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REVIEW 4 major objections 4 minor 69 references

Nonlocality Without Entanglement: Quantum Theory and Beyond

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Nonlocality without entanglement is generic across generalized probabilistic theories, and quantum theory's version is strictly weaker than the pentagon model's.

desk verdict A genuinely new GPT construction of nonlocality-without-entanglement with a clean quantum comparison, but Theorem 2 rests on an unproved optimality claim in the pentagon model. read the letter →

arxiv 1908.10676 v1 pith:MNRBEW3E submitted 2019-08-28 quant-ph math-phmath.MP

Some Sankar Bhattacharya , Sutapa Saha , Tamal Guha , Manik Banik This is my paper · ORCID
classification quant-phmath-phmath.MP
keywords generalizedprobabilistictheoriesnonlocalitywithoutentanglementstatediscriminationlocaloperationsandclassicalcommunicationpolygonalmodelssignalingdimensionproductstatesquantumspacecontinuity
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

The paper asks whether 'nonlocality without entanglement'—the fact that some sets of product states can be perfectly identified only by a joint measurement, never by separate local measurements with classical communication—is a special feature of quantum theory. It answers no: polygonal generalized probabilistic theories, which replace the quantum state space by a regular polygon, exhibit the same phenomenon with product states and separable product-effect measurements. It then matches the elementary systems of quantum theory and the pentagon model by their signaling dimension, and proves that the pentagon model's gap between global and local success probabilities is strictly larger than quantum theory's gap. The authors conclude that quantum nonlocality without entanglement is limited, and link that limitation to the continuity of the quantum state space.

What carries the argument

The load-bearing object is the polygonal GPT model $\mathrm{Ply}(n)$, whose single-party state space is a regular $n$-gon and whose effects are convex combinations of the unit effect and pairs of extremal effects; the pentagon case is self-dual and has exactly five extremal two-outcome measurements. Within $\mathrm{Ply}(5)^{\otimes 3}_{\min}$, the eight product states and the separable effects $\{E_i\}$ realize perfect global discrimination while every one-way local protocol leaves ambiguity. For fair comparison across theories, the paper uses signaling dimension—the least classical dimension that can simulate the system's input-output behavior—to match the pentagon elementary system with a qubit, and quantifies NWE strength as $\Delta = 1 - P_L$, where $P_L$ is the optimal local success probability.

What would settle it

Search over all one-way local protocols in $\mathrm{Ply}(5)^{\otimes 3}_{\min}$, allowing each party any mixture of the allowed effects; if any protocol exceeds 7/8 success on the octet, then $\Delta[\mathrm{Pentagon}]<1/8$ and the strict inequality of Theorem 2 fails.

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Extended reading notes

Core claim

The central claim is that nonlocality without entanglement is generic in operational theories, and its strength is theory-dependent. In the minimal tripartite composition of the pentagon model $\mathrm{Ply}(5)^{\otimes 3}_{\min}$, the eight product states $\{\phi_1,\ldots,\phi_8\}$ are perfectly distinguishable by a separable measurement whose effects are products of single-party effects (the effects $E_1,\ldots,E_8$ satisfy $\sum_i E_i = u^{\otimes 3}$ and $p(E_i|\phi_j) = \delta_{ij}$), yet no local protocol can discriminate them perfectly. The same construction works in hexagon and heptagon models. After matching elementary systems by signaling dimension, the paper proves that for the analogous three-qubit product ensemble the gap is $\Delta[\mathrm{QT}] \le \frac{1}{8}(4-\sqrt{10})$, strictly smaller than $\Delta[\mathrm{Pentagon}] = \frac{1}{8}$. This limited behavior in quantum theory is ascribed to the continuity of the state space, which allows a continuous reversible transformation between any two pure states.

Load-bearing premise

The comparison rests on the unproved assertion that no local protocol in the pentagon model can do better than 7/8 on the eight states, even though each party is allowed to use any mixture of the five basic measurement pairs.

Editorial extensions

If this is right

  • Nonlocality without entanglement, including asymmetric local discrimination and separable but locally unimplementable measurements, is not a signature of Hilbert-space structure; it arises generically in generalized probabilistic theories.
  • For tripartite systems with equal signaling dimension, the pentagon model has strictly stronger NWE than quantum theory: $\Delta[\mathrm{Pentagon}] = 1/8$ while $\Delta[\mathrm{QT}] \le (4-\sqrt{10})/8$.
  • The quantity $\Delta$ provides a common scale for comparing NWE strength across different theories, making the phenomenon quantitatively testable.
  • Because limited NWE in quantum theory is tied to continuity of the state space, limited NWE could serve as a candidate principle in axiomatic derivations of quantum theory.
  • The ordering $\Delta[\mathrm{Quantum}] < \Delta[\mathrm{polygon}]$ holds also for biased priors over the eight states, not only for the uniform distribution.

Reading between the lines

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

  • A numerical search over all local protocols in the pentagon model, allowing convex mixtures of the five basic measurement pairs, would settle the one unproved optimality assertion and either confirm or reduce the strict inequality in Theorem 2.
  • If the pentagon value is confirmed, the octet becomes a compact NWE witness: any GPT with signaling dimension 2 whose state space is a polygon of at least five sides should show the same $1/8$ gap, yielding a family of testable models.
  • Constructing a signaling-dimension-3 GPT with a continuous but non-quantum state space would separate the effect of continuity from the effect of Hilbert-space structure, testing the paper's proposed explanation for quantum theory's limited NWE.
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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 argues that nonlocality without entanglement (NWE) is not peculiar to quantum theory but appears in generalized probabilistic theories (GPTs). It constructs an eight-state product ensemble in the tripartite minimal composition of the pentagon model Ply(5)^3, exhibits a separable product-effect measurement that discriminates it perfectly, and claims that no local protocol can do so. It then proposes a quantitative measure of NWE strength, Δ = 1 − P_L, the gap between global and optimal local success probability, with elementary systems matched by signaling dimension. The central quantitative result is Theorem 2, which states Δ[QT] ≤ (1/8)(4−√10) < 1/8 = Δ[Pentagon], and a biased-prior version is claimed to show the same ordering for all prior weights. The authors attribute the limited NWE of quantum theory to the continuity of the quantum state space.

Significance. If the proofs were complete, the paper would make a valuable conceptual contribution: it would show that a phenomenon previously considered quantum is generic in GPTs, and it would provide a quantitative criterion for comparing NWE strength across theories. The construction of a product-effect measurement for the pentagon octet is explicit and checkable, and the quantum one-parameter optimization giving P_err = (1/8)(4−√10) is analytic and correct. The framework using signaling dimension to put different elementary systems on an equal footing is natural and potentially useful. However, the paper's main quantitative claim rests on an unproved optimality assertion for the pentagon model, and several auxiliary proofs are incomplete or inconsistent at the level of internal references. These issues are load-bearing rather than cosmetic.

major comments (4)
  1. [Appendix B.2, Eqs. (B6)–(B7)] The equality Δ[Pentagon] = 1/8 is asserted, not proved. The flow-chart protocol achieves P_succ = 7/8, but optimality over all local protocols is justified only by the sentence 'It follows from the proof of Proposition-2, any other strategy is no good', and Proposition-2 is never stated in the manuscript. Since E(5) is the convex hull of {0, u, e_i, ¯e_i}, a first-round measurement may use any effect f ∈ E(5), not only the extremal dichotomies {e_i, ¯e_i}, and a rigorous upper bound must rule out all such strategies and all later-round choices. Without a proof that P_L[Pentagon] ≤ 7/8 (or at least P_L[Pentagon] < 1 − (4−√10)/8 ≈ 0.8953), the strict inequality in Theorem 2 is not established.
  2. [Theorem 1 (main text)] The local indistinguishability part of Theorem 1 is not proved. The sentence 'It is not hard to see that whichever measurement Alice starts with no perfect discrimination is possible' substitutes for an argument that must cover arbitrary effects in E(5), not merely the five extremal measurements M_i. Since the octet is perfectly discriminated by a global separable measurement, the entire NWE phenomenon in the pentagon model depends on this claim. Please provide a complete proof or a precise reference to one.
  3. [Lemma 1 (main text)] The described local protocol for the four states in Ply(4)^2 appears to be incorrect as written. When Alice measures M_0 = {e_0, e_2}, the outcome e_0 on Alice's subsystem also has nonzero probability for the state ω_1⊗ω_0, so it does not uniquely select {ω_0⊗ω_0, ω_0⊗ω_3} as the proof claims. Consequently Bob's subsequent measurement cannot perfectly resolve all cases in the way described. Please verify the transition probabilities p(e_0|ω_i) for the squit model and either correct the protocol or adjust the state set and proof.
  4. [Appendix B.1–B.2] The appendix refers repeatedly to 'Proposition-2' and 'Proposition-3', but no such propositions are stated in the paper. This makes the claimed local indistinguishability of the hexagon and heptagon constructions, and the optimality claim for the pentagon protocol, impossible to verify. The numbering of results must be made consistent, and the referenced proofs must either be included or given explicit locations.
minor comments (4)
  1. [Lemma 1 proof] There is a typo in the last sentence: 'perfetly' should be 'perfectly'.
  2. [Appendix B.3] In the biased-prior paragraph, 'p4 = p4 = p' should presumably read 'p4 = p5 = p'; please correct the notation.
  3. [Figure 3] The flow chart is not described algorithmically in the text, which makes the claimed success probability 7/8 hard to reproduce. A short verbal description of the decision tree would improve verifiability.
  4. [Theorem 2 statement] The theorem states an upper bound Δ[QT] ≤ (1/8)(4−√10), which is an upper bound on the gap and hence a lower bound on the optimal success probability; this directional language is correct, but it would help to state explicitly that no optimality of the quantum protocol is claimed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pentagon construction and the quantum comparison are self-contained, with the only self-citation being motivational rather than load-bearing.

full rationale

The central derivations are explicit and verified in the paper rather than imported from its own conclusions. Theorem 1 defines the octet and the eight product effects and directly checks p(E_i|phi_j)=delta_ij; local indistinguishability is argued from the polygonal effect structure, and the global measurement is given in closed form. The comparison in Theorem 2 uses the independently defined signaling dimension of Dall'Arno et al. to match the qubit and pentagon elementary systems, then evaluates explicit local protocols. No parameter is fitted to the quantity being predicted, and no prediction is equivalent by construction to an input. The only self-citation, Ref. [21], appears in the introduction to motivate that GPTs can contain indistinguishable pure states, but the pentagon model's indistinguishability properties are re-derived in Appendix A.2 from the model geometry, so this citation is not load-bearing. Appendix B.2's assertion that the 7/8 pentagon protocol is optimal ('It follows from the proof of Proposition-2, any other strategy is no good') is an unproved step and therefore a correctness risk, but it is not circular: the claimed optimum is neither a fitted input nor a restatement of any definition. The quantum side correctly reports an upper bound on Delta[QT] from an explicit theta-optimized protocol, which is the right logical direction for the inequality. For these reasons the derivation chain is not circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central constructions are self-contained and checkable; there are no fitted parameters and no invented entities. The ledger's main entries are framework axioms (standard GPT assumptions) plus one ad hoc assertion: the exhaustiveness of extremal measurements in the pentagon optimality argument, which is the weakest link in the comparison theorem.

free parameters (2)
  • θ (single-qubit measurement angle) = tan⁻¹(1/3)
    Variational parameter in the 1-way LOCC bound of Appendix B.2; value fixed by exact minimization of Perr(θ), not fitted to data. Listed for transparency; it is an optimization variable, not an ad hoc free parameter.
  • p (biased prior weight) = varies in (0, 1/2)
    External prior parameter in the biased-ensemble extension (Appendix B.3); not part of the uniform-prior central claim.
assumptions (5)
  • domain assumption GPT composition obeys no-signaling and local tomography; valid composites lie between minimal and maximal tensor products (Appendix A.1.D).
    Standard GPT framework (Hardy 2001; Barrett 2007), invoked in Definitions 4-5 of the appendix.
  • domain assumption Self-consistency (SC) condition for compositions: any valid composition of systems, states, effects, and transformations yields non-negative conditional probabilities.
    Taken from Dall'Arno et al. 2017 [37]; used to admit the five squit compositions referenced in Lemma 1.
  • domain assumption In the minimal composition, only product states and product effects exist.
    Definition of ⊗min; Theorem 1 is stated in Ply(5)⊗3_min.
  • ad hoc to paper Checking only extremal measurements {e_i, ¯e_i} suffices to conclude that no local protocol can perfectly discriminate the pentagon octet and that 7/8 is optimal.
    Asserted in Theorem 1 proof ('It is not hard to see...') and Appendix B.2 ('any other strategy is no good'); the convex hull of effects includes non-extremal measurements, and the argument does not cover them.
  • domain assumption Local protocols in GPT need no post-measurement update rule; outcome logic is 'conclusive elimination' (a state is identified or excluded).
    The paper notes update rules are not natural in GPTs (Ref [42]) and all protocols operate by elimination; this operational choice underlies Lemma 1 and Theorem 1.

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Pith. "Pith review of Nonlocality Without Entanglement: Quantum Theory and Beyond." pith.science (2026). https://pith.science/paper/MNRBEW3E

@misc{pith2026190810676,
  author       = {Pith},
  title        = {Pith review of: Nonlocality Without Entanglement: Quantum Theory and Beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNRBEW3E}},
  note         = {Machine review of arXiv:1908.10676}
}
read the original abstract

Quantum nonlocality without entanglement (Q-NWE) encapsulates nonlocal behavior of multipartite product states as they may entail global operation for optimal decoding of the classical information encoded within. Here we show that the phenomena of NWE is not specific to quantum theory only, rather a class of generalized probabilistic theories can exhibit such behavior. In fact several manifestations of NWE, e.g., asymmetric local discrimination, suboptimal local discrimination, notion of separable but locally unimplementable measurement arise generically in operational theories other than quantum theory. We propose a framework to compare the strength of NWE in different theories and show that such behavior in quantum theory is limited, suggesting a specific topological feature of quantum theory, namely, the continuity of state space structure. Our work adds profound foundational appeal to the study of NWE phenomena along with its information theoretic relevance.

Discussion (0). Continue with ORCID to comment.

Reference graph

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