanglement-assisted commu- nication scenarios, New Journal of Physics25, 113004 (2023)

    2023

  13. [13]

    Alimuddin, A

    M. Alimuddin, A. Chakraborty, G. L. Sidhardh, R. K. Patra, S. Sen, S. R. Chowdhury, S. G. Naik, and M. Banik, Advan- tage of hardy’s nonlocal correlation in reverse zero-error chan- nel coding, Phys. Rev. A108, 052430 (2023)

    2023

  14. [14]

    S. Rout, A. Chaturvedi, S. S. Bhattacharya, and P. Horodecki, Facets of non-locality and advantage in entanglement-assisted classical communication tasks (2025), arXiv:2507.10830 [quant-ph]

    Pith/arXiv arXiv 2025

  15. [15]

    Gallego, N

    R. Gallego, N. Brunner, C. Hadley, and A. Acín, Device- independent tests of classical and quantum dimensions, Phys. Rev. Lett.105, 230501 (2010)

    2010

  16. [16]

    Lörwald and G

    S. Lörwald and G. Reinelt, Panda: a software for polyhedral transformations, EURO Journal on Computational Optimiza- tion3, 297 (2015)

    2015

  17. [17]

    One has P m,x tr Am|x ⊗B x|mψAB ≤P m,x λmax(Am|x) tr Bx|mψB ≤ P m,x tr Bx|mψB =P m tr(ψB) =d

  18. [18]

    Code for optimal entanglement-assisted classical communica- tion protocols (2026)

    2026

  19. [19]

    Note that the(d,|X|,|B|) = (2,4,3)scenario is uninteresting because classical polytope can effectively be reduced to that in the scenario(d,|X|,|B|) = (2,3,3)

  20. [20]

    C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on einstein-podolsky-rosen states, Phys. Rev. Lett.69, 2881 (1992)

    1992

  21. [21]

    Tavakoli, J

    A. Tavakoli, J. Pauwels, E. Woodhead, and S. Pironio, Correla- tions in entanglement-assisted prepare-and-measure scenarios, PRX Quantum2, 10.1103/prxquantum.2.040357 (2021)

    work page doi:10.1103/prxquantum.2.040357 2021

  22. [22]

    Pauwels, A

    J. Pauwels, A. Tavakoli, E. Woodhead, and S. Pironio, Entan- glement in prepare-and-measure scenarios: many questions, a few answers, New Journal of Physics24, 063015 (2022)

    2022

  23. [23]

    Piveteau, J

    A. Piveteau, J. Pauwels, E. Håkansson, S. Muhammad, M. Bourennane, and A. Tavakoli, Entanglement-assisted quan- tum communication with simple measurements, Nature Com- munications13, 7878 (2022)

    2022

  24. [24]

    Piveteau, A

    A. Piveteau, A. A. Abbott, S. Muhammad, M. Bourennane, and A. Tavakoli, Weak entanglement improves quantum communi- cation using only product measurements, Phys. Rev. Appl.21, 034053 (2024)

    2024

  25. [25]

    Bakhshinezhad, M

    P. Bakhshinezhad, M. Mehboudi, C. R. i. Carceller, and A. Tavakoli, Scalable entanglement certification via quantum communication, PRX Quantum5, 020319 (2024)

    2024

  26. [26]

    Zhang, J.-L

    C. Zhang, J.-L. Miao, X.-M. Hu, J. Pauwels, Y . Guo, C.-F. Li, G.-C. Guo, A. Tavakoli, and B.-H. Liu, Quantum stochastic communication via high-dimensional entanglement, Phys. Rev. Lett.135, 120802 (2025)

    2025

  27. [27]

    Oszmaniec, L

    M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Simulat- ing positive-operator-valued measures with projective measure- ments, Phys. Rev. Lett.119, 190501 (2017)

    2017

  28. [28]

    M. J. Renner, Compatibility of generalized noisy qubit mea- surements, Phys. Rev. Lett.132, 250202 (2024)

    2024

  29. [29]

    Zhang and E

    Y . Zhang and E. Chitambar, Exact steering bound for two-qubit werner states, Phys. Rev. Lett.132, 250201 (2024)

    2024

  30. [30]

    M. J. Renner, A. Tavakoli, and M. T. Quintino, Classical cost of transmitting a qubit, Phys. Rev. Lett.130, 120801 (2023)

    2023

  31. [31]

    A. Acín, S. Pironio, T. Vértesi, and P. Wittek, Optimal ran- domness certification from one entangled bit, Phys. Rev. A93, 040102 (2016)

    2016

  32. [32]

    Mironowicz and M

    P. Mironowicz and M. Pawłowski, Experimentally feasi- ble semi-device-independent certification of four-outcome positive-operator-valued measurements, Phys. Rev. A100, 030301 (2019)

    2019

  33. [33]

    Tavakoli, D

    A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling computa- tion of correlation bounds for finite-dimensional quantum sys- tems via symmetrization, Phys. Rev. Lett.122, 070501 (2019)

    2019

  34. [34]

    Tavakoli, M

    A. Tavakoli, M. Smania, T. Vértesi, N. Brun- ner, and M. Bourennane, Self-testing nonprojective quantum measurements in prepare-and-measure ex- periments, Science Advances6, eaaw6664 (2020), https://www.science.org/doi/pdf/10.1126/sciadv.aaw6664

    doi:10.1126/sciadv.aaw6664 2020

  35. [35]

    Martínez, E

    D. Martínez, E. S. Gómez, J. Cariñe, L. Pereira, A. Delgado, S. P. Walborn, A. Tavakoli, and G. Lima, Certification of a non-projective qudit measurement using multiport beamsplit- ters, Nature Physics19, 190 (2023)

    2023

  36. [36]

    Prevedel, P

    R. Prevedel, P. Walther, F. Tiefenbacher, P. Böhi, R. Kaltenbaek, T. Jennewein, and A. Zeilinger, High-speed linear optics quan- tum computing using active feed-forward, Nature445, 65 (2007)

    2007

  37. [37]

    X.-S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor, B. Wittmann, A. Mech, J. Kofler, E. Anisimova, V . Makarov, T. Jennewein, R. Ursin, and A. Zeilinger, Quantum teleportation over 143 kilometres using active feed-forward, Nature489, 269 (2012)

    2012

  38. [38]

    Carrera Vazquez, C

    A. Carrera Vazquez, C. Tornow, D. Ristè, S. Woerner, M. Takita, and D. J. Egger, Combining quantum processors with real-time classical communication, Nature636, 75 (2024)

    2024

  39. [39]

    Sakaguchi, S

    A. Sakaguchi, S. Konno, F. Hanamura, W. Asavanant, K. Takase, H. Ogawa, P. Marek, R. Filip, J.-i. Yoshikawa, E. Huntington, H. Yonezawa, and A. Furusawa, Nonlinear feed- forward enabling quantum computation, Nature Communica- tions14, 3817 (2023)

    2023

  40. [40]

    Tavakoli, E

    A. Tavakoli, E. Zambrini Cruzeiro, J. Bohr Brask, N. Gisin, and N. Brunner, Informationally restricted quantum correla- 7 tions, Quantum4, 332 (2020)

    2020

  41. [41]

    Chaturvedi and D

    A. Chaturvedi and D. Saha, Quantum prescriptions are more on- tologically distinct than they are operationally distinguishable, Quantum4, 345 (2020)

    2020

  42. [42]

    Tavakoli, E

    A. Tavakoli, E. Zambrini Cruzeiro, E. Woodhead, and S. Piro- nio, Informationally restricted correlations: a general frame- work for classical and quantum systems, Quantum6, 620 (2022)

    2022

  43. [43]

    Tavakoli, A

    A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Araújo, Semidefinite programming relaxations for quantum corre- lations, Reviews of Modern Physics96, 10.1103/revmod- phys.96.045006 (2024)

    doi:10.1103/revmod- 2024

  44. [44]

    Pironio, All Clauser–Horne–Shimony–Holt polytopes, Jour- nal of Physics A: Mathematical and Theoretical47, 424020 (2014)

    S. Pironio, All Clauser–Horne–Shimony–Holt polytopes, Jour- nal of Physics A: Mathematical and Theoretical47, 424020 (2014)

    2014

  45. [45]

    G. M. Ziegler,Lectures on Polytopes, Graduate Texts in Math- ematics, V ol. 152 (Springer, New York, 1995). Appendix A: The second facet in the 4-input-4-output scenario We introduce the second facet associated the (d,|X|,|B|) = (2,4,4)scenario. Alice encodes her in- putsx∈ {0,1,2,3}into a single bit sent to Bob, who upon receiving the message decodes it ...

    1995

  46. [46]

    Let Alice and Bob share the max- imally entangled state|ϕ +⟩

    Quantum communication By using a qubit message, we can enhance the adaptive communication advantage. Let Alice and Bob share the max- imally entangled state|ϕ +⟩. Alice applies unitariesXand Yto her qubit whenx= 0andx= 1, respectively, and send the resulting state to Bob. Whenx= 2andx= 3Al- ice measures her local state with observableZ. In the former case...

  47. [47]

    In general, these are convex combinations of determin- istic response functions

    Classical correlations The correlations of the adaptive prepare-and-measure- scenario admits a classical model if they can be decomposed on the formp(b|x) = P λ p(λ)pλ(b|x), wherep(λ)is a prob- ability distribution and pλ(b|x) = X m=0,1 p(m|x, λ)p(b|m, λ).(B2) Here,p(m|x, λ)andp(b|m, λ)are known as response func- tions. In general, these are convex combin...

  48. [48]

    Using thatS (n) is linear in the correlations, the functional reaches its maximum value at the vertices of the polytopeL

    The classical bound We show thatS (n) cannot exceed unit for all classical cor- relations. Using thatS (n) is linear in the correlations, the functional reaches its maximum value at the vertices of the polytopeL. This implies that S (n) = 1 2(n−1) X λ p(λ) n−1X i=0 (n−1)p λ(i|i) +p λ(⊥ |i) ≤max λ 1 2(n−1) n−1X i=0 (n−1)p λ(i|i) +p λ(⊥ |i) . (B5) Since Ali...

  49. [49]

    inconclusive

    Linearly independent strategies Through an explicit construction, we now show that there existsn 2 linearly independent vertices for whichS (n) = 1. We denote the set of these strategies by ˜P. As already mentioned above, there isndifferent ways to play the second strategy in which Bob only discriminates one of Alice’s inputs and outputs "inconclusive" in...

  50. [50]

    In this case the parties perform the following strategy:f(x) =δ(x∈ {0, k})andg(0) = 1 andg(1) =k, withk∈ {2, ..., n−1}

    Next, consider that Bob correctly distinguishx= 1, k, withk∈ {2, ..., n−1}. In this case the parties perform the following strategy:f(x) =δ(x∈ {0, k})andg(0) = 1 andg(1) =k, withk∈ {2, ..., n−1}. This yields that S (n) = 1 2(p(1|1) +p(k|k)) = 1andp(k|0) = 1for allk >1. At this point, we have found(n−1) 2 + (n−1) +n= n2 linearly independent vertices that s...

  51. [51]

    Since the collection of points in ˜Psaturates the inequality S (n) ≤1, these points lies in a face ofL[45]

    Proof of facet We now show that the set of vertices in ˜Pspans a facet. Since the collection of points in ˜Psaturates the inequality S (n) ≤1, these points lies in a face ofL[45]. The di- mension of the face ˜Pis the dimension of its affine hull, i.e., dim(aff( ˜P)), where aff( ˜P) ={ n2 X i=1 λixi | n2 X i=1 λi = 1},(B16) for some coefficientsλ i. Specif...

  52. [52]

    2p(1|0) +p(2|0) +p(3|0) +p(2|1) +p(3|1) + 2p(4|1)−p(1|2)−p(3|2)−p(4|2)−p(1|3)−p(2|3)−p(4|3)≤2 2.2071

    2071

  53. [53]

    2p(1|0) + 2p(2|0) +p(3|0) +p(3|1) + 2p(4|1)−2p(1|3)−2p(2|3)−p(3|3)−2p(4|3)≤2 2.1547

  54. [54]

    2p(1|0) +p(2|0) +p(3|0) +p(2|1) +p(3|1) + 2p(4|1)−2p(1|3)−p(2|3)−p(3|3)−2p(4|3)≤2 2.1547

  55. [55]

    2p(1|0) + 2p(2|0) +p(3|0) +p(3|1) + 2p(4|1)−p(1|2)−p(2|2)−p(4|2)−p(1|3)−p(2|3)−p(3|3)−p(4|3)≤2 2.1784

  56. [56]

    2p(1|0) +p(2|0) +p(3|0) +p(2|1) + 2p(4|1) +p(3|2)−2p(1|3)−p(2|3)−2p(3|3)−2p(4|3)≤2 2.1784

  57. [57]

    2p(1|0) +p(2|0) +p(3|0) +p(2|1) +p(3|1) + 2p(4|1)−p(1|2)−p(4|2)−p(1|3)−p(2|3)−p(3|3)−p(4|3)≤2 2.1784

  58. [58]

    p(1|0) +p(2|0) +p(3|0) +p(1|1) +p(2|1) +p(4|1) +p(3|2) +p(4|2)−p(1|3)−p(2|3)−p(3|3)−p(4|3)≤2 2.2071

    2071

  59. [59]

    4p(1|0) + 2p(2|0) +p(3|0) + 2p(2|1) +p(3|1) + 4p(4|1) + 2p(3|2)−4p(1|3)−2p(2|3)−3p(3|3)−4p(4|3)≤4 4.3094

  60. [60]

    2p(1|0) +p(2|0) +p(2|1) + 2p(3|1)−p(1|2)−p(3|2)−p(1|3)−p(2|3)−p(3|3)≤2 2.1784

  61. [61]

    p(1|0) +p(2|0) +p(1|1) +p(3|1) +p(2|2) +p(3|2)−p(1|3)−p(2|3)−p(3|3)≤2 2.2071

    2071

  62. [62]

    3p(1|0) +p(2|0) +p(2|1) + 3p(3|1) +p(2|2) + 3p(4|2)−3p(1|3)−2p(2|3)−3p(3|3)−3p(4|3)≤3 3.1432

  63. [63]

    2p(1|0) + 2p(2|0) +p(3|0) +p(1|1) +p(3|1) + 2p(4|1) +p(2|2) +p(3|2)+ +p(4|2)−2p(1|3)−2p(2|3)−p(3|3)−2p(4|3)≤3 3.1784

  64. [64]

    2p(1|0) +p(2|0) +p(3|0) +p(1|1) +p(2|1) + 2p(4|1) +p(2|2) + 2p(3|2)+ p(4|2)−2p(1|3)−p(2|3)−2p(3|3)−2p(4|3)≤3 3.1784

  65. [65]

    4p(1|0) + 2p(2|0) +p(3|0) + 2p(2|1) +p(3|1) + 4p(4|1)−p(1|2) + 2p(3|2)− p(4|2)−3p(1|3)−2p(2|3)−3p(3|3)−3p(4|3)≤4 4.3202

  66. [66]

    3p(1|0) + 3p(2|0) + 2p(3|0) + 3p(1|1) + 2p(3|1) + 3p(4|1) + 3p(2|2)+ 2p(3|2) + 3p(4|2)−3p(1|3)−3p(2|3)−2p(3|3)−3p(4|3)≤6 6.6213

  67. [67]

    3p(1|0) + 2p(2|0) +p(3|0) +p(1|1) + 2p(3|1) +p(4|1) + 2p(2|2) +p(3|2)+ 3p(4|2)−3p(1|3)−2p(2|3)−2p(3|3)−3p(4|3)≤4 4.3371

  68. [68]

    4p(1|0) + 3p(2|0) + 2p(3|0) + 2p(1|1) + 2p(3|1) + 2p(4|1) + 3p(2|2) + 2p(3|2)+ 4p(4|2)−4p(1|3)−3p(2|3)−2p(3|3)−4p(4|3)≤6 6.5298

  69. [69]

    4p(1|0) + 3p(2|0) + 2p(3|0) + 2p(1|1) + 2p(3|1) + 3p(4|1) + 3p(2|2) + 2p(3|2)+ 3p(4|2)−4p(1|3)−3p(2|3)−2p(3|3)−4p(4|3)≤6 6.5733 TABLE I:List over non-trivial facet inequalitites in the prepare-and-measure scenario in which Alice’s receivesx∈ {0,1,2,3}and Bob produce outputsb∈ {0,1,2,3,4}