arxiv: 2605.04931 · v2 · submitted 2026-05-06 · 🪐 quant-ph

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Quantum Realizability of Probabilistic Theories Stable under Teleportation

Miguel A. A. Lisboa

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Pith reviewed 2026-05-11 01:17 UTC · model grok-4.3

classification 🪐 quant-ph

keywords general probabilistic theoriesCHSH inequalityquantum realizabilityteleportationentanglement swappingrepresentation theoryKlein four-groupdihedral group

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The pith

Exactly two of seven GPT families stable under teleportation admit quantum realizations.

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

The paper resolves an open question left by the prior classification of general probabilistic theories whose CHSH value remains unchanged under arbitrary teleportation and entanglement swapping. That classification produced seven families indexed by characters of the Klein four-group, the cyclic group of order four, and the dihedral group of order eight. By applying elementary representation theory to the indexing characters, the work proves that only the two families labeled χ^{K_4}_{1234} and χ^{D_4}_{125} can be realized inside ordinary quantum mechanics, while the remaining five families cannot.

Core claim

Using elementary representation theory, we prove that exactly two families are quantum-realizable, namely χ^{K_4}_{1234} and χ^{D_4}_{125}, while the remaining five admit no quantum realization.

What carries the argument

The representation theory of the groups K_4, Z_4 and D_4 applied to the characters that index the seven families; it determines which families can be embedded into the quantum formalism.

If this is right

Only two of the seven teleportation-stable GPTs can appear inside quantum theory.
The other five families describe stable non-quantum behaviors that cannot be reproduced by any quantum system.
Quantum mechanics satisfies the stability condition only for the two specific indexed families.
Any experimental test of CHSH stability after many rounds of teleportation can at most realize one of the two quantum families.

Where Pith is reading between the lines

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

The result isolates a precise algebraic signature that separates quantum from non-quantum stable theories.
It supplies a concrete criterion for deciding whether a new candidate GPT family is quantum or not without constructing an explicit Hilbert-space model.
The same representation-theoretic test could be applied to other stability conditions, such as those involving more parties or different Bell inequalities.

Load-bearing premise

The earlier exhaustive list of exactly seven families is taken as complete, and quantum realizability is fully decided by the representation theory of the indexing groups.

What would settle it

An explicit quantum model realizing any one of the five families labeled non-realizable, or a concrete counter-example showing that representation theory fails to capture quantum embeddability for one of the two families labeled realizable.

read the original abstract

The classification of general probabilistic theories (GPTs) whose CHSH value is stable under arbitrary rounds of teleportation and entanglement swapping was obtained in Dmello and Gross work and consists of seven families, indexed by characters of the Klein four-group $K_4$, the cyclic group $\mathbb{Z}_4$, and the dihedral group $D_4$. The question of which of these families admits a realization within standard quantum mechanics was left open. In this work we resolve this question completely. Using elementary representation theory, we prove that exactly two families are quantum-realizable, namely $\chi^{K_4}_{1234}$ and $\chi^{D_4}_{125}$, while the remaining five admit no quantum realization.

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.

Desk Editor's Note private letter to a colleague

This paper claims to settle the open question on quantum realizability for the seven Dmello-Gross teleportation-stable GPT families by showing only two work.

read the letter

The paper resolves the open question left by Dmello and Gross on which of their seven families of teleportation-stable GPTs can be realized in quantum mechanics. It concludes that only two families work, specifically those indexed by χ^{K_4}_{1234} and χ^{D_4}_{125}, while the other five do not. What is new is the application of elementary representation theory to decide quantum realizability for each family. The paper takes the prior classification as given and checks which characters admit unitary representations that satisfy the stability conditions under teleportation and entanglement swapping. This gives a clean yes-or-no for each of the seven. The work is straightforward in its approach and fills a specific gap in the literature on generalized probabilistic theories. If the derivations hold, it narrows down the possible quantum models in this setting. The main soft spot is the dependence on the Dmello-Gross enumeration being exhaustive. The no-go results apply only within those seven families. If there are other GPTs stable under teleportation that fall outside this list, or if a quantum realization for one of the five families exists via a different representation not captured by the given characters, then the conclusion would need adjustment. The abstract presents the proof as complete, but the mapping from group characters to the absence of quantum embeddings needs careful checking to ensure no cases were missed. This paper is for researchers in quantum foundations who work with GPTs and classification results. Someone following the Dmello-Gross paper would find it directly relevant. It deserves a serious referee because it provides a mathematical resolution to an open problem, even though the argument builds on prior work. I would recommend sending it for peer review, with referees asked to verify the representation theory steps in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript resolves an open question left by Dmello and Gross by determining which of the seven GPT families (indexed by characters of K_4, Z_4, and D_4) whose CHSH value remains invariant under arbitrary teleportation and entanglement swapping admit realizations in standard quantum mechanics. Using elementary representation theory, the authors prove that precisely two families—those indexed by χ^{K_4}_{1234} and χ^{D_4}_{125}—are quantum-realizable, while the remaining five families admit no quantum realization.

Significance. If the central proof holds, the result cleanly settles the quantum embeddability question for this class of stable GPTs, providing a concrete demarcation between quantum and non-quantum theories within a mathematically well-defined family. The approach leverages standard tools of finite-group representation theory to obtain an exhaustive no-go result for five families, which is a strength when the derivations are fully explicit and self-contained.

major comments (2)

[§3] §3 (Quantum realizability via characters): the definition of quantum realizability is tied to the existence of a unitary representation of the indexing group that satisfies the stability equations derived from teleportation invariance. It is not shown that every possible quantum model of such a GPT must arise from a representation equivalent to one of the given characters; an alternative embedding (e.g., via a different Hilbert-space construction not reducible to the character) could in principle realize one of the five excluded families. This assumption is load-bearing for the claim that exactly two families are realizable.
[§2] §2 (Adoption of Dmello-Gross classification): the manuscript takes the prior enumeration of exactly seven families as exhaustive for GPTs with CHSH stability under teleportation and swapping. While the focus is on the quantum question, the central count of “exactly two” quantum-realizable families inherits any incompleteness in the prior list; a short self-contained argument or explicit citation showing no additional families exist outside the seven would be required to make the no-go statements for the five families fully rigorous.

minor comments (2)

Notation for the characters χ^{K_4}_{1234} and χ^{D_4}_{125} is introduced without an explicit table or equation listing the values on group elements; adding such a reference table would improve readability when the representation-theoretic conditions are applied later.
The abstract states that the proof is “complete” via elementary representation theory, yet the main text contains only sketches for the non-realizability arguments of the five families. Expanding at least one of these sketches into a fully written derivation (with explicit matrix or character orthogonality checks) would make the manuscript self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (Quantum realizability via characters): the definition of quantum realizability is tied to the existence of a unitary representation of the indexing group that satisfies the stability equations derived from teleportation invariance. It is not shown that every possible quantum model of such a GPT must arise from a representation equivalent to one of the given characters; an alternative embedding (e.g., via a different Hilbert-space construction not reducible to the character) could in principle realize one of the five excluded families. This assumption is load-bearing for the claim that exactly two families are realizable.

Authors: We agree that the manuscript should clarify why quantum realizability is fully captured by the character-based unitary representations. In the context of these GPTs, the teleportation invariance and stability conditions impose specific algebraic relations on the observables and states that any quantum realization must satisfy. These relations correspond exactly to the representation theory of the underlying group (K_4, Z_4, or D_4). Any alternative Hilbert space construction would still need to realize the same correlation functions and stability under teleportation, which forces the existence of a unitary representation equivalent to one of the characters. We will add a detailed explanation in §3, including a proof sketch showing that the stability equations imply the representation must be of this form, thereby ruling out alternative embeddings for the excluded families. revision: yes
Referee: [§2] §2 (Adoption of Dmello-Gross classification): the manuscript takes the prior enumeration of exactly seven families as exhaustive for GPTs with CHSH stability under teleportation and swapping. While the focus is on the quantum question, the central count of “exactly two” quantum-realizable families inherits any incompleteness in the prior list; a short self-contained argument or explicit citation showing no additional families exist outside the seven would be required to make the no-go statements for the five families fully rigorous.

Authors: The classification into exactly seven families is taken from the prior work of Dmello and Gross, which provides a complete enumeration based on the possible characters satisfying the stability conditions. To address this, we will include an explicit reference to the relevant theorem in Dmello and Gross (with section or theorem number) that proves these are all possible families. Additionally, we will add a brief self-contained summary in §2 outlining why no other families exist, drawing from the group-theoretic constraints on the CHSH value under teleportation. This will make the no-go results for the five families fully rigorous within our manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: standard representation theory applied to external prior classification

full rationale

The paper takes the exhaustive list of seven GPT families from the independent Dmello-Gross classification as an external premise and applies elementary group representation theory to decide which admit quantum realizations. No equations or steps reduce the conclusion to the paper's own inputs by construction, no parameters are fitted and relabeled as predictions, and the cited classification is not a self-citation. The derivation chain is a direct mathematical check against an assumed external list and therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof depends on the completeness of the seven-family classification from prior work and on standard mathematical facts from finite group representation theory; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption The classification of GPTs with CHSH value stable under arbitrary teleportation and entanglement swapping into exactly seven families indexed by characters of K_4, Z_4, and D_4 is complete.
Invoked as the starting point for applying the new representation-theoretic test.
standard math Elementary representation theory of finite groups determines whether a GPT indexed by a group character admits a quantum realization.
The core tool used to prove which families are realizable.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

[1]

Quantum Realizability of Probabilistic Theories Stable under Teleportation

The seven characters are χK4 1234, χ Z4 1234, χ D4 125, χ D4 135, χ D4 145, χ D4 12345, χ D4 123452.(1) The first two are the regular characters ofK 4 andZ 4 re- spectively; the remaining five are characters ofD 4 spec- ified by the multiplicities of its five irreducible represen- tations. ∗ miguel.lisboa@venturus.org.br The question, posed in Ref. [1] an...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

It is realized by takingχ U =χ E1 (or equivalently χE3), the two-dimensional irreducible characters ofD 8 listed in Sec. II. Proof.By Corollary 1 applied to the Schur coverD 8, the representationU:D 8 →U(H) must be irreducible. Among the irreducible representations ofD 8, those in whichζ 4 acts as−1are exactlyχ E1 andχ E3 (χE2 acts trivially on⟨ζ 4⟩, as n...

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[3]

The familyχ K4 1234 requiresm 5 = 0, but herem 5 = 1

The familiesχ D4 135 andχ D4 145 are not produced (they would require different multiplicitiesm 3 orm 4). The familyχ K4 1234 requiresm 5 = 0, but herem 5 = 1. The familiesχ D4 12345 andχ D4 123452 are excluded by Corollary 2. The familyχ Z4 1234 is addressed in Proposition 3. VII. THE REMAINING CASES It remains to rule outχ Z4 1234,χ D4 135, andχ D4 145....

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For an alternative direct argument that does not use Corollary 1, we show that the multiplicity ofχ5 inχ conj is always even

Sinceχ D4 135 and χD4 145 are different families, they are not realized in the trivial class. For an alternative direct argument that does not use Corollary 1, we show that the multiplicity ofχ5 inχ conj is always even. Writeχ U =aχ 1+bχ2+cχ3+dχ4+eχ5 with a, b, c, d, e∈N 0. By Lemma 1 and the remark following it,χ conj =χ U ⊗ χU. Since all irreducible cha...

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We now exhibit explicit quantum systems realizing each. 7 Both constructions useH A =H C =C 2 (single-qubit spaces) and the maximally entangled Bell state |Φ+⟩= 1√ 2 |00⟩+|11⟩ ∈C 2 ⊗C 2.(51) We also use the four Pauli operators onC 2: σ0 =1= 1 0 0 1 , σ 1 =σ x = 0 1 1 0 , σ2 =σ y = 0−i i0 , σ 3 =σ z = 1 0 0−1 .(52) The Pauli operators satisfyσ 2 i =1,σ † ...

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The four operators{σ 0, σ1, σ2, σ3}form a com- plete set of coset representatives forP 1/Z(P1) ∼= K4

To make the group struc- ture explicit: the (single-qubit) Pauli group is P1 :=⟨i1, σ x, σy, σz⟩={i kσj :k∈ {0,1,2,3}, j∈ {0,1,2,3} }, (64) of order 16, with centerZ(P 1) =⟨i1⟩={±1,±i1}of order 4. The four operators{σ 0, σ1, σ2, σ3}form a com- plete set of coset representatives forP 1/Z(P1) ∼= K4. Conjugation by an element ofP 1 depends only on its coset ...

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Equivalently, this conjugation represen- tation lifts to the linear representationχ 5 ofD 4 via the descentD 4 ↠D 4/⟨r2⟩ ∼= K4, which is the form used in Proposition 1. 8 C. The familyχ D4 125: a POVM construction The phase operator is S:= 1 0 0i ,(65) with powersS 2 =σ z,S 3 =S −1 = diag(1,−i), andS 4 =

work page
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Together withσ x,Sgenerates a finite subgroup of U(2). Direct computation gives σxSσx = i0 0 1 =iS −1 =iS 3.(66) Thus the relationsrs −1 =r −1 ofD 4 is satisfied only up to the phasei: (σx)S(σ x)S=iS −1 ·S=i·1̸=1.(67) This non-trivial cocycle shows that the assignmentr7→ S,s7→σ x defines a projective representation ofD 4 onC 2 in the non-trivial cohomolog...

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Charlie applies the inverse of the operator appearing on his side: V(b,k) = (σ† k)† =σ k, V (a,k) = ((Sσk)†)† =Sσ k, (79) restoring|Φ +⟩A1C1 exactly. The eight corrections {σk}3 k=0 ∪ {Sσ k}3 k=0 ⊂U(2) (80) generate a finite subgroup of U(2) that, modulo the cen- tral phase⟨i1⟩, is isomorphic toD 4, withr7→[S] and s7→[σ x] in the projective quotient. Sinc...

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