Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations

Daniel Centeno; Elie Wolfe; Luca Mrini; Maria Ciudad Ala\~n\'on; Ralph Jason Costales; Soham Bhattacharyya; Yasamin Panahi

REVIEW 4 major objections 4 minor 1 cited by

Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In N-partite Bell scenarios where the first N-1 parties have two settings and two outcomes, all convexly extremal quantum correlations are realizable with qubits for those parties and dimension $2^{N-1}$ for the last.

desk verdict Useful dimension bounds for a class of Bell scenarios, but the key bipartite lemma has a purification gap that propagates; worth a serious referee with revision. read the letter →

arxiv 2505.20519 v1 pith:CG42N332 submitted 2025-05-26 quant-ph

Yasamin Panahi , Maria Ciudad Alañón , Daniel Centeno , Ralph Jason Costales , Luca Mrini , Soham Bhattacharyya , Elie Wolfe This is my paper · ORCID

classification quant-ph MSC 81P4052A35 PACS 03.65.Ud03.67.-a

keywords BellscenariosquantumcorrelationsHilbertspacedimensionqubitboundsSchmidtdecompositionCarathéodory'stheoremwithcommunicationconvexextremalpoints

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 establishes finite upper bounds on the Hilbert space dimensions needed to reproduce quantum correlations in Bell experiments. Its central result is that in any N-partite Bell scenario where the first N-1 parties each have two measurement settings and two outcomes, every convexly extremal quantum correlation can be realized with qubits for those N-1 parties and a local Hilbert space of dimension $2^{N-1}$ for the remaining party. The paper then uses Carathéodory's theorem to convert this extremal-point bound into a bound valid for every quantum correlation, multiplying each local dimension by the affine dimension of the quantum set. The same bounds are shown to transfer to Bell scenarios with communication, because such scenarios are projections of ordinary Bell scenarios. A sympathetic reader should care because these are among the few multi-setting scenarios where finite-dimensional sufficiency is known, and the bounds make convex optimization over quantum correlations and the membership problem tractable.

What carries the argument

The machinery has three parts. First, a known proposition compresses any party with two settings and two outcomes to a qubit, regardless of the rest of the scenario. Second, a Schmidt-decomposition argument compares dimensions across parties: in the bipartite case the second party's needed dimension never exceeds the first's, and the multipartite generalization bounds the last party's dimension by the product of the preceding parties' dimensions. Third, convex geometry converts these extremal-point bounds into all-correlation bounds: because the convex hull of the finite-dimensional quantum set is the full quantum set, Carathéodory's theorem multiplies each local dimension by the affine dimension of the quantum set. For Bell+ scenarios, the maximal-interruption projection carries these bounds over unchanged.

What would settle it

In the bipartite scenario where Alice has two settings and two outcomes and Bob has, say, three outcomes per setting, run a standard hierarchy of semidefinite relaxations to test whether every convexly extremal quantum correlation is attainable by a two-qubit state; a single extremal correlation requiring Bob's local dimension greater than 2 would disprove the central bound for N=2.

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Extended reading notes

Core claim

The paper claims two theorems. Theorem 6 states that in an N-partite Bell scenario where the first N-1 parties have two settings and two outcomes, the maximum of any convex function over all quantum correlations equals the maximum over correlations generated by a pure state whose first N-1 local Hilbert space dimensions are 2 and whose final local Hilbert space dimension is $2^{N-1}$. Theorem 7 extends this to every quantum correlation: a correlation is quantum if and only if it can be realized with the first N-1 local dimensions equal to $2 \cdot \mathrm{AffineDimension}(Q_\infty)$ and the last equal to $2^{N-1} \cdot \mathrm{AffineDimension}(Q_\infty)$, where the affine dimension of the quantum set is given by a closed formula in terms of the settings and outcomes. The same dimension bounds are shown to carry over to Bell+ scenarios, where measurement settings may depend on other parties' outputs.

Load-bearing premise

The argument's load-bearing premise is that a mixed bipartite quantum correlation can be analyzed as a pure state on the same two local Hilbert spaces without introducing a third purification register; if that compression fails for mixed states, the Schmidt-projection proof of the dimension comparison does not go through.

Editorial extensions

If this is right

In any bipartite Bell scenario where one party has two settings and two outcomes, all convexly extremal quantum correlations can be produced by two-qubit states, no matter how many settings or outcomes the other party has.
Maximizing any convex function over quantum correlations in the covered scenarios, such as computing maximal Bell violations, reduces to an optimization over a finite-dimensional Hilbert space of known dimension.
Every quantum correlation in the covered scenarios, not just extremal ones, has an explicit finite-dimensional realization; the dimension formula gives a route to a finite algorithm for the quantum-membership problem.
The same finite-dimension bounds apply to nonstandard Bell scenarios with communication, including instrumental scenarios, since projection cannot increase the required dimension.

Reading between the lines

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

The theorems leave open whether $2^{N-1}$ is tight for large N; for N=2 the bound matches known qubit results, but for larger N the true minimum for specific functionals may be smaller.
The affine-dimension multiplier grows quickly with the number of settings and outcomes, so the finite bounds for non-extremal correlations may be too large for direct numerics; a likely fruitful direction is finding smaller multipliers by exploiting additional structure of the finite-dimensional quantum set.
The proof technique suggests a general recipe: whenever a scenario's extremal quantum correlations are finitely realizable, physically mixing those realizations automatically yields finite realizations for all correlations, with the Carathéodory number of the finite set as the overhead.
Because the Bell+ step uses only the projection property, the results should transfer to any causal scenario obtained by interrupting communication links between parties, not just the examples discussed.

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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

The paper gives explicit Hilbert-space dimension upper bounds for a useful family of Bell scenarios: where the first N−1 parties have two dichotomic settings, the last party can be bounded by 2^(N−1) for extremal correlations, and a Carathéodory-based bound handles all (including non-extremal) correlations. The transfer to Bell+ communication scenarios via maximal interruption is clean and practical. The writing is clear and the prior literature is handled honestly, with Lemma 4 explicitly credited to Sikora et al. [20]. That is real credit where due.

The main new content is Theorem 6 (multipartite 2^(N−1)) and Theorem 7 (Carathéodory extension). These are plausible and likely correct, but they inherit a real gap in the proof of Lemma 4. The proof says "without loss of generality" work with a pure state, then applies Schmidt projectors inside H_A⊗H_B. For a mixed state, purification requires a reference system R, and the Schmidt decomposition between A and BR does not give vectors in H_B alone. So the projector Π as defined cannot leave the purified state invariant unless R is trivial. This is a genuine reasoning gap, not a stylistic quibble. It propagates to Lemma 5 and therefore to Theorems 6 and 7. The result may still be true—Ref [20] likely proves the needed compression, and the paper even notes the lemma was previously observed there—but the derivation as written does not.

A second soft spot is more minor: Theorem 7 uses AffineDimension(Q_d) = AffineDimension(Q_∞), citing Ref [21, Theorem 1]. That reference gives the affine dimension for the full quantum set, not for the restricted set Q_d. The equality needs justification for the specific bounded dimensions, since for small d the affine span can shrink. This is likely fixable, but it is asserted rather than shown.

The bipartite qubit result (both parties qubits when one is 2×2) is already contained in Masanes plus Sikora; the paper does not overclaim this, but the novelty rests on the multipartite and Carathéodory parts.

Who this is for: people working on dimension bounds, convex optimization over the quantum set, and Bell+ scenarios. It deserves a serious referee. The gaps are repairable and the central claims are probably true, so I would accept for peer review with a clear request to fix Lemma 4's purification step and to prove or properly cite the affine-dimension equality.

Referee Report

4 major / 4 minor

Summary. The paper claims finite upper bounds on the local Hilbert space dimensions sufficient to reproduce quantum correlations in certain Bell scenarios. For scenarios in which the first N-1 parties have two settings and two outcomes, it asserts (Theorem 6) that the maximum of any convex function over all quantum correlations is attained on a pure state with local dimensions 2 for the first N-1 parties and 2^(N-1) for the last party. For non-extremal correlations, Theorem 7 multiplies these dimensions by the affine dimension of the quantum set, using a Carathéodory/Fenchel argument. The authors also translate these bounds to Bell scenarios with communication through a maximal-interruption projection, yielding Corollaries 9 and 10.

Significance. The question of sufficient Hilbert space dimension for quantum correlations is important for numerical optimization, dimension witnesses, and the foundational classification of quantum resources. The paper addresses a meaningful problem and builds on solid ingredients: Masanes' qubit-reduction theorem, Schmidt decomposition, and the maximal-interruption construction for causal scenarios. If the proof gaps were repaired, the results would provide explicit and reasonably general finite-dimensional sufficiency bounds, and the Bell+ extension is a useful conceptual transfer. However, the current manuscript has a purification gap in Lemmas 4-5 and a convexity inconsistency in Section II.D, so the central claims are not established by the arguments as written.

major comments (4)

[II.B, Lemma 4, Eqs. (3)-(4)] The proof's first step, 'Without loss of generality we will consider pure states,' is not valid for mixed states. A purification of a mixed state on H_A⊗H_B is a pure state |ψ⟩ in H_A⊗H_B⊗H_R, not in H_A⊗H_B. The Schmidt decomposition of Lemma 3 and the projector Π of Eq. (3) are therefore not directly applicable to |ψ⟩, and the trace identities in Eq. (4) do not follow. The lemma may be true and is attributed to Ref. [20], but the proof as written leaves this step unjustified for non-pure correlations.
[II.C, Lemma 5] The proof of Lemma 5 invokes 'the same reasoning as in the bipartite case' to justify working with pure states. Since Lemma 4's purification step is unsupported for mixed states, Lemma 5's bound d_N ≤ d_1×...×d_{N-1} is likewise unsupported for general non-extremal correlations. This bound is load-bearing for Theorems 6 and 7, so the gap must be repaired or the proof must explicitly rely on a prior result.
[II.D, Eqs. (5)-(7)] The set Q_⃗d is defined as the set of all correlations realizable with local Hilbert space dimensions at most ⃗d. This set is convex, since a convex combination of density operators with local dimensions at most ⃗d is again a density operator with the same local dimensions. Therefore ConvexHull(Q_⃗d)=Q_⃗d and CathNum(Q_⃗d)=1, and the Carathéodory/Fenchel multiplication leading to Eq. (7) and Theorem 7 is not justified. If the authors intended Q_⃗d to denote a smaller nonconvex set, such as correlations generated by pure states, that definition must be stated explicitly, and Eq. (5b) would then require the additional nontrivial claim that Q∞ is the convex hull of its extreme points, which is not established and is delicate in light of the known non-closure of Q∞ cited in the Introduction.
[II.C, Theorem 6] The theorem compares a maximum over Q∞ with a maximum over a finite-dimensional set. Because Q∞ is not assumed closed, and is in general not closed, the text must argue that the maximum is attained and that it is attained at a convexly extremal point. No such argument is provided; if the maximizer lies in the closure of Q∞ but outside Q∞, the stated equality can fail. This is not merely a technicality, since the Introduction itself cites examples requiring infinite Hilbert space dimension for extremal correlations.

minor comments (4)

[II.B] The word 'Schimdt' appears in the sentence 'Combining the use of the Schimdt decomposition'; it should read 'Schmidt'.
[II.D] The notation CathNum(S) is nonstandard and should be defined in a way that makes clear whether it refers to the Carathéodory number of the set S itself or of its convex hull; the current usage is ambiguous.
[III, Proposition 8] The proof of Proposition 8 is essentially a citation plus a geometric statement; a short explicit argument that the maximal-interruption projection preserves the required dimension bound would improve readability and self-containment.
[II.D, Theorem 7] The phrase 'p admits some quantum realization if and only if p admits a quantum realization where...' is logically redundant; the nontrivial content is an upper bound on sufficient dimensions, and the 'only if' direction holds by definition of Q∞.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the dimension bounds are derived from external theorems (Masanes, Schmidt decomposition, Carathéodory/Fenchel) and prior independent work; no step reduces to its own input.

full rationale

The paper's derivation chain is not circular. The central results (Lemma 4, Theorem 6, Theorem 7) are obtained by combining external propositions: Masanes' Proposition 2 (Ref [8]), the Schmidt decomposition (Lemma 3), and the previously noted dimension-compression result of Ref [20]. The bound d_B <= d_A in Lemma 4 is a direct consequence of the Schmidt decomposition for pure states, not an assumption equivalent to the conclusion; for mixed states the paper invokes purification, and although the argument as written omits the reference system and is therefore a genuine proof gap in Section II.B (Lemma 4 proof, Eqs. (3)-(4)), this is a correctness defect rather than a circular reduction. Theorem 6 follows by applying Masanes' theorem and Lemma 5 to convexly extremal points, and Theorem 7 follows from Carathéodory/Fenchel convex-hull arguments; none of these steps defines the target quantity in terms of itself. The Bell+ section relies on the maximal-interruption relation from Refs [23,24], one of which shares a co-author (E. Wolfe), but that cited result is an independent published theorem and is not used to establish the standard Bell scenario bounds; the self-citation is not load-bearing for the main claim. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in via citation. The score is therefore 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard results from convex geometry and prior quantum information theorems, plus one unproven connectedness assertion used for the Fenchel refinement.

assumptions (7)

standard math Schmidt decomposition for bipartite pure states (Lemma 3)
Used in Lemma 4 and Lemma 5 to restrict dimensions to the minimum of the two subsystem dimensions.
domain assumption Masanes' Proposition 2: a party with two dichotomic measurements can be reduced to a qubit for convexly extremal correlations
Quoted from Refs [8,9]; this is the key prior result for reducing the first N-1 parties.
domain assumption Sikora et al. Lemma 4: bipartite Schmidt bound for mixed states
Cited from Ref [20]; the paper's own proof has a purification gap, so the result is taken as a black box.
standard math Carathéodory and Fenchel convex geometry bounds
Used to bound the number of extremal points needed to express any correlation as a convex combination.
domain assumption Q_d is pathwise connected
Asserted without proof in Section II.D; needed for the Fenchel refinement to reduce the Carathéodory number from A+1 to A.
domain assumption Affine dimension of Q_∞ is given by Eq. (8) from Pironio [21]
Used to compute the Carathéodory number in Theorem 7. The paper also asserts that the affine dimension of Q_d equals that of Q_∞ for the specific d.
domain assumption Maximal interruption construction (Prop 8)
Standard result from Refs [23,24] that Bell+ scenarios are projections of standard Bell scenarios; used to transfer bounds.

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Pith. "Pith review of Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations." pith.science (2026). https://pith.science/paper/CG42N332

@misc{pith2026250520519,
  author       = {Pith},
  title        = {Pith review of: Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CG42N332}},
  note         = {Machine review of arXiv:2505.20519}
}

read the original abstract

We introduce novel upper bounds on the Hilbert space dimensions required to realize quantum correlations in Bell scenarios. We start by considering bipartite cases wherein one of the two parties has two settings and two outcomes. Regardless of the number of measurements and outcomes of the other party, the Hilbert space dimension of the first party can be limited to two while still achieving all convexly extremal quantum correlations. We then leverage Schmidt decomposition to show that the remaining party can losslessly also be restricted to a qubit Hilbert space. We then extend this idea to multipartite scenarios. We also adapt our results to provide upper bounds of local Hilbert space dimensions to achieve any quantum correlation, including convexly non-extremal correlations, by utilizing Caratheodory's theorem. Finally, we generalize our results to nonstandard Bell scenarios with communication. Taken together, our results fill in several previously unresolved aspects of the problem of determining sufficient Hilbert space dimensionality, expanding the collection of scenarios for which finite-dimensional quantum systems are known to be sufficient to reproduce any quantum correlation.

Figures

Figures reproduced from arXiv: 2505.20519 by the authors.

**Figure 2.** FIG. 2. Circuit interpretation for bipartite Bell scenario. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Instrumental scenario [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Maximal interruption for the instrumental scenario [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [20]

MinimumDimensionofaHilbertSpaceNeededtoGener- ateaQuantumCorrelation,

Jamie Sikora, Antonios Varvitsiotis, and Zhaohui Wei, “MinimumDimensionofaHilbertSpaceNeededtoGener- ateaQuantumCorrelation,”Phys. Rev. Lett.117,060401 (2016)

work page 2016

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