{"id":"3df4bb35-a248-4610-82c9-30e63c73c612","arxiv_id":"2412.17119","paper_version":4,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For a new quantum analogue of empirical coordination, the paper proposes single-letter rate formulas but leaves a central converse unproven.","lead":"The paper introduces a new task, empirical coordination, for quantum networks and derives candidate rate formulas for simulating quantum states on average. The optimality claims for classical-link networks rest on a converse proof that does not establish the stated lower bound.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse for classical-link theorems bounds only accessible information, which can be strictly below the claimed infimum over separable extensions; the capacity formulas are unproven.","rationale":"The central claim is exact capacities for classical-link networks. Achievability via Lemma 7 is plausible and the quantum-link results have independent support from strong-coordination prior work. But the converse for Theorem 2 is the linchpin for Theorem 3 and Corollary 4, and it is invalid as written. The measurement reduction maps the quantum problem to a classical coordination problem whose rate constraint is the accessible information; since the theorem's rate is an infimum over classical labels that can be more informative than any measurement of the quantum outputs, the lower bound does not follow. This is not merely a missing epsilon: for non-orthogonal conditional states the gap is quantitative. The example with |0> and |+> shows inf = 1 while accessible info < 1. The paper acknowledges no such limitation; the converse section presents the measurement step as if it were innocuous. Because the main advertised results (Theorems 2 and 3) are unsupported, rejection is appropriate, although the achievability half and the quantum-link theorems may survive a repair.","tokens_in":34240,"tokens_out":12475,"duration_ms":126414,"concrete_test":"Compute the claimed capacity and the accessible information for the two-qubit example above (or Example 2 with p = 0.2): evaluate inf_{sigma in S2-node} I(X;Y)_sigma analytically (it equals H(X) when the conditional states are distinct pure states) and numerically optimize a POVM on B to find the accessible information; verify accessible < inf. Additionally, re-derive the two-node converse replacing the measurement by the quantum mutual information bound nR >= I(X^n; B^n)_rho; the resulting lower bound is chi({p_X, omega^x_B}), not inf_sigma I(X;Y)_sigma, which confirms that the proof in Sec. VIII-A proves a strictly weaker statement than Theorem 2.","verdict_should_be":"REJECT","load_bearing_attack":"The converse proofs of Theorems 2 and 3 (Sec. VIII-A/B) measure Bob/Charlie outputs in fixed product bases and then apply the classical coordination converse to the resulting classical variables (Eqs. 121-123 and 128-132). This can only yield a lower bound in terms of accessible information. For any extension sigma in S2-node, X -> Y -> B is a Markov chain, so I(X;Y)_sigma >= I(X;B)_omega; hence inf_sigma I(X;Y)_sigma >= chi({p_X, omega^x_B}). When the states {omega^x_B} are non-orthogonal, the accessible information is strictly less than chi, so no measurement-based argument can reach the claimed infimum. The proof never constructs an extension whose label distribution matches the measured pi nor proves I(X;Y)_pi >= inf. Concretely, take A trivial and omega_XB = (1/2)|x=0><x=0| otimes |0><0| + (1/2)|x=1><x=1| otimes |+><+|. Then S2-node forces Y=X (each conditional state is a distinct pure state), so inf I(X;Y) = 1, while any projective measurement on B gives accessible info < 1 (about 0.39 for the optimal POVM). Thus the converse in Sec. VIII-A cannot establish the claimed bound. The quantum-link theorems are not affected by this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces empirical coordination for quantum correlations, where the goal is for a network of quantum nodes to produce a time-averaged joint state close to a target state, and characterizes the minimal classical or quantum communication rates for several small networks. For classical links, it treats a two-node network, a three-node cascade, and an isolated-node special case, giving rate formulas as infima over separable state extensions; it also proves that common randomness does not change the achievable rates. For quantum links, it gives the two-node capacity H(B)_ω and a broadcast-network region with rates H(B|X) and H(C|Y). The paper includes achievability via covering arguments, converse arguments, examples, and a discussion of applications to nonlocal games and CHSH.","tokens_in":34501,"tokens_out":13108,"duration_ms":135387,"significance":"If the main theorems are correct, the paper provides exact minimal communication resources for average-state simulation in simple quantum networks, a natural quantum analogue of classical empirical coordination. The quantum-link results are appealing and the achievability side is built on standard covering arguments and on the authors' published strong-coordination results, which are appropriately cited. The discussion connecting empirical coordination to nonlocal games is interesting and gives a concrete operational interpretation. However, the converse proofs for the classical-link theorems (Theorems 2 and 3) contain a load-bearing gap: they lower-bound the rate by the accessible information of a fixed projective measurement on Bob's output, which can be strictly smaller than the claimed infimum over separable extensions. Because this gap affects the central capacity claims for classical links, the paper is not yet ready for acceptance.","major_comments":[{"comment":"The converse for Theorem 2 measures Bob's output in a fixed projective basis and then applies the classical coordination converse, obtaining R ≥ I(XJ;YJ) for the induced classical distribution. This argument can only lower-bound the rate by the accessible information of the ensemble {p_X, ω^x_B}, whereas the claimed capacity is the infimum of I(X;Y) over extensions in S2-node(ω). For non-orthogonal conditional states these quantities differ: the accessible information is at most the Holevo quantity, while inf_{σ∈S2-node} I(X;Y)_σ is at least the Holevo quantity because X→Y→B is a Markov chain for every extension. A concrete failure is the state ω_XB = (1/2)|0⟩⟨0|⊗|0⟩⟨0| + (1/2)|1⟩⟨1|⊗|+⟩⟨+| with A trivial: S2-node forces Y=X, so the claimed infimum is I(X;Y)=1, but any projective measurement on B gives strictly less than 1 (about 0.399 for the optimal POVM). Thus the proof in Section VIII-A does not establish the claimed converse. The same gap affects Theorem 3 through Eqs. (128)-(132) in Section VIII-B. A valid converse would need to construct an extension, for example from the code's internal message, rather than measuring Bob's output in a fixed basis.","section":"Section VIII-A, Eqs. (117)-(123)"},{"comment":"The formal definition of the cascade code gives Bob the conditional distribution p_{M2→3|X^n M1→2}, which means Bob has access to Alice's source sequence X^n. This contradicts the network model in Figure 5, where only Alice receives X^n, and it also contradicts the achievability scheme in Section VII-C, where Bob decodes Y^n and Z^n from the bin indices without seeing X^n. Moreover, the converse in Section VIII-B relies on the Markov chain X^n → M1→2 → (Y^n,Z^n); if Bob's message M2→3 depends on X^n, the inequality nR1→2 ≥ I(X^n;Y^nZ^n) in Eq. (128) is no longer justified (one would only get a bound involving n(R1→2+R2→3)). The definition should be p_{M2→3|M1→2}, and Eq. (53) should be updated accordingly.","section":"Section IV-C, formal definition and Eq. (53)"}],"minor_comments":[{"comment":"There are notational typos in the non-signaling conditions: (77b) should state ω_AB^(x,y) = ω_AB^(x,y') for all y,y', rather than identifying the state with a state that depends only on y'; and in (77c), the quantifier 'x,x′ ∈ Y' should read 'x,x′ ∈ X'.","section":"Section V-B, Eq. (77)"},{"comment":"The sentence preceding Eq. (78) lists the rate constraints as 'Q1→2 ≥ H(B|X)ω, Q1→2 ≥ H(C|Y)ω'; the second inequality should be Q1→3 ≥ H(C|Y)ω, as in the displayed region (78).","section":"Section V-B, text before Theorem 6"},{"comment":"The description of the improved decomposition is confusing: the sentence 'Y = X with probability 1' applies to the first decomposition, not to the improved one. For the improved rate 0.3112, Y is not a deterministic function of X; rather p_{Y|X}(0|0)=1 and p_{Y|X}(0|1)=p_{Y|X}(+|1)=1/2, yielding H(Y)=0.8113 and H(Y|X)=0.5.","section":"Example 1, Eq. (45)"},{"comment":"There is a typographical artifact in Eq. (65): the displayed state ends with a comma and a period ('... , .'). This should be cleaned up.","section":"Eq. (65)"},{"comment":"The lemma statement and proof are essentially standard, but the dependence on δ is stated only through γ(δ) in Eq. (93) and the rate condition (105); it would help readers to state explicitly that the rate condition is asymptotically tight as δ→0 and n→∞.","section":"Section VII-A, Lemma 7"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the converse for the classical-link theorems. The measurement-based argument cannot prove the claimed infimum-over-extensions formulas, as the stress-test example shows. I would be willing to accept a revised version if the authors supply a correct converse, for instance by using the code's message as the classical extension variable and symmetrizing over time to handle block-coupling. If no such converse is provided, the paper should not be accepted, because Theorems 2 and 3 are the main classical-link results. The quantum-link sections appear more solid, but the applicability of the cited strong-coordination results to the empirical criterion should also be checked carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take: the paper introduces a genuinely new operational framework—empirical coordination for quantum correlations—and the quantum-link rate formulas look right. But the two headline capacity theorems for classical-link networks (Theorems 2 and 3) rest on converse proofs that don't establish the claimed bounds. The flaw is specific and load-bearing: the converses in Sections VIII-A/B measure the receivers' outputs in fixed bases and then apply the classical coordination converse to the resulting classical variables. That route only gives a bound in terms of accessible information, and accessible information can be strictly smaller than the infimum over the separable extensions that appear in the capacity formulas.\n\nThe concrete picture: take ω_XB = 1/2 |0><0|⊗|0><0| + 1/2 |1><1|⊗|+><+|. Any valid extension in S2-node forces Y to be a deterministic function of X (the conditional states are distinct pure states), so the infimum of I(X;Y) is 1. But any projective measurement on B gives at most about 0.39 bits of mutual information. The converse in Section VIII-A would conclude R ≥ 0.39, not R ≥ 1. The gap is the difference between accessible information and the infimum over extensions; the proof never constructs an extension that attains the infimum, nor shows the measured π is one. The same problem appears in the cascade converse.\n\nWhat's good: the definition itself is natural and worth having, and the optimization over state extensions is a real new feature absent from classical coordination. The achievability arguments are standard covering arguments and look plausible. The quantum-link results (Theorem 5, H(B); Theorem 6, the broadcast rates) are straightforward adaptations of Schumacher compression and the authors' prior strong-coordination work, and their converses appear sound. Theorem 1 (CR doesn't help) also seems fine.\n\nWho should read it: people working on coordination theory and quantum state simulation. The framework is promising, but the paper in its current form overclaims. As it stands, I would not accept it: the central classical-link capacity theorems are unproven. That said, the errors are repairable, and the novel framework and sound quantum-link sections justify sending it to a serious referee rather than desk-rejecting. I'd recommend a revise-and-resubmit with the converse proofs as the focus.","headline":"New framework for empirical coordination of quantum correlations, but the headline classical-link capacity theorems are not proven: the converse bounds only accessible information, which can be strictly below the claimed infimum over extensions.","tokens_in":34986,"tokens_out":5728,"would_cite":false,"duration_ms":54683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","94A17"],"pacs":["03.67.-a","03.67.Hk"],"model":"deepseek-v4-flash","headline":"Empirical coordination of quantum correlations has exact capacity formulas: over classical links the rates are an optimization over separable state extensions, and over a quantum link the two-node capacity is the von Neumann entropy…","keywords":["empirical coordination","quantum correlations","coordination capacity","cascade network","separable extensions","nonlocal games","quantum information theory","Schumacher compression"],"falsifier":"Search over separable two-qubit target states for one where the infimum over extensions of $I(X;Y)$ strictly exceeds the Holevo information of the induced ensemble; for such a target, the converse's measurement step cannot certify the claimed capacity. The decisive check is then to run a two-node empirical-coordination protocol at a rate below the claimed formula and test whether the empirical average state still converges to the target.","tokens_in":34032,"feed_emoji":"⚛️","tokens_out":8254,"duration_ms":84493,"temperature":0.7,"pith_summary":"This paper introduces empirical coordination as a task for quantum networks: parties connected by rate-limited classical or quantum links must produce outputs whose time-averaged joint state approaches a desired target state. The authors prove exact minimum communication rates for the basic building-block networks: a two-node link, a three-node cascade with classical links, and a broadcast configuration with quantum links. The classical-link formulas are minima or regions over separable extensions of the target state, a genuinely quantum feature with no classical analogue. The two-node quantum-link capacity equals the von Neumann entropy $H(B)_\\omega$, so the optimal strategy is simply Schumacher compression of Bob's half of the state. These results give the precise communication cost of simulating quantum correlations on average, with direct consequences for the resources needed to realize nonlocal games.","feed_headline":"Exact rates found for simulating quantum states on average","feed_subtitle":"Capacity formulas say how many bits or qubits three parties need to mimic a quantum correlation.","key_machinery":"The load-bearing object is the set of separable classical-quantum extensions, written $S(\\omega)$: a decomposition $\\sigma_{XYZABC}=\\sum_{x,y,z} p_{XYZ}(x,y,z) |x\\rangle\\langle x|\\otimes|y\\rangle\\langle y|\\otimes|z\\rangle\\langle z|\\otimes\\sigma^x_A\\otimes\\sigma^y_B\\otimes\\sigma^z_C$ whose $X$-register marginal reproduces the target $\\omega_{XABC}$. The capacity formulas are infima or unions over this set. The achievability side runs through a generic random-binning and covering lemma, combined with rate-splitting for the cascade; the converse side measures the output systems projectively and applies classical entropy-continuity bounds, with quantum continuity bounds in the quantum-link cases.","core_discovery":"The paper's central claim is that empirical coordination capacities for quantum states can be characterized exactly. For a two-node classical-link network, the capacity is $\\inf_{\\sigma} I(X;Y)_\\sigma$ over all separable classical-quantum extensions of the target state, and coordination is impossible when the target is entangled. For the three-node cascade, the capacity region is the union, over separable extensions $\\sigma_{XYZABC}$ with $\\sigma_{XABC}=\\omega_{XABC}$, of rate pairs $(R_{12},R_{23})$ satisfying $R_{12} \\ge I(X;YZ)_\\sigma$ and $R_{23} \\ge I(X;Z)_\\sigma$. With a quantum link, the two-node capacity is $H(B)_\\omega$, and in the broadcast network with receivers' side information the region is $Q_{12} \\ge H(B|X)_\\omega$, $Q_{23} \\ge H(C|Y)_\\omega$. A further theorem states that shared randomness before transmission does not change any of these optimal rates.","pith_inferences":["Editorial inference: the appearance of optimization over extensions suggests a general principle for classical-link quantum coordination: the capacity problem is a convex or variational computation over state decompositions, so the next practical bottleneck is algorithmic rather than conceptual.","Editorial inference: the results are asymptotic, and the finite-blocklength behavior of empirical coordination may require a different figure of merit than trace distance; a one-shot version of these capacity formulas is a natural open problem.","Editorial inference: the CHSH example indicates that even a small amount of shared entanglement, below the Bell-violation threshold, can sharply raise the winning probability because the payoff gradient is steep near zero entanglement; this suggests a testable prediction that coordination rates near the threshold suffice for near-maximal advantage in games with steep payoff functions.","Editorial inference: the broadcast theorem assumes Alice is ignorant of the receivers' questions; a variant where Alice receives the question side information before encoding would relax the non-signaling constraints (77) and likely change the rate region."],"forward_implications":["Any separable bipartite state can be coordinated over a single classical link at the minimal rate $\\inf I(X;Y)$ over extensions, while any entangled state cannot be coordinated at all.","In the cascade network, the second link only needs to cover the $X$-$Z$ correlation, while the first link must cover the full $X$-$YZ$ correlation, and the two constraints cannot be reduced to a single Markov chain condition.","Common randomness is never necessary for empirical coordination: every CR-assisted protocol can be derandomized without increasing the required rates.","For a two-node quantum link, the optimal rate is exactly the von Neumann entropy of Bob's reduced state, so classical compression wisdom carries over to average-state simulation.","In the nonlocal-game application, the resources needed to implement a game strategy are the conditional entropies from the broadcast theorem, and for the CHSH example a Bell violation appears once the coordination rate exceeds about $0.2643$ qubits per round."],"supporting_citations":[{"why":"Defines classical empirical coordination and supplies the random-binning and covering lemma on which the generic achievability scheme (Lemma 7) is built.","marker":"[16]"},{"why":"The authors' earlier strong-coordination results provide the achievability constructions used for the direct parts of the quantum-link theorems.","marker":"[60]"},{"why":"The original quantum noiseless coding theorem gives the achievability side of the two-node quantum-link capacity $H(B)_\\omega$.","marker":"[105]"},{"why":"Provides the entropy and continuity background used in the classical-link converse rate bounds.","marker":"[113]"},{"why":"Its Lemma 2.7 supplies the entropy-continuity estimate used to pass from measured output distributions to the target mutual information in the classical-link converses.","marker":"[114]"},{"why":"Provides the tight uniform continuity bound used in the quantum-link converse proofs.","marker":"[115]"}],"fun_headline_variants":["Exact coordination rates found for quantum networks","Cascade quantum coordination capacity characterized","Empirical coordination: optimal rates for quantum states","Shared randomness can't boost quantum coordination rates","Quantum coordination: exact rates for classical and quantum links"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classical-link converse proofs presume that measuring Bob's and Charlie's outputs in a fixed projective basis and then applying classical coordination inequalities can certify the true quantum rate; if non-orthogonal output states allow the real minimum to fall below the extension-optimization formula, those lower bounds would not be tight.","fun_headline_variants_meta":{"raw":{"variants":["Exact coordination rates found for quantum networks","Cascade quantum coordination capacity characterized","Empirical coordination: optimal rates for quantum states","Shared randomness can't boost quantum coordination rates","Quantum coordination: exact rates for classical and quantum links"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1321,"prompt_tokens":991,"completion_tokens":330,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":263}},"tokens_in":607,"tokens_out":330,"duration_ms":3505,"temperature":1.0,"reasoning_tokens":263,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:49:32.547289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search over separable two-qubit target states for one where the infimum over extensions of $I(X;Y)$ strictly exceeds the Holevo information of the induced ensemble; for such a target, the converse's measurement step cannot certify the claimed capacity. The decisive check is then to run a two-node empirical-coordination protocol at a rate below the claimed formula and test whether the empirical average state still converges to the target.","supporting_citations":[{"cited_title":"Quantum coordination rates in multi-user networks,","cited_arxiv_id":null,"evidence_quote":"The authors' earlier strong-coordination results provide the achievability constructions used for the direct parts of the quantum-link theorems."},{"cited_title":"Quantum coding,","cited_arxiv_id":null,"evidence_quote":"The original quantum noiseless coding theorem gives the achievability side of the two-node quantum-link capacity $H(B)_\\omega$."},{"cited_title":"A mathematical theory of communication,","cited_arxiv_id":null,"evidence_quote":"Provides the entropy and continuity background used in the classical-link converse rate bounds."},{"cited_title":"Csisz ´ar and J","cited_arxiv_id":null,"evidence_quote":"Its Lemma 2.7 supplies the entropy-continuity estimate used to pass from measured output distributions to the target mutual information in the classical-link converses."},{"cited_title":"Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints,","cited_arxiv_id":null,"evidence_quote":"Provides the tight uniform continuity bound used in the quantum-link converse proofs."}],"review_version":1}