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REVIEW 4 major objections 4 minor 37 references

Characterising the set of deterministic quantum correlations in prepare-and-measure scenarios

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In prepare-and-measure scenarios, a correlation is classical for a chosen pair of inputs exactly when an adversary with access to the shared randomness can predict the measurement outcome with certainty.

desk verdict A sound reframing of classicality as predictability, but the bounded-overlap SDP has a load-bearing linearization error that needs fixing. read the letter →

arxiv 2608.09667 v1 pith:RBZTQVPE submitted 2026-08-10 quant-ph

Nicola D'Alessandro , Oliver Karlsson , Carles Roch i Carceller This is my paper · ORCID
classification quant-ph
keywords prepare-and-measurescenariosdeterministiccorrelationsclassicalitycertificationadversarialguessingprobabilitysemidefiniteprogrammingcommunicationrestrictionsstatediscrimination
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 proposes a unified way to certify non-classical correlations in prepare-and-measure experiments: instead of asking whether the data fits a classical message model, ask whether an adversary holding the classical side-information could predict Bob's outcome perfectly. The paper establishes that, for any fixed choice of Alice's and Bob's settings, these two questions are the same—perfect predictability is exactly equivalent to the existence of a deterministic explanation. It then turns this equivalence into a computational tool, formulating the search for a deterministic explanation as a semidefinite-programming feasibility problem under three different communication restrictions: a fixed ensemble of prepared states, lower bounds on pairwise state overlaps, and bounds on observable expectation values. A sympathetic reader would care because the approach can inherit a large body of semidefinite-programming tools from randomness certification, and it works without assuming a fixed Hilbert-space dimension.

What carries the argument

The load-bearing object is the adversarial guessing probability $p_g^{(x,y)}$ and the equivalence in Observation 1. The certification machinery consists of three SDP relaxations: a block-moment matrix for the fixed-ensemble restriction, a Gram-matrix hierarchy for bounded overlaps, and a tracial moment-matrix hierarchy for restricted observables; in each, the observed statistics, the restriction, and the perfect-predictability condition are encoded as linear constraints on a positive semidefinite matrix. The paper also uses the non-orthogonality graph of the prepared states—vertices are preparations and edges connect non-orthogonal states—to prove that, under a fixed pure-state ensemble, deterministic models force identical outcomes across the whole connected graph, yielding the linear-witness bound of Result 1.

What would settle it

For a fixed $n\ge 5$, compute numerically the largest state-discrimination success probability compatible with the perfect-predictability constraint and pairwise overlaps at least $\beta$, and compare it with the right-hand side of Eq. (13); any feasible configuration exceeding the bound would disprove it, and evaluating the per-realization optimal success probability as a function of the overlap at intermediate $c$ values would test the concavity step.

Watch

Extended reading notes

Core claim

The central claim is Observation 1: correlations produced in a prepare-and-measure scenario for a concrete choice of input settings $(x,y)$ admit a classical model if and only if the optimal guessing probability $p_g^{(x,y)}=1$, where $p_g^{(x,y)}$ is the maximum, over all quantum realizations compatible with the communication restriction, of $\sum_\lambda q(\lambda)\max_b p_\lambda(b|x,y)$. The forward direction is immediate from the definition of a deterministic model; the converse holds because a guessing probability of one forces every hidden-variable branch to have a deterministic outcome distribution. Consequently, showing that no admissible quantum realization allows an adversary to predict perfectly is logically equivalent to certifying that the correlations are non-classical. The paper uses this equivalence to certify non-classicality by proving the infeasibility of an SDP that imposes the linear constraint of perfect predictability, and it illustrates the method with three restrictions, including a closed-form bound for state discrimination with bounded overlaps.

Load-bearing premise

The analytical bounded-overlap bound assumes, without proof, that the best per-realization discrimination probability is concave in the overlap and that an optimal solution can be chosen with rank-one projective measurements invariant under cyclic relabeling of the states; if either assumption fails, Eq. (13) and its numerical illustration do not follow.

Editorial extensions

If this is right

  • Any communication restriction that can be written as linear constraints on the moments fits the same certification template, so the three SDP programs are instances of a general method rather than isolated tests.
  • For a fixed pure-state ensemble whose non-orthogonality graph is connected, every linear witness $W=\sum_{b,x,y} c_{bxy}p(b|x,y)$ satisfies the deterministic bound $W\le \sum_y \max_b\sum_x c_{bxy}$.
  • For the mutually-unbiased-bases witness $W_{n,d}$, the quantum maximum is $nd$ and the deterministic bound is $n$, so any measured value above $n$ certifies non-classicality with detector efficiency above $1/d$.
  • For $n$ states with pairwise overlaps at least $\beta$ and a single deterministic input, the success probability is bounded by $\frac{1}{n}(1+\Delta^2/(n-1))$, and for $n=2$ this coincides with a known non-contextual bound.
  • Classicality becomes relative to a chosen subset of inputs, producing a nested hierarchy in which a correlation can be deterministic for some settings and non-classical for others.

Reading between the lines

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

  • The same SDP templates should apply to any other communication restriction expressible as linear constraints, such as energy bounds or information-content bounds, making the three worked examples a proof of principle rather than an exhaustive list.
  • If the size of the input subset for which perfect predictability holds is treated as a quantifier, the approach defines a graded measure of determinism that could be used to compare how non-classical different correlations are.
  • A numerical audit of the two auxiliary assumptions used in deriving Eq. (13)—concavity of the per-realization optimal success probability and the existence of rank-one projective measurements with cyclic symmetry—would establish how broadly that closed-form bound holds.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces an adversarial operational definition of classicality in prepare-and-measure scenarios: a correlation is classical for a chosen set of inputs if a hypothetical adversary with classical side-information can perfectly predict the measurement outcomes. The central observation (Observation 1) equates the existence of a deterministic model with unit guessing probability, and the authors use it to turn non-classicality certification into an SDP infeasibility problem. They instantiate this idea for three communication restrictions: fixed ensemble, bounded overlaps, and restricted observables, giving an SDP for each. They also prove a general linear-witness bound under a connected non-orthogonality graph (Result 1) and provide numerical illustrations, including a closed-form bound for bounded-overlap state discrimination in Appendix B.

Significance. The conceptual framework is attractive and potentially useful: it connects non-classicality certification in prepare-and-measure scenarios to a broad body of QRNG and SDP-hierarchy techniques, and it has the virtue of not being tied to dimension assumptions. Observation 1 and Result 1 are correct under the intended reading, and the presentation is generally clear. However, several load-bearing technical points are not established: the definition of the guessing probability is not written correctly, the bounded-overlap relaxation is not a valid outer approximation in general, the fixed-ensemble block-moment normalization is inconsistent as stated, and the Appendix B analytic bound relies on unproved concavity and symmetry assumptions. These issues prevent me from endorsing the current version, but they are substantial rather than trivial and require a major revision.

major comments (4)
  1. [Eq. (6) and Observation 1] The definition of p_g^{(x,y)} is not faithful to the preceding sentence. Eq. (6) writes p_g^{(x,y)} = max_{p∈Q_{S_i}} Σ_λ q(λ) max_b p_λ(b|x,y), with no constraint that the optimized realization reproduces the observed distribution p(b|x,y). As written, p_g depends only on the set Q_{S_i}, not on the observed correlation; if Q_{S_i} contains any deterministic distribution, then p_g=1 for every observed p, and the 'only if' direction of Observation 1 is false. The intended optimization is over decompositions {q(λ), ρ_x^λ, M_{b|y}^λ} compatible with S_i and with the fixed p(b|x,y). This must be written explicitly, because Observation 1 is the foundation of all three certification methods.
  2. [App. A2 and Eq. (8)] The replacement of the overlap assumption S_β by the constraint G_{|Ψ_x>,|Ψ_{x'}>} ≥ β_{xx'} is not a legitimate WLOG reduction. The assumption is Σ_λ q(λ)|⟨ψ_x^λ|ψ_{x'}^λ⟩| ≥ β_{xx'}, while G_{|Ψ_x>,|Ψ_{x'}>} is the λ-averaged complex inner product. A valid model can have this complex average with real part below β_{xx'}, or even negative, because of λ-dependent phases. The residual freedom to rotate each |ψ_x^λ> by a λ-independent phase does not in general make all averaged inner products real and nonnegative; for that one would need the phases to form a coboundary on the non-orthogonality graph, which fails in generic complex realizations (already for three states). Consequently the feasible set of Eq. (8) is not a superset of the exact set C^{(x,y)}_{S_β}, and infeasibility of this SDP can falsely certify classical correlations as non-classical. This affects the bounded-overlap method and the numerical illustration in Fig. 2b. Please either prove the reduction for the restricted symmetric cases actually used, or replace Eq. (8) by a genuine outer relaxation that is guaranteed to contain all S_β-compatible models.
  3. [App. A1, after Eq. (A2)] The normalization statement for the block-moment matrix is inconsistent. The text sets Γ^λ_{ρ_x,1} = q(λ) ρ_x^λ and then asserts tr(Γ^λ_{ρ_x,1}) = 1. From the definition, tr(Γ^λ_{ρ_x,1}) = q(λ) tr(ρ_x^λ) = q(λ). The correct normalization of a valid fixed-ensemble decomposition is Σ_λ tr(Γ^λ_{ρ_x,1}) = 1, equivalently tr(Γ^λ_{ρ_x,1}) = q(λ). If the individual trace constraint were imposed as written, the SDP Eq. (7) would over-constrain the feasible set and could exclude valid fixed-ensemble models, leading to false non-classicality certification. Please correct Appendix A1 and state explicitly which trace constraints are enforced in the numerical implementation of Eq. (7).
  4. [Appendix B, Eqs. (B2)-(B10)] The derivation of the closed-form bound Eq. (13) relies on three unproved assumptions: (i) the function P(c_λ) is concave in c_λ; (ii) saturation of the averaged overlap constraint implies per-realization saturation |⟨ψ_x^λ|ψ_{x'}^λ⟩| = β for all λ; and (iii) an optimal solution exists with rank-1 projective measurements and with the cyclic label-permutation invariance used in Eqs. (B5)-(B7). These assertions are used to replace the optimization over all λ-dependent preparations by a single symmetric optimization, so they are load-bearing for Eq. (13). For example, concavity is not evident because increasing c_λ shrinks the feasible set, and equality in a Jensen step would require additional strictness arguments. Please supply proofs for these steps or explicitly label Eq. (13) as a conjectured bound rather than a derived one.
minor comments (4)
  1. [Eq. (4)] The notation D_λ(b|x,y) ∈ Q_{S_i} is unclear, because D_λ is a deterministic probability table while Q_{S_i} is a set of averaged correlations. It would be clearer to say that the deterministic table D_λ arises from a realization compatible with S_i.
  2. [Section V.B] The phrase 'under the assumption that at most |X*| = n* inputs contribute deterministically' is confusing: the selected inputs in X* are deterministic by construction of the predictability constraint. Please rephrase to describe the cardinality of the selected input subset.
  3. [App. A2, Eq. (A8)] The Gram matrix in Eq. (A8) is presented as a block matrix, but the block indexing and the precise definition of the vector set S are not given in the displayed equation; a short explanation of how blocks correspond to elements of S would improve readability.
  4. [Eq. (7) and App. A1] The symbol λ is used both for the hidden variable and for the deterministic assignment {λ_xy}; in equations such as Γ^λ_{ρ_x, M_{λ_{xy}|y}} this double use is confusing. A distinct notation for the assignment, e.g. a bold lambda index, would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is a direct reformulation, and self-citations are to tool hierarchies rather than to the target result.

full rationale

The derivation chain is self-contained. Observation 1 equates membership in the classical set C^{(x,y)}_{S_i}, defined by deterministic hidden-variable assignments, with the value p_g^{(x,y)}=1 of the guessing probability; this is a direct equivalence between two formulations of the same condition, not a prediction derived from an input that already contains the conclusion. The three SDP relaxations in Eqs. (7), (8), and (9) apply established moment-matrix and Gram-matrix hierarchies as computational tools, and the self-citations to Refs. [22], [27], and [29] refer to those hierarchies, not to the paper's classicality claim, so they are not load-bearing. Result 1 is proven from Observation 1 and the connectivity of the non-orthogonality graph without importing an external uniqueness or fitted parameter. The bounded-overlap analysis in Appendix B contains asserted but unproved technical assumptions (concavity of P(c_lambda) and existence of a cyclic-invariant rank-one projective optimum), and the Appendix A.2 replacement of the absolute-value overlap constraint by a real Gram-entry constraint is a correctness risk, but these are gaps or possible errors in the proof, not circular reductions of the conclusion to the premises. No fitted quantity is renamed as a prediction, and no load-bearing step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters. The only genuinely ad hoc premise is the concavity and permutation-symmetry reduction in Appendix B. Eve is a standard hypothetical adversary, not an invented physical entity.

assumptions (4)
  • standard math Moment-matrix PSD relaxation correctly outer-approximates the quantum set Q_S and remains valid when the linear predictability constraint is added.
    Assumed throughout Appendix A to turn the non-linear feasibility problem into an SDP.
  • domain assumption Without loss of generality, preparations are pure and all mixtures are absorbed into the hidden variable lambda; Eve has perfect knowledge of lambda.
    Section III, Eq. (5). This is standard in hidden-variable models.
  • ad hoc to paper The per-realization maximum discrimination probability P(c) is concave in the overlap c, and an optimal solution exists with rank-1 projective measurements invariant under cyclic label permutations.
    Appendix B, Eqs. (B2)-(B7). Asserted, not proved; this is the load-bearing assumption for Eq. (13).
  • standard math In a fixed pure ensemble, a deterministic outcome assignment that is compatible with the POVM must assign the same outcome to non-orthogonal preparations.
    Used in Result 1 to propagate equality along the non-orthogonality graph.

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Cite this review

Pith. "Pith review of Characterising the set of deterministic quantum correlations in prepare-and-measure scenarios." pith.science (2026). https://pith.science/paper/RBZTQVPE

@misc{pith2026260809667,
  author       = {Pith},
  title        = {Pith review of: Characterising the set of deterministic quantum correlations in prepare-and-measure scenarios},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBZTQVPE}},
  note         = {Machine review of arXiv:2608.09667}
}
read the original abstract

Correlations that do not admit a deterministic explanation are a central feature of quantum theory and a key resource for quantum information processing. Identifying and certifying such correlations, however, remains a fundamental challenge. In this work, we advance on this problem by considering deterministic correlations in the prepare-and-measure scenario consistent with a wide range of communication restrictions. To certify correlations incompatible with such deterministic models, we ask whether they admit a scenario in which measurement outcomes can be perfectly predicted by an adversary equipped with classical side-information. We show the usefulness of this approach and propose semidefinite programming relaxations tailored to three representative communication restrictions: fixed ensemble, bounded overlaps and restricted observables.

Figures

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Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Reference graph

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