{"id":"6348b3d0-7b01-4b78-82ba-4d2704cdd18c","arxiv_id":"2508.08092","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Classical and quantum agents can rank the same two adaptive strategies in opposite order of memory-based complexity, and the paper provides sufficient conditions for when this happens.","lead":"This paper shows that which of two strategies counts as more complex can flip depending on whether the agent executing it stores memory classically or quantum mechanically. It defines a quantity called channel excess entropy that lower-bounds the memory cost of any such agent, and gives simple conditions that certify when classical and quantum agents must disagree on the ordering.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (21) for Q_B is internally inconsistent: at α=1 it yields nonzero complexity for a detector that never registers a particle, so Scenario A's numerical results are unverified.","rationale":"The paper's core structural result — E^A_I = E_J − E_I ≤ Q^A_I ≤ C^A_I (Result 2) and the sufficient conditions Results 3 and 4 — appear sound; the reader's algebraic check and our own examination support this. The reader's weakest-assumption concern (restricted quantum agent class and fidelity-based optimality) is legitimate but not fatal to the existence claim, because some demonstrations use the computed Q only as an upper bound in the direction that is robust to further lowering (e.g., Scenario A's Q_A > Q_B remains if Q_B is an upper bound). The overstrong 'any agent' phrasing in the abstract is a separate issue. However, our independent check found a concrete internal inconsistency in the central example: Eq. (21) gives a nonzero quantum complexity for the α=1 limit where the process is trivial and Q_B must be 0. Direct diagonalization using the paper's own quantum states and stationary distribution disagrees with Eq. (21) (e.g., 0.516 vs 0.700 bits at r=0.2, α=0.5). This does not by itself falsify the existence of ambiguity — a corrected calculation still shows opposite signs in parts of parameter space — but it invalidates the numerical values, the plotted ranges, and any directional statement that depends on the exact Q_B formula. The right response is to keep the CONDITIONAL verdict: the manuscript must correct Eq. (21) and Appendix F and regenerate all Scenario A figures before the quantitative claims can be accepted.","tokens_in":29376,"tokens_out":37592,"duration_ms":385453,"concrete_test":"Evaluate Eq. (21) at α=1, r∈(0,1) and check whether it returns 0; a correct formula must, because with α=1 the noisy detector always outputs 0 and has a single causal state. Then independently recompute Q_B as the von Neumann entropy of ρ = b|σ1⟩⟨σ1|+(1−b)|0⟩⟨0| with b=1/(1+(1−r)(1−α)), |σ1⟩=√α|0⟩+√(1−α)|1⟩, and compare with Eq. (21) across the (α,r) grid used in Figs. 10–12; if the two disagree, regenerate the figures and re-evaluate Result 5's direction with the corrected values.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central concrete demonstration (Scenario A) relies on the closed-form quantum complexity of Bob's dead-time detector, Eq. (21): Q_B = h((c − √(c+d))/(2c)) with c=2+r−αr and d=−4r(2−4α+2α^2). This formula fails a trivial limit: at α=1 the detector never outputs 1 (it always 'misses' and stays in state 1), so the strategy is the constant-zero process, whose ε-transducer has one state and whose classical and quantum complexities are both 0. Substituting α=1 into Eq. (21) gives c=2, d=0, argument=(2−√2)/4≈0.146, so Q_B≈0.61 bits. Recomputing from the paper's own states (Appendix F), the average memory is ρ = b|σ1⟩⟨σ1|+(1−b)|0⟩⟨0| with |σ1⟩=√α|0⟩+√(1−α)|1⟩, b=1/(1+(1−r)(1−α)); its smaller eigenvalue is (1+s−√((1−s)^2+4αs))/(2(1+s)) with s=(1−r)(1−α). For r=0.2, α=0.5, this yields Q_B≈0.516, not the ≈0.700 from Eq. (21). Consequently the plotted Q_B curves, the stated ambiguity ranges α∈0.3–0.68 (Fig. 11) and r∈0.12–0.26 (Fig. 12), and any directional claim such as Result 5 are not supported as written. The general inequalities (Result 2) and sufficient conditions (Results 3–4) are unaffected; the issue is the example arithmetic, which is nevertheless the paper's primary quantitative evidence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the channel excess entropy E^A_I for causal input-output processes and proves (Propositions 3–5, Result 2) that it lower-bounds both the classical statistical complexity C^A_I and the quantum complexity Q^A_I (in the quantum-agent class of Elliott et al.) of an agent executing a strategy A under input I. Using the decomposition E^A_I = E_J − E_I and the inequalities E ≤ Q ≤ C, the authors derive sufficient conditions (Results 3 and 4) for classical and quantum agents to rank two strategies in opposite orders. They demonstrate the phenomenon in three analytic scenarios: two detectors driven by a biased coin (Result 5), one investor-type transducer driven by two IID inputs (Result 7), and an agent versus its operational inverse (Result 9). A final section shows that the agent-level ambiguity need not be reflected in the statistics of the output processes alone.","tokens_in":29727,"tokens_out":18941,"duration_ms":214895,"significance":"If the technical results stand, the paper makes a useful conceptual contribution: relative memory-based complexity of strategies is not intrinsic but depends on whether the executing agent stores classical or quantum information, and the channel excess entropy gives a universal lower bound on the required memory. The sufficient conditions in Results 3 and 4 are elegant and practically useful, since they let one detect ambiguity using only excess entropies and one known quantum complexity. The analytic constructions are transparent, and the paper explicitly acknowledges the prior thesis [32] in defining E^A_I. However, the primary quantitative demonstration in Scenario A is undermined by an incorrect closed-form expression for Q_B (Eq. (21)), so the paper as written does not support the stated numerical ranges and plots. The general framework and the other scenarios are not affected by this arithmetic error, but the example needs to be corrected before the claims can be accepted as presented.","major_comments":[{"comment":"The closed-form formula for Bob's quantum complexity is incorrect. At α=1, the detector never outputs 1 and its stationary state is the single state |σ1>=|0>, so both classical and quantum complexities must vanish. Substituting α=1 into Eq. (21) gives c=2, d=0, and Q_B=h((2−√2)/4)≈0.61 bits. Also at α=0 the formula does not reduce to Q_B=C_B=h(b) with b=1/(1+(1−r)(1−α)), as it must for orthogonal states. From the states in Eq. (20) and the stationary weight b, the correct eigenvalues of ρ=b|σ1><σ1|+(1−b)|0><0| are (1±√(1−4b(1−b)(1−α)))/2, so Q_B=h((1−√(1−4b(1−b)(1−α)))/2). This error propagates to Figures 10–13 and the stated ambiguity intervals α∈0.3–0.68 and r∈0.12–0.26. The sufficient-condition framework of Section III is not affected, but the example's quantitative evidence must be recomputed.","section":"§IV.A, Eq. (21); Appendix F, Eq. (F8)"}],"minor_comments":[{"comment":"In the proof of Proposition 3, the entropy terms should involve the causal-state variable S: the right-hand side should read H[S|⇀X]−H[S|⇀X,⇀Y], and the following line should be H[S|⇀X] ≤ H[S]. As written, the joint past ↼(X,Y) appears where S is intended.","section":"Appendix C, Eq. (C9)"},{"comment":"The statement 'C_B⃗q_I = Q_B⃗q_I = C_A^I − Q_A^I + ε, for all ε∈[Q_A^I,C_A^I]' is dimensionally inconsistent: as ε ranges over [Q_A^I,C_A^I], the quantity C_A^I−Q_A^I+ε ranges over [C_A^I, C_A^I+C_A^I−Q_A^I], not [Q_A^I,C_A^I]. It should be formulated as 'for every value v∈[Q_A^I,C_A^I] there is a parameter choice with C=Q=v' (or 'Q_A^I+ε with ε∈[0,C_A^I−Q_A^I]').","section":"Result 6"},{"comment":"The statement that for the previous section's examples 'the complexities of the output process follow the complexities of the input-output processes for IID inputs' appears incorrect at least for Alice's delay detector: the output process is an IID biased coin with zero statistical complexity, while C_A^I=h(r)>0. Please clarify what is meant by 'follow'.","section":"§V, first paragraph"},{"comment":"The claims about 'any agent' and 'quantum agents' are stated very broadly. The universal lower bound E^A_I≤Q^A_I is robust, but the specific quantum complexities in the examples are optimized only within the restricted agent class of Ref. [3] (pure memory states in one-to-one correspondence with causal states, preserved input tape, projective measurements). A sentence qualifying this scope would improve precision.","section":"Abstract and §II.D"}],"recommendation":"major_revision","confidential_remarks":"The skeptical report's Eq. (21) concern is valid; I checked the α=1 and α=0 limits and the formula is clearly inconsistent with the paper's own encoding. The core results (channel excess entropy bound, decomposition, sufficient conditions) appear sound and are not invalidated by this arithmetic error, so the paper is fixable. The authors should re-derive Q_B, update the affected figures and text, and carefully re-check the other numerical claims before acceptance. No concerns about attribution or novel disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the core idea is good and the structural results are worth a look, but the main quantitative example has an arithmetic error that invalidates the reported numbers.\n\nWhat is actually new: channel excess entropy E^A_I, the decomposition E^A_I = E_J - E_I for causal channels, and the sufficient conditions in Results 3 and 4. Generalizing the ambiguity-of-simplicity phenomenon from passive stochastic processes to adaptive input-output processes is a natural step, and the paper is honest about inheriting the epsilon-transducer framework and the quantum-agent class from prior work. The lower bound E^A_I ≤ Q^A_I ≤ C^A_I is a useful addition.\n\nThe soft spot: Eq. (21) for Q_B in Scenario A is not the entropy of the paper's own quantum state. At α=1, Bob's detector always outputs 0 and stays in state 1, so the process is a constant zero and Q_B must be 0. The formula gives h((2−√2)/4) ≈ 0.61 bits. Recomputing from their |σ1⟩, |σ2⟩ at r=0.2, α=0.5 gives Q_B ≈ 0.516 bits, not the ≈0.700 reported. That makes the plotted curves, the ambiguity ranges, and Result 5 as demonstrated unsupported. The structural inequalities and sufficient conditions do not depend on this formula, but Scenario A is the paper's flagship illustration, so this is a major flaw, not a typo.\n\nSecondary issues: the abstract claims the bound applies to 'any agent', but the proof is for the restricted quantum-agent class of assumptions (i)-(iv) in Sec. II.D. Appendix C's proof of Proposition 3 has a conditional-DPI step that misstates equalities; the intended inequality chain works. Result 6's statement seems to contradict its own range (the text says 'for any value in [Q^A_I, C^A_I]' but the formula says C^A_I − Q^A_I + ε, which would exceed the upper range); Appendix H supports the construction, so this is fixable. Scenarios B and C I did not fully recheck; the reader's independent checks suggest they may hold, but all numerical examples need a careful redo after Scenario A's error.\n\nVerdict: the paper is not desk-reject material. The conceptual claim is important enough and the structural results seem sound. It needs major revision before publication.","headline":"Good structural idea, but the flagship numerical example has an arithmetic error that invalidates the plotted results.","tokens_in":30336,"tokens_out":4110,"would_cite":false,"duration_ms":41350,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that adaptive strategies' relative memory complexity can reverse between classical and quantum agents, with channel excess entropy as a universal lower bound that diagnoses when reversal occurs.","keywords":["quantum complexity","statistical complexity","input-output processes","adaptive agents","channel excess entropy","quantum memory advantage","classical-quantum ambiguity","complexity ordering"],"falsifier":"Take Bob's noisy dead-time detector and search over all quantum circuits, not just the fidelity-saturating two-state encoding, for a faithful implementation whose average memory entropy falls below the claimed $Q^B_I = h\\!\\left(\\frac{c-\\sqrt{c+d}}{2c}\\right)$. Finding one would refute the claimed optimality in that example; proving that every faithful implementation has entropy at least this value would confirm it.","tokens_in":29147,"feed_emoji":"⚛️","tokens_out":8017,"duration_ms":89290,"temperature":0.7,"pith_summary":"The paper tries to establish that \"which of two adaptive strategies is more complex\" has no substrate-independent answer: an agent storing classical bits can rank strategies one way, while an agent storing quantum information can rank them the opposite way. To make this precise, the authors introduce the channel excess entropy, $E^A_I = E_J - E_I$, the mutual information between a strategy's joint input-output past and its future output given future input, and prove it lower-bounds both the classical statistical complexity and the quantum complexity of any causal strategy. They then derive a sufficient condition for reversal—if $C^A > C^B$ but $E^B > Q^A$, then $Q^A < Q^B$—and exhibit explicit reversals for two agents facing the same input, one agent facing two inputs, and an agent versus its operational inverse. If correct, complexity rankings are not intrinsic; they depend on the physics of the memory doing the computation, and the channel excess entropy is a universal floor on required memory.","feed_headline":"Quantum memory reverses which strategy looks more complex","feed_subtitle":"Channel excess entropy, a new lower bound, predicts when classical and quantum memory costs give opposite rankings.","key_machinery":"The central object is the channel excess entropy, $E^A_I = I[\\text{joint past}; \\text{future output} \\mid \\text{future input}]$, which for causal channels decomposes as $E^A_I = E_J - E_I$. It functions as a universal lower bound on both classical and quantum memory costs, so the interval $[E^A_I, C^A_I]$ is the room in which quantum encodings can act. The optimal quantum encodings are built by minimising the von Neumann entropy of the average memory state subject to a maximum-fidelity constraint on the overlaps between quantum causal states; saturating that constraint is how each example's claimed quantum complexity is certified.","core_discovery":"The paper's central claim is that the relative memory cost of executing an adaptive strategy is not an intrinsic property of the strategy: it shifts when the agent's memory is quantum. Formally, for any causal input-output process $A$ driven by an input process $I$, the channel excess entropy $E^A_I = E_J - E_I$ satisfies $E^A_I \\leq Q^A_I \\leq C^A_I$, where $C^A_I$ is the Shannon entropy of the strategy's causal-state distribution and $Q^A_I$ is the von Neumann entropy of the average quantum memory state. Because the two complexities can sit at different heights inside this interval, the order of two strategies can be $C^A_I > C^B_I$ while $Q^A_I < Q^B_I$. The authors prove this can happen","pith_inferences":["The result implies that common complexity comparisons—which behavior is more sophisticated—are ambiguous unless the storage substrate is specified; a complexity label may flip under an upgrade to quantum memory.","The same machinery could be applied to reward-driven agents: the memory needed to attain a given expected reward might exhibit the same classical-quantum reversal, since the paper notes this as an open direction.","The single-agent two-input example suggests that even relabelling or reweighting an input distribution can reverse quantum-classical rankings, so experimental demonstrations could use a fixed physical device and only change the input statistics.","Channel excess entropy is substrate-independent, so it offers a candidate scalar measure for comparisons when classical and quantum rankings conflict."],"forward_implications":["For any two strategies $A$ and $B$ driven by inputs, if $C^A_I > C^B_I$ and $E^B_I > Q^A_I$, the order flips: $Q^A_I < Q^B_I$.","Strategies with deterministic transitions, such as delay detectors, have equal classical and quantum complexities for every input, making them stable reference points in comparisons.","A single strategy can be judged simpler under one input and more complex under another, depending only on whether the agent uses classical or quantum memory.","Executing a strategy and executing its operational inverse can reverse complexity ranking between classical and quantum agents.","No agent, classical or quantum, can execute a causal strategy with less memory than its channel excess entropy; quantum memory can close but not breach that gap."],"supporting_citations":[{"why":"Defines input-output processes as channels and the epsilon-transducer, the minimal classical presentation whose Shannon entropy is the classical complexity $C$.","marker":"[1]"},{"why":"Supplies the quantum encoding of causal states used to construct quantum agents for input-output processes.","marker":"[2]"},{"why":"Defines the restricted quantum-agent class and the maximum-fidelity optimality criterion used to certify each example's quantum complexity.","marker":"[3]"},{"why":"Establishes the analogous ambiguity of simplicity for stochastic-process prediction, which this paper generalises to adaptive strategies.","marker":"[4]"},{"why":"Provides experimental and theoretical support that classical and quantum statistical complexity can diverge, motivating the ambiguity phenomenon.","marker":"[5]"},{"why":"Establishes that statistical complexity lower-bounds excess entropy for stochastic processes, the template for the channel version.","marker":"[8]"},{"why":"Shows quantum memory can reduce the complexity of classical models and introduces the excess-entropy lower bound for quantum models.","marker":"[13]"},{"why":"Relates distinguishability of quantum states to von Neumann entropy, used to compute quantum complexity from Gram-matrix eigenvalues.","marker":"[21]"},{"why":"Supplies the information-theoretic bound on quantum messages used to show channel excess entropy lower-bounds quantum complexity.","marker":"[22]"},{"why":"Supplies the data-processing inequality used to prove channel excess entropy lower-bounds classical complexity.","marker":"[24]"}],"fun_headline_variants":["Quantum memory flips strategy complexity order","Quantum agents invert which strategy is more complex","Quantum memory reverses complexity ranking of strategies","Quantum memory changes which strategy needs more memory","Quantum memory flips classical complexity ranking"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The arguments assume inputs cannot be influenced by future outputs (causality) and that the best quantum agent is one of the specially restricted class of pure-state memory devices considered here; if a quantum device outside that class uses less memory, some example numbers change.","fun_headline_variants_meta":{"raw":{"variants":["Quantum memory flips strategy complexity order","Quantum agents invert which strategy is more complex","Quantum memory reverses complexity ranking of strategies","Quantum memory changes which strategy needs more memory","Quantum memory flips classical complexity ranking"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000564,"raw_usage":{"total_tokens":2479,"prompt_tokens":681,"completion_tokens":1798,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":1735}},"tokens_in":425,"tokens_out":1798,"duration_ms":15046,"temperature":1.0,"reasoning_tokens":1735,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T21:43:34.907652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take Bob's noisy dead-time detector and search over all quantum circuits, not just the fidelity-saturating two-state encoding, for a faithful implementation whose average memory entropy falls below the claimed $Q^B_I = h\\!\\left(\\frac{c-\\sqrt{c+d}}{2c}\\right)$. Finding one would refute the claimed optimality in that example; proving that every faithful implementation has entropy at least this value would confirm it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines input-output processes as channels and the epsilon-transducer, the minimal classical presentation whose Shannon entropy is the classical complexity $C$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the quantum encoding of causal states used to construct quantum agents for input-output processes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the restricted quantum-agent class and the maximum-fidelity optimality criterion used to certify each example's quantum complexity."},{"cited_title":"Note that this algorithm will give an input-output pro- cess that mapsOtoIbut there may be undefined tran- sitions, which can are essentially free parameters","cited_arxiv_id":null,"evidence_quote":"Establishes the analogous ambiguity of simplicity for stochastic-process prediction, which this paper generalises to adaptive strategies."},{"cited_title":"Error-tolerant witnessing of divergences in classical and quantum statistical complexity","cited_arxiv_id":"1711.03661","evidence_quote":"Provides experimental and theoretical support that classical and quantum statistical complexity can diverge, motivating the ambiguity phenomenon."},{"cited_title":"Appendix J: Excess entropies for scenario A We derive the excess entopies for scenario A in Section IV A","cited_arxiv_id":null,"evidence_quote":"Establishes that statistical complexity lower-bounds excess entropy for stochastic processes, the template for the channel version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows quantum memory can reduce the complexity of classical models and introduces the excess-entropy lower bound for quantum models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Relates distinguishability of quantum states to von Neumann entropy, used to compute quantum complexity from Gram-matrix eigenvalues."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the information-theoretic bound on quantum messages used to show channel excess entropy lower-bounds quantum complexity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the data-processing inequality used to prove channel excess entropy lower-bounds classical complexity."}],"review_version":1}