Contextuality in the $n$-qubit Pauli group

Markus Frembs

REVIEW 3 major objections 5 minor 38 references

Contextuality in the $n$-qubit Pauli group

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Contextuality in the n-qubit Pauli group collapses to a single exceptional case: two qubits.

desk verdict Genuinely useful classification paper—Theorem 2 looks sound, but Lemma 11's proof is garbled and Lemma 13 leans on a deep cited theorem. read the letter →

arxiv 2608.11993 v1 pith:FWFWYX6E submitted 2026-08-12 quant-ph math.SG

Markus Frembs This is my paper · ORCID

classification quant-phmath.SG MSC 81P1381P6851A5005B25 PACS 03.65.Ta03.67.-a

keywords contextualityKochen-SpeckertheoremPauligroupstabiliserstatesframefunctionssymplecticpolarspacesCameron-Lieblersets

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

This paper proves that the n-qubit Pauli group admits no nontrivial 'noncontextual properties' beyond the two-qubit case, and it gives a complete classification of the Boolean-valued frame functions that encode them. The central claim is that for n≥3, every Boolean-valued frame function on stabiliser states is constant, meaning the only nontrivial noncontextual properties occur for n=2. The paper further shows that even the underlying symplectic theory, stripped of the Pauli group's phase structure, is contextual, though it admits a restricted set of nonconstant frame functions. If correct, this sharpens the boundary between contextual and noncontextual behaviour in the most widely used discrete quantum setting.

What carries the argument

The central objects are Boolean-valued frame functions—maps f from stabiliser states (or symplectic vectors) to {0,1} that assign a fixed number k of 1s in every maximal context—and the context poset (Iso(V),⊆) of isotropic subspaces of the symplectic vector space V=$Z_2^{{2n}}$. The classification proceeds via Fourier analysis over Lagrangian subspaces (Parseval's identity, Eq. B3), reduction of frame functions to lower-dimensional eigenspaces (Lemma 5), and the use of quadratic refinements and their Witt indices to constrain weight-2 and weight-4 frame functions.

What would settle it

Find an explicit Boolean-valued frame function on stabiliser states for n=3 with weight k=4 that is not constant, or construct a weight-2 symplectic frame function on $S_symp^{3}$ that is not of the form f_{q,α}. Either would directly contradict Theorem 2 or Theorem 3.

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Extended reading notes

Core claim

For n≥3, the context poset of the Pauli group C(\tilde{P}_n) admits no nontrivial noncontextual properties; equivalently, every Boolean-valued frame function on the stabiliser states S_stab^n is constant. The only nontrivial case is n=2, where weight-2 noncontextual properties exist as in Proposition 1. In the symplectic theory, nonconstant Boolean-valued frame functions do exist for all n, but only in restricted forms: they are either evaluations of linear functionals combined with the quadratic refinement q_W, or they are supported on level sets of quadratic refinements of Witt index n-1 (Theorem 3). The paper establishes this by characterizing all Boolean-valued frame functions on both the Pauli group and its symplectic projection.

Load-bearing premise

The classification of weight-2 frame functions in the symplectic case rests on the unproven (in this paper) universality of the natural projective embedding of the elliptic polar space Q^-(2n-1,2); if that universality fails, additional weight-2 frame functions beyond the listed ones could exist.

Editorial extensions

If this is right

For n≥3, stabiliser quantum theory admits no hidden-variable model that is noncontextual even for constants, in a generalised sense; only the two-qubit Pauli group has any noncontextual properties.
The nonexistence of valuations in quaternionic quantum mechanics for dimension 2^{n-1} (n≥3) follows from a discrete argument using only the context stabiliser poset, without relying on the full Gleason theorem.
The classification of Boolean-valued frame functions in the symplectic theory completely determines all Cameron–Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space, yielding parameters x∈{0,1,2,2^n-2,2^n-1,2^n} for n≥3.
Contextuality in the Pauli group is not merely a consequence of the nontrivial phase cocycle; the underlying symplectic geometry itself forces contextuality for n≥3.

Reading between the lines

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

One could test the robustness of the classification by numerically searching for Boolean-valued frame functions on stabiliser states for n=4 with weights other than the forbidden values; the paper's proof implies none exist, so a search that finds one would refute the central claim.
The paper implies that for odd-prime qudits, the analogous Cameron–Liebler classification problem remains open and could yield a richer family of noncontextual properties; this is a natural next step the author does not explicitly pursue.
The result suggests that the two-qubit case is the only possible resource for noncontextual properties within Pauli-based quantum computation, potentially limiting the role of such properties in magic-state distillation schemes for n≥3.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Markus, this paper is a real contribution and it deserves a serious referee, but it needs cleanup before it is publishable as-is.

The new idea is simple and productive: a noncontextual property is a relaxation of a valuation that keeps a non-atomic event fixed under the holonomy group of a context connection. Theorem 1 shows this is exactly equivalent to a Boolean-valued frame function, and that equivalence is what makes the paper work. The headline results are genuinely strong: for n>=3, every Boolean frame function on n-qubit stabiliser states is constant, so the two-qubit Mermin-Peres weight-2 family is the only nontrivial case in the Pauli group. For the symplectic theory, the classification lists all nonconstant Boolean frame functions explicitly, and the translation to Cameron–Liebler sets of maximal totally isotropic flats in the affine binary symplectic space is a nice payoff—it solves the binary case of a problem that was open in that form.

The proof of Theorem 2 is clean: odd weights are ruled out by the usual no-valuation argument, weight 2 by a quadratic-refinement/Mermin-Peres argument, weight 4 by the nonexistence of 1-ovoids, and everything else by induction with Lemma 5. I checked the logic and found no gap there. The symplectic part is much more technical, and that is where the soft spots are.

First, the proof of Lemma 11 contains a corrupted passage: about ten lines of the computation are gibberish. The intended construction is recoverable from Lemma 1 and the context, but not verifiable as written. That must be fixed.

Second, Lemma 13 is load-bearing: it extends certain additive maps on the elliptic quadric to linear functionals, via universality of the projective embedding of Q^-(2n-1,2), citing Refs. [64,65]. The result is standard and I have no reason to doubt it, but the paper's central classification depends on it without an independent argument. A referee will want a precise statement of the embedding theorem and its applicability, or a direct proof.

The absence of machine-checkable certificates for the finite computations in Lemma 16 is a minor issue; the lists are explicit enough to audit by hand, and crediting the LLM that found them is transparent. The citation pattern is fine—the self-references are to the author's own framework, but the classification itself is derived from definitions, not fitted to data.

My recommendation: send this to peer review. Anyone working on Kochen-Specker contextuality, stabiliser theory, or Cameron-Liebler sets will get value from it. It is mathematically substantial, the central no-go for Pauli groups appears sound, and the two issues named above are fixable with reasonable effort. I would like to see a revised version before signing off, but the paper deserves referee time.

Referee Report

3 major / 5 minor

Summary. The paper introduces a generalized notion of Kochen-Specker noncontextuality, called a "noncontextual property," defined via context connections whose holonomy stabilizes a nontrivial event. Theorem 1 equates such properties with nonconstant Boolean-valued frame functions. The main results are: Theorem 2 shows that for n≥3 every Boolean-valued frame function on stabiliser states of the n-qubit Pauli group is constant, so no nontrivial noncontextual properties exist; Proposition 1 constructs the exceptional weight-2 noncontextual properties for n=2; Theorem 3 classifies all nonconstant Boolean-valued frame functions in the underlying symplectic theory for n≥3, showing they are either valuations from linear functionals and quadratic refinements (weight 1), weight-2 functions built from a quadratic refinement of Witt index n−1 and a linear functional, or complements of these. Corollary 5 translates this classification into a classification of binary affine-symplectic Cameron–Liebler sets of maximal totally isotropic flats.

Significance. If the identified gaps are repaired, this is a substantial contribution. The paper gives a clean conceptual bridge between holonomy-based Kochen–Specker noncontextuality and Boolean frame functions, proves a complete no-go theorem for the stabiliser Pauli theory for n≥3, and identifies the two-qubit case as the unique nontrivial exception. The symplectic classification is connected to a concrete open problem in finite geometry, the binary affine-symplectic Cameron–Liebler problem, and the authors provide explicit constructions and exhaustive finite-geometric arguments including the weight-4 case for n=3. The proof strategy—Fourier analysis over GF(2), ovoid nonexistence, polar-space embedding theory, and explicit Lagrangian computations—is coherent and, apart from the issues below, internally consistent. The paper also contains useful applications to quaternionic valuations and to the Λ-polytope literature. These are genuine conceptual and technical advances, provided the corrupted passages are repaired and the cited universality input in Lemma 13 is properly documented.

major comments (3)

[Appendix D, proof of Lemma 11] The proof of Lemma 11 contains an unreadable corrupted passage immediately after the equality "p_{U|C_{\tilde U}} = ...", consisting of the strings "⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟...". This is the central restriction computation that establishes compatibility of the selected elements p_U under coarse-graining. Since Lemma 11 is essential for the existence part of Theorem 3, this passage must be rewritten as a clean, verifiable derivation.
[Appendix D, Lemma 13] The proof of Lemma 13 asserts injectivity of the canonical map Φ:E_Q→V by citing the universality of the natural projective embedding of the elliptic polar space Q^-(2n-1,2) (Refs. [64,65]), but no precise statement of the invoked theorem is given and no independent derivation is supplied. Because Lemma 14, and therefore the weight-2 classification in Theorem 3 and Corollary 5, depends critically on this injectivity—if it failed, additional additive maps on Q not induced by linear functionals on V would survive—the authors should state the exact universality result from [64,65] and prove that it applies to this embedding, or give a self-contained argument.
[Appendix F, Proposition 2] The correspondence between symplectic stabiliser states and affine Lagrangians is written with corrupted symbols ("∼ /leftr⫯g⊸tl⫯ne→", "/leftfootl⫯ne→"), which obscures the bijection Φ and the map f ↦ L_f. Since Proposition 2 and Corollary 5 rest on this bijection, the text must be repaired and the bijection stated explicitly and cleanly.

minor comments (5)

[Title and abstract] "then-qubit Pauli group" should read "the n-qubit Pauli group" (two occurrences); the repeated article appears to be a LaTeX or extraction artifact.
[Definition 3] The notation "(0,1≠) p_0 ∈ C_0" is nonstandard; consider writing "p_0 ∉ {0,1}" or "a nontrivial event p_0".
[Appendix A, proof of Theorem 1, Eq. (A5)] The expression "1_p" is undefined; clarify whether it denotes the indicator of p or something else in the decomposition C = C_0^f ⊕ C_1^f.
[Corollary 5] The phrase "this completes the binary classification in all ranks" should be reconciled with the exceptional n=2 case, which is handled separately in Proposition 1 and requires the arbitrary function s:q^{-1}(0)→Z_2.
[Section IV and Lemma 9] The main text refers to Lemma 9 before it is stated in Appendix D; add a forward pointer or restructure so that the lemma appears before first citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the frame-function classifications are derived from definitions, linear algebra, and external finite-geometry theorems; the author's self-citations are not load-bearing for the main results.

full rationale

The paper's main claims are Theorem 2 (constant Boolean-valued frame functions on n-qubit stabiliser states for n≥3) and Theorem 3 (symplectic classification). Theorem 2 is proved in App. B from Lemma 2 (odd-weight frame functions imply valuations, ruled out by the external Mermin-Peres arguments), Lemma 3 (weight-2 functions force a Mermin-Peres arrangement), and Lemma 6 (weight-4 functions force a 1-ovoid, whose nonexistence is cited to external work). No fitted parameter is renamed as a prediction, and no definition smuggles the conclusion: the noncontextual-property/frame-function equivalence (Theorem 1) is proved directly in App. A by coarse-graining arguments. Theorem 3 is proved in App. D from linear-algebra lemmas (Lemmas 9, 14, 17) together with Lemma 13, which cites the universality of the projective embedding of the elliptic polar space Q^-(2n-1,2) to Refs. [64] (Cardinali-Giuzzi-Pasini) and [65] (Tits). That cited universality is an external mathematical theorem, not an assertion of this paper; reliance on it is an auditability and completeness concern, but not circularity. The paper itself also notes in App. F that a general Cameron-Liebler classification is not presently known, which is an honest scope statement. The author's self-citations (Refs. [5,6] for the context-connection framework and Ref. [20] for unextendible stabiliser bases) are used to set up terminology or to note additional properties of stabiliser posets; they are not used to force either classification. In particular, removing Ref. [20] would not change Theorem 2 or Theorem 3, and Theorem 1's proof is self-contained even without Refs. [5,6]. The final identification with Cameron-Liebler sets in App. F uses the external equivalence from Guo-Wan [25] to translate the already-proved classification, and the paper explicitly states that Guo-Wan had only partial results, so this is not a renaming of a known classification. Overall, the derivation chain is self-contained apart from standard external theorems, and no circular reduction is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper is a pure mathematics work with no fitted parameters. Its central claims rest on the author's earlier context-connection framework (Refs. [5,6]), on a deep embedding-universality theorem from finite geometry (Lm. 13), on standard facts about ovoids and the Mermin-Peres square, and on a self-cited result about unextendible stabiliser bases (Ref. [20]). The main new invented entity is the definitional notion of noncontextual property.

assumptions (5)

domain assumption Equivalence between KS noncontextuality and trivial contextual holonomy, and the context-connection formalism of Refs. [5,6].
Taken as the foundational framework; the paper extends it to non-atomic events via Def. 3 without re-deriving the equivalence Eq. (2).
domain assumption Universality of the projective embedding of the elliptic polar space Q^-(2n-1,2) (Ref. [64], Tits [65]).
Load-bearing in Lm. 13 to extend local characters on q^{-1}(0) to a global linear functional; cited, not proved.
domain assumption Nonexistence of 1-ovoids in the symplectic polar space W(2n-1,2) for n≥3, and the bound on maximal sets of mutually anti-commuting Pauli operators (Ref. [38], footnote 57).
Used in Lm. 6 and Lm. 15 to exclude weight-4 stabiliser frame functions and odd-weight symplectic frame functions.
standard math The Mermin-Peres square admits no valuation (Refs. [13-15]).
Used in Lm. 2 and Lm. 3 to rule out odd-weight and weight-2 frame functions by reducing to a no-valuation argument.
domain assumption P_stab^n is not a partial Boolean algebra for n≥4, and there exist unextendible sets of orthogonal stabiliser states (Ref. [20], same author).
Used to describe the difference between C(P̃n) and C(P_stab^n); not load-bearing for the main classification, but shapes the poset framework.

invented entities (1)

Noncontextual property (Def. 3)
purpose: Generalises the notion of valuation from atoms to arbitrary events of an event algebra; equivalent to Boolean-valued frame functions by Thm 1.
A definitional concept, not a physical entity; it has no external falsifiable handle, but it is the core object of the paper.

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Cite this review

Pith. "Pith review of Contextuality in the $n$-qubit Pauli group." pith.science (2026). https://pith.science/paper/FWFWYX6E

@misc{pith2026260811993,
  author       = {Pith},
  title        = {Pith review of: Contextuality in the $n$-qubit Pauli group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWFWYX6E}},
  note         = {Machine review of arXiv:2608.11993}
}

abstract

The $n$-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the $n$-qubit Pauli group. For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with $n\geq 2$, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the $n$-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all $n$, yet only in restricted form. By comparison, this shows that contextuality in the $n$-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.

Figures

Figures reproduced from arXiv: 2608.11993 by the authors.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

38 extracted references · 30 canonical work pages

[30]

Planat and M

M. Planat and M. Saniga, On the Pauli graphs ofN- qudits, Quantum Inf. Comput.8, 127 (2008)

work page 2008
[31]

Havl´ ıˇ cek, B

H. Havl´ ıˇ cek, B. Odehnal, and M. Saniga, Factor-group- generated polar spaces and (multi-)qudits, SIGMA5, 096 (2009)

work page 2009
[32]

Thas, The geometry of generalized pauli operators of n-qudit hilbert space, and an application to mubs, Euro- physics Letters86, 60005 (2009)

K. Thas, The geometry of generalized pauli operators of n-qudit hilbert space, and an application to mubs, Euro- physics Letters86, 60005 (2009)

work page 2009
[33]

L´ evay, M

P. L´ evay, M. Saniga, and P. Vrana, Three-qubit opera- tors, the split Cayley hexagon of order two, and black holes, Phys. Rev. D78, 124022 (2008)

work page 2008
[34]

Holweck, M

F. Holweck, M. Saniga, and P. L´ evay, A notable relation betweenN-qubit and 2 N−1-qubit Pauli groups via binary LGr(N,2N), SIGMA10, 041 (2014)

work page 2014
[35]

An algebraically closed family of informational n-qubit purity invariants

M. Frembs, G. Natale, C. S. P. Wever, and P. A. Hoehn, An algebraically closed family of informational n-qubit purity invariants (2026), arXiv:2607.24987 [quant-ph]

work page Pith review arXiv 2026
[36]

Maximal complementarity in the n-qubit Pauli group

M. Frembs, G. Natale, C. S. P. Wever, and P. A. Hoehn, Maximal complementarity in the n-qubit pauli group (2026), arXiv:2607.24988 [quant-ph]

work page Pith review arXiv 2026
[37]

Arf, Untersuchungen ¨ uber quadratische Formen in K¨ orpern der Charakteristik 2., Journal f¨ ur die reine und angewandte Mathematik1941, 148 (1941)

C. Arf, Untersuchungen ¨ uber quadratische Formen in K¨ orpern der Charakteristik 2., Journal f¨ ur die reine und angewandte Mathematik1941, 148 (1941)

work page 1941

Show all 38 references

[38]

Sarkar and E

R. Sarkar and E. van den Berg, On sets of maximally commuting and anticommuting Pauli operators, Res. Math. Sci.8, 14 (2021)

2021
[39]

For a thorough comparison of these notions, see Ref. [6]
[40]

Greechie, On the structure of orthomodular lattices satisfying the chain condition, J

R. Greechie, On the structure of orthomodular lattices satisfying the chain condition, J. Comb. Theory4, 210 (1968)

1968
[41]

Harding and M

J. Harding and M. Navara, Subalgebras of orthomodular lattices, Order28, 549 (2011)

2011
[42]

Kochen, A reconstruction of quantum mechanics, Foundations of Physics45, 557 (2015)

S. Kochen, A reconstruction of quantum mechanics, Foundations of Physics45, 557 (2015)

2015
[43]

Harding, C

J. Harding, C. Heunen, B. Lindenhovius, and M. Navara, Boolean subalgebras of orthoalgebras, Order36, 563 (2019). 6

2019
[44]

Cannon and A

S. Cannon and A. D¨ oring, A generalisation of stone du- ality to orthomodular lattices, inReality and Measure- ment in Algebraic Quantum Theory, edited by M. Ozawa, J. Butterfield, H. Halvorson, M. R´ edei, Y. Kitajima, and F. Buscemi (Springer, Singapore, 2018) pp. 3–65

2018
[45]

Cabello, Specker’s fundamental principle of quan- tum mechanics, arXiv e-prints (2012), arXiv:1212.1756 [quant-ph]

A. Cabello, Specker’s fundamental principle of quan- tum mechanics, arXiv e-prints (2012), arXiv:1212.1756 [quant-ph]

2012 arXiv
[46]

Fritz, A

T. Fritz, A. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Ac´ ın, Local orthogonality as a multi- partite principle for quantum correlations, Nature Com- munications4, 10.1038/ncomms3263 (2013)

2013 doi
[47]

All Boolean algebras are assumed finite and atomic
[48]

A Boolean algebra defines a Boolean ring byp⋅q∶=p∧q andp+q∶=(p∧q ⊥)∨(p ⊥∧q), and vice versa viap ⊥ ∶=1+p, p∧q∶=pq, andp∨q∶=p+q+pq
[49]

Moreover, it holds (by construction) for the context poset of then-qubit Pauli groupC( ̃Pn)in Eq

Clearly, this is the case for the poset of all contexts of a quantum system, denoted byC(H). Moreover, it holds (by construction) for the context poset of then-qubit Pauli groupC( ̃Pn)in Eq. (8), as well as the context poset generated by stabiliser states (see Eq. (10)) below
[50]

van den Berg and C

B. van den Berg and C. Heunen, Noncommutativity as a colimit, Applied Categorical Structures20, 393 (2012)

2012
[51]

Abramsky and R

S. Abramsky and R. S. Barbosa, The Logic of Con- textuality, in29th EACSL Annual Conference on Com- puter Science Logic (CSL 2021), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 183, edited by C. Baier and J. Goubault-Larrecq (Schloss Dagstuhl – Leibniz-Zentr...

2021
[52]

J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys.38, 447 (1966)

1966
[53]

Yu and C

S. Yu and C. H. Oh, State-independent proof of kochen- specker theorem with 13 rays, Phys. Rev. Lett.108, 030402 (2012)

2012
[54]

Zurel, C

M. Zurel, C. Okay, and R. Raussendorf, Hidden variable model for universal quantum computation with magic states on qubits, Phys. Rev. Lett.125, 260404 (2020)

2020
[55]

Raussendorf, J

R. Raussendorf, J. Bermejo-Vega, E. Tyhurst, C. Okay, and M. Zurel, Phase-space-simulation method for quan- tum computation with magic states on qubits, Physical Review A101, 10.1103/physreva.101.012350 (2020)

2020 doi
[56]

S. Ipek, A. T. Yucel, F. Shahi, C. Ozdemir, and C. Okay, Phase space tableau simulation for quantum computation (2025), arXiv:2506.04033

2025 arXiv
[57]

Yet, maximal sets of mutually anti-commuting Pauli operators have cardinality 2n+1 (see Ref

Note that a 1-ovoid translates into a set of mutually anti- commuting Hermitian Pauli operators, whose cardinality is that of a partition of ̃Pn into disjoint maximal Abelian subgroups, that is, 2n+1. Yet, maximal sets of mutually anti-commuting Pauli operators have cardinalit...
[58]

A. M. Gleason, Measures on the closed subspaces of a hilbert space, inThe Logico-Algebraic Approach to Quan- tum Mechanics: Volume I: Historical Evolution, edited by C. A. Hooker (Springer, Dordrecht, 1957) pp. 123– 133

1957
[59]

From a physical point of view, Π Θ may be interpreted as the projection onto a Kramers pair
[60]

We choose the normalisation of the{e v}v∈V to match that of the Weyl representation in Eq. (4)
[61]

Dehaene and B

J. Dehaene and B. De Moor, Clifford group, stabilizer states, and linear and quadratic operations over GF(2), Phys. Rev. A68, 042318 (2003)

2003
[62]

Aaronson and D

S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A70, 052328 (2004)

2004
[63]

Note that this is different to a valuation, which de- mands linearity with respect to the projective represen- tation ofVin terms of Pauli operators (see Lm

There are many such linear functionals, indeed,∣V ∗∣= 22n. Note that this is different to a valuation, which de- mands linearity with respect to the projective represen- tation ofVin terms of Pauli operators (see Lm. 10)
[64]

Cardinali, L

I. Cardinali, L. Giuzzi, and A. Pasini, The generating rank of a polar grassmannian, Advances in Geometry21, 515 (2021)

2021
[65]

Tits,Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol

J. Tits,Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386 (Springer, 1974) chapter 8; the exceptional rank-2 cases are 8.6(II)(a),(b)

1974
[66]

Gross, Hudson’s theorem for finite-dimensional quan- tum systems, J

D. Gross, Hudson’s theorem for finite-dimensional quan- tum systems, J. Math. Phys.47, 122107 (2006)

2006
[67]

Filmus and F

Y. Filmus and F. Ihringer, Boolean degree 1 functions on some classical association schemes, J. Combin. Theory Ser. A162, 241 (2019). Appendix A: Context posets, Kochen-Specker contextuality and proof of Thm. 1 We review the formal setting underlying Sec. II, Kochen-Specker no...

2019

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [30]

Planat and M

M. Planat and M. Saniga, On the Pauli graphs ofN- qudits, Quantum Inf. Comput.8, 127 (2008)

work page 2008

[2] [31]

Havl´ ıˇ cek, B

H. Havl´ ıˇ cek, B. Odehnal, and M. Saniga, Factor-group- generated polar spaces and (multi-)qudits, SIGMA5, 096 (2009)

work page 2009

[3] [32]

Thas, The geometry of generalized pauli operators of n-qudit hilbert space, and an application to mubs, Euro- physics Letters86, 60005 (2009)

K. Thas, The geometry of generalized pauli operators of n-qudit hilbert space, and an application to mubs, Euro- physics Letters86, 60005 (2009)

work page 2009

[4] [33]

L´ evay, M

P. L´ evay, M. Saniga, and P. Vrana, Three-qubit opera- tors, the split Cayley hexagon of order two, and black holes, Phys. Rev. D78, 124022 (2008)

work page 2008

[5] [34]

Holweck, M

F. Holweck, M. Saniga, and P. L´ evay, A notable relation betweenN-qubit and 2 N−1-qubit Pauli groups via binary LGr(N,2N), SIGMA10, 041 (2014)

work page 2014

[6] [35]

An algebraically closed family of informational n-qubit purity invariants

M. Frembs, G. Natale, C. S. P. Wever, and P. A. Hoehn, An algebraically closed family of informational n-qubit purity invariants (2026), arXiv:2607.24987 [quant-ph]

work page Pith review arXiv 2026

[7] [36]

Maximal complementarity in the n-qubit Pauli group

M. Frembs, G. Natale, C. S. P. Wever, and P. A. Hoehn, Maximal complementarity in the n-qubit pauli group (2026), arXiv:2607.24988 [quant-ph]

work page Pith review arXiv 2026

[8] [37]

Arf, Untersuchungen ¨ uber quadratische Formen in K¨ orpern der Charakteristik 2., Journal f¨ ur die reine und angewandte Mathematik1941, 148 (1941)

C. Arf, Untersuchungen ¨ uber quadratische Formen in K¨ orpern der Charakteristik 2., Journal f¨ ur die reine und angewandte Mathematik1941, 148 (1941)

work page 1941