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Limits of Classical correlations and Quantum advantages under (Anti-)Distinguishability constraints in Multipartite Communication

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Pith's one-line read Quantum communication can beat every classical strategy in multipartite tasks where each message is constrained only by how much it reveals (or deliberately misreports) its sender's input—and the advantage for joint antidistinguishing…

desk verdict Useful facet framework, but the exponential quantum advantage depends on silently excluding classical shared randomness, so the central claim does not hold as stated. read the letter →

arxiv 2506.07699 v2 pith:PSECPELC submitted 2025-06-09 quant-ph

Ankush Pandit , Soumyabrata Hazra , Satyaki Manna , Anubhav Chaturvedi , Debashis Saha This is my paper · ORCID
classification quant-ph MSC 81P4581P15 PACS 03.67.-a
keywords multipartitecommunicationdistinguishabilityantidistinguishabilityquantumadvantageprepare-and-measurescenariosfacetinequalitiesPBRtheoremepistemicincompleteness
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 asks whether quantum communication helps when each sender's message is constrained not by size but by how much the receiver can identify, or deliberately misidentify, that sender's input. It shows that with several independent senders and a single receiver, quantum strategies that share no entanglement can outperform all classical strategies under such constraints. The sharpest result is for the task of deliberately misidentifying the joint input string: a PBR-based quantum protocol reaches perfect success while the comparable classical success is bounded by $1-\prod_{i=1}^N(1-A_i)$, giving a quantum-over-classical ratio that grows like $2^N$ as the number of senders grows. Along the way, the paper provides a systematic method for listing the facet inequalities of the classical correlation polytope in such scenarios and exhibits explicit qubit protocols that violate them.

What carries the argument

The PBR joint measurement: a $2^N$-outcome POVM on the tensor product of $N$ qubit states, built so each outcome has zero probability for one of the $2^N$ input strings. This measurement delivers $\tilde{S}_Q=1$. On the classical side the carrying bound is the product formula $\tilde{S}_C \le 1-\prod_{i=1}^N(1-A_i)$, which is derived from the per-sender antidistinguishability constraints. For the small scenarios the machinery is the extended classical polytope $D_C^+$ (or $A_C^+$), built as a product of per-sender encoding polytopes and the receiver's decoding polytope; its facet inequalities are enumerated and then probed by a semidefinite SeeSaw optimization over quantum states and measurements.

What would settle it

Fix two binary senders with equal per-sender antidistinguishability $A$, enumerate every deterministic classical encoding and decoding, and compute the best probability of deliberately misidentifying the joint input; any value exceeding $1-(1-A)^2$ would refute the factorised upper bound that supports the claimed exponential quantum advantage.

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Extended reading notes

Core claim

On the task where the receiver must deliberately misidentify the joint input string $(x_1,\dots,x_N)$, the paper proves an upper bound on every classical strategy: $\tilde{S}_C \le 1-\prod_{i=1}^N(1-A_i)$, where $A_i$ is the maximal probability that the receiver can be wrong about sender $i$'s individual input. A quantum strategy using $N$ qubits and the PBR joint measurement attains $\tilde{S}_Q=1$ while keeping each sender's individual antidistinguishability at $A_Q=(1+\sin\theta)/2<1$, so the advantage ratio is $(A_C/A_Q)^N=(2/(1+\sin\theta))^N>1$, and the best allowed $\theta$ pushes this toward $2^N$ as the number of senders grows. In the small scenarios, the paper enumerates all facet inequalities of the classical correlation polytopes and exhibits qubit protocols that violate three of them, establishing quantum advantage without shared entanglement.

Load-bearing premise

The load-bearing step is that the receiver's best chance of misidentifying the joint input can be minimized one sender at a time; if that decomposition fails, the classical bound that the quantum protocol beats is not established.

Editorial extensions

If this is right

  • In the joint-antidistinguishing task, every classical protocol satisfies $\tilde{S}_C \le 1-\prod_{i=1}^N(1-A_i)$, so to reach perfect joint antidistinguishing classically every sender must have $A_i=1$.
  • The PBR-inspired quantum protocol reaches $\tilde{S}_Q=1$ with $A_Q=(1+\sin\theta)/2<1$, yielding a quantum-classical ratio $(2/(1+\sin\theta))^N$ that tends to $2^N$ as $N$ grows.
  • In two-sender scenarios with no receiver input, facet inequalities of the classical polytope are violated by explicit qubit protocols, so entanglement-free quantum advantage already appears at small sizes.
  • Because the constraints bound distinguishability or antidistinguishability rather than Hilbert-space dimension, the demonstrated advantage is not an artefact of sending larger quantum systems.
  • Any such advantage implies epistemic incompleteness of quantum theory under the preparation-independence assumption.

Reading between the lines

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

  • The same factorised bound suggests that the exponential advantage should survive when senders use different input alphabets, as long as a joint measurement can antidistinguish the resulting product states.
  • The facet-inequality construction could be repurposed as a semi-device-independent privacy witness: exceeding an antidistinguishability facet certifies that the communicated states cannot be explained by an epistemically complete, preparation-independent classical model.
  • A brute-force enumeration of deterministic classical strategies for small $N$ would independently verify the classical bound and isolate precisely where the PBR construction is needed.
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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

2 major / 4 minor

Summary. The paper introduces multipartite communication tasks in which each sender's message is constrained only by a bound on its distinguishability or antidistinguishability. It defines classical and quantum correlation sets, derives facet inequalities for several two-sender scenarios using vertex enumeration, and uses SeeSaw SDP methods to find quantum violations. The central theoretical results are Theorem 1, an upper bound on the classical success of antidistinguishing the joint input string, and Theorem 2, a PBR-inspired protocol claimed to give an exponential quantum-over-classical advantage in that task as the number of senders grows. A sufficient condition for quantum advantage (Theorem 3) and a connection to epistemic incompleteness are also given.

Significance. If the main claim held as stated, the paper would provide an exponential quantum advantage in multipartite communication without shared entanglement and under only (anti-)distinguishability constraints, which would be a striking result. The paper has concrete strengths: explicit qubit strategies for inequalities I1 and I6 that can be checked by direct calculation, valid SDP relaxations that give lower bounds on the quantum value, a public repository of extremal strategy vertices, and facet computations for several previously uncharacterized scenarios. However, the classical model underlying the main theorems is nonstandard: Eq. (16) excludes shared randomness between senders, and under the standard classical model in which shared randomness is allowed, the claimed exponential advantage of Theorem 2 reverses. This makes the significance of the central claim highly conditional on an unstated and possibly unintended restriction.

major comments (2)
  1. [§II, Eq. (16); §IV, Theorems 1 and 2] The paper's classical model is defined by Eq. (16), a product of independent sender encodings p_e(m_i|x_i), which silently excludes shared randomness between senders. This is not the standard notion of a classical strategy in multipartite communication or Bell-type scenarios, where a shared random seed is usually allowed. The exclusion is load-bearing: if shared randomness is allowed, Theorem 1's bound (53) is false. For N=2 with binary inputs, let lambda in {0,1} be shared uniformly; if lambda=0, sender 1 sends m1=x1 and sender 2 sends m2=0, and if lambda=1, sender 1 sends m1=0 and sender 2 sends m2=x2, while the receiver outputs (1-m1,1-m2). Then eS_C=1 for every input pair, and the marginals give A_1=A_2=3/4, violating eS_C <= 1-(1-A_1)(1-A_2)=15/16. Generalizing with a random designated perfect sender gives A_i=1/2+1/(2N) with eS_C=1, whereas the PBR protocol of Theorem 2 has A_Q approximately 1/2+(ln 2)/N for large N; hence (A_C/A_Q)^N < 1 for large N, and the advertised exponential advantage is an artifact of the no-shared-randomness restriction. Moreover, the polytope construction in Section II appears to be the convex hull of deterministic strategies (which is the shared-randomness model), while the proof of Theorem 1 applies only to the product model of Eq. (16); the manuscript needs to state which model is intended, justify that choice, and reconcile the two.
  2. [§IV, Theorem 1 proof, Eq. (56)] The proof of Theorem 1 invokes the general identity min_i ∏_j alpha_{i,j} = ∏_j min_i alpha_{i,j}, which is false for generic non-negative matrices; for example, with alpha=[[1,2],[2,1]] the left side is 2 and the right side is 1. In the specific application, alpha_{x,j}=q_{x_j} p_e(m_j|x_j) depends only on the j-th coordinate of the input string x, so the minimization over the product string does factorize coordinate-wise and the theorem can be repaired. However, the proof as written states an incorrect identity and does not supply the needed coordinate-wise argument. Since Eq. (53) is used in Theorems 2 and 3, the corrected argument should be inserted before the bound is relied upon.
minor comments (4)
  1. [Appendix C, Table V] The column labels of Table V list pe(4|3) four times; the intended labels are pe(1|3), pe(2|3), pe(3|3), and pe(4|3).
  2. [Appendix C, Table VIII] Table VIII repeats the entry p(2|2,1) and omits the intended p(2|2,2) or a similar distinct label; please correct the header row.
  3. [§III, Fig. 5 caption] The phrase 'precessional error' should probably read 'precision error' or 'numerical error' in the caption of Fig. 5.
  4. [§II, after Eq. (23)] The claim that it is sufficient to consider n_mi = 2(n_xi - 1) messages is cited to [35] and asserted to extend to antidistinguishability, but the extension is not shown; a brief proof or a precise reference to the analogous argument would improve rigor.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central bounds and quantum protocols are anchored in external theorems (PBR, Helstrom, Johnston-Russo-Sikora) and the paper's explicit private-coin classical model.

full rationale

The load-bearing chain is not circular. Theorem 1's classical bound eS_C ≤ 1 − Π_i(1−A_i) is derived from the paper's own constraints (21) and (55). Although the general identity (56) is false as stated, in the specific application the factors α_{i,j} = q_{x_j} p_e(m_j|x_j) depend on the joint index only through the coordinate x_j, so the min-product factorization is valid; the bound is therefore not an input restated as an output. Theorem 2's quantum protocol is supplied by the PBR theorem (Ref. [36]) and the per-sender A_Q value by the Helstrom formula; the comparison value A_C = 1 follows from Theorem 1's bound, not from a fitted or recycled constant. Theorem 3 invokes external necessary and sufficient antidistinguishability conditions from Ref. [39]. Self-citations (e.g., Refs. [23], [24], [26], [27]) are confined to background notions such as bounded ontological distinctness and epistemic incompleteness, and the main advantage claims do not depend on those citations being true. A proof-presentation flaw exists: Eq. (56) is asserted for arbitrary non-negative matrices, which is false; however, the subsequent application is valid because the factors factorize over coordinates, so this is a correctness or exposition issue, not circularity. A separate model-scope caveat is that Eq. (16) defines classical correlations without shared randomness; the quantum advantages are therefore claims about the private-coin classical polytope, and the robustness of the advantage under shared randomness is a substantive correctness question, not an instance of the derivation reducing to its own inputs.

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

The central derivations depend on standard external results (PBR, Helstrom, CFS) and on a few domain assumptions about independence and convex hulls. No new physical entities are postulated; the only free choices are protocol parameters theta and example state angles, chosen by hand rather than fitted to data.

free parameters (2)
  • PBR angle theta = theta = 2 arctan(2^(1/N)-1) for optimal advantage; general theta in the range giving A_Q = (1+sin(theta))/2
    Chosen to saturate the PBR antidistinguishability condition (59); the advantage ratio (2/(1+sin(theta)))^N depends directly on it.
  • Example state angles 5*pi/18 and 19*pi/60 = 5*pi/18 and 19*pi/60
    Hand-picked in the (n=3, N=2) example to satisfy Theorem 3's sufficient conditions; verified by SDP to give A_Q=0.9798 and eS_Q=1.
assumptions (5)
  • domain assumption Senders are independent and non-communicating; the joint state is a product state.
    Used in Eq (3) and throughout to factor sender probabilities.
  • domain assumption A global POVM does not enhance (anti)distinguishability of local quantum states.
    Invoked after Eq (37) to reduce per-sender constraints to the joint state.
  • domain assumption The classical achievable set is the convex hull of deterministic encoding and decoding strategies.
    Definitions 4-8 and the facet enumeration treat the classical set as a polytope; shared randomness is implicitly allowed.
  • ad hoc to paper It suffices to consider n_mi = 2(n_xi - 1) messages per sender.
    The paper adopts this bound from Appendix C of Ref [35] without proof; all facet computations rely on it.
  • domain assumption The PBR theorem provides an N-copy antidistinguishing measurement under condition (59).
    Theorem 2's quantum protocol is built directly on this cited result.

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Cite this review

Pith. "Pith review of Limits of Classical correlations and Quantum advantages under (Anti-)Distinguishability constraints in Multipartite Communication." pith.science (2026). https://pith.science/paper/PSECPELC

@misc{pith2026250607699,
  author       = {Pith},
  title        = {Pith review of: Limits of Classical correlations and Quantum advantages under (Anti-)Distinguishability constraints in Multipartite Communication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSECPELC}},
  note         = {Machine review of arXiv:2506.07699}
}
read the original abstract

We consider communication scenarios with multiple senders and a single receiver. Focusing on communication tasks where the distinguishability or antidistinguishability of the sender's input is bounded, we show that quantum strategies-without any shared entanglement-can outperform the classical ones. We introduce a systematic technique for deriving the facet inequalities that delineate the polytope of classical correlations in such scenarios. As a proof of principle, we recover the complete set of facet inequalities for some nontrivial scenarios involving two senders and a receiver with no input. Explicit quantum protocols are studied that violate these inequalities, demonstrating quantum advantage. We further investigate the task of antidistinguishing the joint input string held by the senders and derive upper bounds on the optimal classical success probability. Leveraging the Pusey-Barrett-Rudolph theorem, we prove that when each sender has a binary input, the quantum advantage grows with the number of senders. We also provide sufficient conditions for quantum advantage for arbitrary input sizes and illustrate them through several explicit examples.

Figures

Figures reproduced from arXiv: 2506.07699 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic diagram of a multipartite [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Landscape of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Quantum advantage vs success probability for [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Landscape of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Quantum advantage in terms of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

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Forward citations

Cited by 2 Pith papers

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    Three bipartite product states are globally antidistinguishable yet not LOCC antidistinguishable under the paper's restricted one-pass LOCC model, giving a claimed minimal example of exclusion-based nonlocality.

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