{"id":"174da80f-803c-4e69-8cb9-fd12411a0122","arxiv_id":"2505.18539","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Non-diagonal unitary gates can have unequal multipartite entangling and disentangling powers, with the asymmetry appearing for even versus odd numbers of qubits.","lead":"This paper compares how much a quantum unitary operation can create versus destroy genuine multipartite entanglement, and finds cases where the two powers differ. The effect depends on whether the system has an even or odd number of qubits, which could matter for designing quantum circuits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed E_N≠D_N disparity for non-diagonal unitaries is supported only by unverified numerical optima; if the symmetry reductions in Construction 1 are not tight or ISRES missed the global maximum, the reported gap could vanish.","rationale":"After reading the paper in good faith, I find that the central existence claim (E_N≠D_N for non-diagonal U_ND(λ), with parity dependence) is a numerical observation rather than a proven theorem. The key step is the maximization in Eq. (1)–(2). The authors use ISRES and impose symmetry reductions in Construction 1 without proof. Because E_N and D_N are separate maxima over the same FS set, any asymmetric restriction can lower one more than the other, manufacturing an apparent disparity. This is the single most load-bearing point: the entire even/odd effect rests on the trustworthiness of these numerical optima. I could not find an internal inconsistency that would force the claim to be false; the unitaries are real orthogonal and their adjoints are transposes, and transposition is not a local operation, so an asymmetry is plausible. However, the evidence as presented is insufficient: no code, no data, no global-optimality certificates, and no analytical bounds. The reader's CONDITIONAL verdict is appropriate; the proposed test would either corroborate the claim or reveal a spurious gap. I therefore agree with the reader's assessment and recommend no change to the verdict.","tokens_in":17778,"tokens_out":10748,"duration_ms":94873,"concrete_test":"Recompute E_N and D_N for N=4 and N=6 at representative λ values (e.g., π/3, π/2, 2π/3) using the full separable-state parameterization (2N angles) and a multi-start global search (e.g., 10^5 random restarts followed by local refinement), cross-checked with an independent optimizer such as differential evolution or a dense grid. If the E_N−D_N gap at any of these λ values changes sign or falls below numerical tolerance (e.g., 10^-6), the reported disparity is an artifact of the restricted optimization. Also report and compare the optimized input states with the symmetries assumed in Construction 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—unequal multipartite entangling and disentangling powers for unitaries in Eq. (6) (even N) and Eq. (7) (odd N)—rests entirely on numerical maximization of the GGM over fully separable states. No analytical proof of the inequality is provided. The optimization is performed with ISRES, a stochastic evolutionary algorithm that does not certify global optimality, and the search is further restricted by unproven symmetry reductions: for U_ND(λ) in Construction 1 the optimal input is asserted to have the special form |ψ1> ⊗ (⊗_{i=2}^{2m+1}|ψi>) ⊗ |ψ_{2m+2}> with |ψ1>=|ψ_{2m+2}>, and for U†_ND(λ) the optimum is reduced to two parameters only. Since maximization over a restricted subset can only lower the computed value, an unjustified restriction can create a spurious gap between E_N and D_N if it is not equally tight for both. For instance, if the true global maximum of D_N is higher than the reported two-parameter value, the apparent E_N < D_N (or vice versa) would disappear under full optimization. In addition, the curves in Fig. 2 show no error bars or multi-start verification, and the scaling of the disparity with N is not quantified. Thus the parity-dependent phenomenon is not yet established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines multipartite entangling and disentangling powers of a unitary operator as the maximum GGM generated from fully separable states by U and by U†, respectively. It claims that diagonal unitaries have equal entangling and disentangling powers, while certain non-diagonal unitary families show a disparity that depends on the parity of the number of qubits: the family in Eq. (6) for even N and the family in Eq. (7) for odd N. The paper further claims that this asymmetry can be realized through nearest-neighbor Dzyaloshinskii–Moriya Hamiltonians for even N and a combination of Heisenberg and DM interactions for odd N, and that two-layer random circuits exhibit analogous behavior. The evidence for the non-diagonal and Hamiltonian claims is numerical, based on ISRES maximization of the GGM over fully separable states.","tokens_in":18076,"tokens_out":5961,"duration_ms":50344,"significance":"If the numerical results are correct, the paper identifies a novel and genuinely multipartite phenomenon: unitary operations can have different capacities for generating versus destroying genuine multipartite entanglement, with a qualitative dependence on the parity of the number of parties. This would extend the known bipartite entangling/disentangling disparity to the multipartite setting and connect it to physically realizable spin Hamiltonians. The paper is clearly written and introduces a natural multipartite definition based on the GGM. However, the central claims are not yet rigorously established: the proof for diagonal unitaries is incomplete, and the non-diagonal and Hamiltonian claims rest on unverified numerical optima and unproven symmetry reductions.","major_comments":[{"comment":"The central claim that E_N(U_ND(λ)) ≠ D_N(U_ND(λ)) for even N (and the analogous claim for odd N in Construction 2) rests entirely on ISRES numerical maximization without any global optimality certificate. The asserted symmetry reductions for the optimal inputs (|ψ_1⟩ = |ψ_{2m+2}⟩ for U_ND, and the two-parameter reduction for U_ND†) are not proved; since restricting the search space can only lower the computed maximum, an asymmetric restriction could create a spurious gap. For instance, if the true global maximum of D_N is higher than the reported two-parameter value, the apparent E_N < D_N could disappear under full optimization. Please provide an analytical proof of the inequality (e.g., separate lower and upper bounds on the two quantities) or certified global optimization results such as multi-start statistics, a second independent optimizer, or interval-based bounds.","section":"Sec. III.1, Construction 1 and Fig. 2"},{"comment":"The claimed proof that E_N(U_d,ϕ) = D_N(U_d,ϕ) for the single-parameter diagonal unitaries is incomplete. The coefficients α_k(N), β(N), γ_k(N) in Eq. (B9) are never specified, and the reduction to equal angles θ_1 = ... = θ_N is justified only by a numerical observation ('We numerically find out'). Moreover, for N ≥ 4 the GGM depends on the largest eigenvalues of reduced density matrices over all subsystem sizes up to N/2, whereas the proof analyzes only single-party reduced density matrices ρ_i. Consequently Proposition 2 is not established by the given argument; the authors should either supply a complete derivation or explicitly present this as a numerical conjecture with adequate supporting evidence.","section":"Appendix B 2, Eq. (B9) and Proposition 2"},{"comment":"The parity-dependent asymmetry reported for the Hamiltonian-generated unitaries U_DM and U_DM,H is supported only by plots of Δ_N, without quantitative specification of the optimized values, and the statement in footnote 1 that the optimal fully separable states coincide with those of Construction 1 is unproven. Because the same global-maximization issue applies here, the physical claim that these Hamiltonians simulate the disparity is not yet established. Please report the numerical values, the number of independent optimization runs, and the sensitivity of the optima to starting points, or provide an analytical argument that the reported Δ_N is a rigorous lower bound on |E_N − D_N|.","section":"Sec. III.2, Hamiltonian simulations"}],"minor_comments":[{"comment":"The definition of the generalized geometric measure as 1 − max{...} over sets of maximum eigenvalues of l-site reduced density matrices is ambiguous; the maximum should be taken over all eigenvalues of all reduced density matrices for subsystem sizes l = 1, ..., ⌊N/2⌋. Please restate the formula more precisely.","section":"Appendix A 1"},{"comment":"The definition of U_w is difficult to parse (e.g., 'ω2' appears to be a typesetting error for ω^2, and the orthogonality of |β_t⟩ and |γ_t⟩ is stated tersely). Please provide an explicit matrix representation for U_w or a clearer basis notation to ensure reproducibility.","section":"Construction 2, Eq. (7)"},{"comment":"The sentence about the qubit-qutrit entangling power reaching its maximum while the disentangling power remains lower should cite Ref. [45] at that point, rather than only in the introduction.","section":"Sec. III.1"},{"comment":"The curves for E and D are visually close in some λ-ranges; adding markers or an inset would make the disparity clearer. The absence of any indication of optimization uncertainty (e.g., error bars or number of runs) is also a concern, though this is already noted in the major comments.","section":"Fig. 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a timely and interesting question, and the authors have made a reasonable first attempt by constructing explicit unitary families and connecting them to spin Hamiltonians. However, the central claims are supported almost entirely by numerical optimization without global convergence guarantees, and the analytical proof for diagonal unitaries is visibly incomplete. Given the authors' resources, it should be feasible to strengthen the paper substantially, e.g., by providing an analytical proof for at least one family, or by performing a certified or multi-start global optimization with detailed reporting. I would not recommend rejection at this stage, but the revision must address the global-optimality and proof-completeness issues before the results can be considered established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper reports a genuinely new phenomenon—for certain non-diagonal multipartite unitaries, the maximum GGM generated from product states differs from that generated by the adjoint, and the behavior flips depending on whether N is even or odd. The constructions (Eq. 6 for even N, Eq. 7 for odd N) are explicit, and the even/odd distinction is not in the prior literature. The paper also shows the asymmetry can be realized with DM and Heisenberg Hamiltonians, and with two-layer random circuits. That is a real and interesting contribution to the entangling-power literature.\n\nWhat it does well: the definitions are clean, the GGM-based powers are natural extensions of the bipartite notion, and the numerical work is extensive—diagonal unitaries (where they find equality), the two non-diagonal families, Haar-random unitaries, and time-dependent Hamiltonians. The figures show clear gaps over finite parameter ranges, not just isolated points. The authors are honest that the non-diagonal results are numerical.\n\nWhere it is soft. First, the proof of Proposition 2 (diagonal unitaries in arbitrary N) is incomplete as written: the coefficients α_k(N), β(N), γ_k(N) in Eq. (B9) are never specified, and the 'numerically find that θ1=...=θN' step is not an analytical derivation. This matters less because the diagonal equality is not the paper's main claim, but a proof should be a proof. Second, the central E≠D claim for non-diagonal unitaries is entirely dependent on ISRES finding global maxima over the fully separable set. The symmetry reductions in Construction 1 are asserted, not proven; if the two-parameter reduction for U† is not tight, the computed D could be too low and the gap spurious. The authors give no code, no data, and no multi-start comparison. Random unitary scatter (Fig. 5) and the Hamiltonian results partially mitigate this—the phenomenon seems robust—but I would like to see at least a semi-analytical argument or a certification.\n\nIf I were refereeing, I would ask for: explicit optimized states and their GGM values, multi-start or alternative optimization (e.g., differential evolution with many restarts) for a few representative λ, and a complete or corrected proof of Prop 2. None of these seem impossible; the paper's core idea is sound.\n\nWho is it for: quantum information theorists working on entangling power, circuit design, and entanglement dynamics in spin chains. It deserves a serious referee; the referee should push on the numerical evidence.","headline":"New multipartite entangling/disentangling asymmetry with a parity effect, numerically supported but not yet analytically certified; worth refereeing with requests for stronger evidence.","tokens_in":18606,"tokens_out":3022,"would_cite":true,"duration_ms":27651,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","81P45","81P68"],"pacs":["03.67.Mn","03.65.Ud"],"model":"deepseek-v4-flash","headline":"Certain non-diagonal unitaries have unequal multipartite entangling and disentangling powers, with the construction depending on whether the number of qubits is even or odd.","keywords":["multipartite entanglement","entangling power","disentangling power","genuine multipartite entanglement","generalized geometric measure","Dzyaloshinskii-Moriya interaction","random unitary circuits"],"falsifier":"Evaluate $E_4(U_{ND}(\\lambda))$ and $D_4(U_{ND}(\\lambda))$ at $\\lambda=\\pi/3$ using a dense grid over all four single-qubit Bloch angles followed by local refinement; if the two maxima agree within numerical precision, or if any fully separable state beats the reported value for either quantity, the parity-dependent disparity reported for even $N$ is an artifact of the optimization rather than a property of the unitary.","tokens_in":17566,"feed_emoji":"⚛️","tokens_out":13054,"duration_ms":97909,"temperature":0.7,"pith_summary":"This paper asks whether a unitary operation on $N$ qubits has the same capacity to generate genuine multipartite entanglement from a fully separable state as its adjoint does, a question that matters because entanglement creation and erasure are both resources in quantum information processing. The answer it argues is no in general: a class of diagonal unitaries preserves the equality, but for two constructed families of non-diagonal unitaries the entangling power $E_N(U)$ and the disentangling power $D_N(U)=E_N(U^\\dagger)$ become unequal when the optimization is over fully separable inputs. The new observation is a parity effect: one family realizes the imbalance for even $N$, a different family is needed for odd $N$, and the same behavior is found for Hamiltonian evolutions with nearest-neighbor Dzyaloshinskii-Moriya (even $N$) or Heisenberg plus DM interactions (odd $N$), as well as for two-layer random brickwork circuits.","feed_headline":"Some unitaries entangle more powerfully than they disentangle","feed_subtitle":"Which gates show the gap depends on whether the system has an even or odd number of qubits.","key_machinery":"The load-bearing object is the pair of quantities $E_N(U)$ and $D_N(U)=E_N(U^\\dagger)$, computed from the generalized geometric measure, which for a pure state is $G(|\\psi\\rangle)=1-\\max|\\langle\\phi|\\psi\\rangle|^2$ over all states that are not genuinely multipartite entangled and can be evaluated from the largest Schmidt coefficients of the reduced density matrices. The constructions that break the symmetry are brickwork-style global unitaries built from two-qubit gates: $U_{ND}(\\lambda)$ of Eq. (6) alternates the same two-qubit gate $U_{i,i+1}(\\lambda)$ on odd and even bonds, while the odd-$N$ family of Eq. (7) supplements this with a specially designed two-qubit gate $U_w$ built from non-orthogonal single-qubit states. The paper points to the noncommutativity of the odd and even layers as the source of the directional asymmetry, since for diagonal unitaries no such noncommuting structure exists and the powers match.","core_discovery":"On its own terms, the paper establishes that multipartite entangling and disentangling powers, defined via the generalized geometric measure (GGM) as $E_N(U)=\\max_{|\\psi\\rangle\\in S_N} G(U|\\psi\\rangle)$ and $D_N(U)=E_N(U^\\dagger)$, are equal for diagonal unitaries: Proposition 1 and Proposition 2 prove this for the single-phase family $U_{d,\\phi}=\\mathrm{diag}(1,\\dots,1,e^{i\\phi})$, and numerical sampling supports it for arbitrary diagonal unitaries on three, four, and five qubits. The central discovery is that this equality is not generic. For the non-diagonal family $U_{ND}(\\lambda)$ of Eq. (6), built from alternating layers of the two-qubit gate $U_{i,i+1}(\\lambda)$, the paper finds $E_N^{\\mathrm{even}}\\neq D_N^{\\mathrm{even}}$ for even $N$ (shown for $N=4,6$ and checked up to $N=10$), while the same family gives equal powers for odd $N$. For odd $N$, replacing one layer with the two-qubit gate $U_w$ of Eq. (7) restores the disparity (shown for $N=3,5$). These inequalities are reported as numerical results, obtained by maximizing GGM over fully separable states, with no analytical proof presented.","pith_inferences":["If the numerical inequalities survive more thorough optimization, a practical consequence follows: circuit designers who want to use a gate to erase genuine multipartite entanglement cannot assume that running the inverse evolution is equivalent to reversing the resource dynamics.","The parity dependence may be tied to the bipartition structure of the brickwork geometry; a testable extension is to ask whether the odd-$N$ effect disappears if $U_w$ is replaced by any local-unitary equivalent, which would indicate that only the global layer structure matters.","A natural next check is whether the same even-odd asymmetry appears when the generalized geometric measure is replaced by another genuine-multipartite-entanglement measure, such as one based on von Neumann entropy; if it does not, the disparity is measure-dependent rather than an operational feature of the unitaries.","The Hamiltonian simulations suggest a concrete experimental signature: in a chain with DM interactions, the amount of genuine multipartite entanglement generated by $e^{-iHt}$ and by $e^{+iHt}$ from the best product states should differ for times in specific windows, which could be tested in cold-atom or trapped-ion platforms."],"forward_implications":["Diagonal unitary operators do not distinguish entangling from disentangling direction, so resource generation and resource erasure are equally easy for them.","For even numbers of qubits, two identical noncommuting layers of the same two-qubit gate are enough to create the disparity; for odd numbers, the two layers must differ.","The disparity can be produced by physical Hamiltonians: nearest-neighbor Dzyaloshinskii-Moriya interactions for even $N$, and a combination of Heisenberg and DM interactions for odd $N$, at evolution times away from multiples of $\\pi/2$ (or $\\pi/4$ in the mixed case).","Random two-qubit Haar gates arranged in two alternating layers reproduce the effect, meaning the inequality is not an artifact of a specially fine-tuned gate alone.","In the random-circuit setting, even-$N$ systems show the asymmetry with identical gate sets in both layers, while odd-$N$ systems require distinct gate sets, sharpening the parity contrast."],"supporting_citations":[{"why":"Supplies the original definition of entangling power that the paper generalizes to the multipartite setting.","marker":"[42]"},{"why":"Provides the bipartite example of unequal entangling and disentangling powers that this work extends, including the $U_w$ gate construction.","marker":"[45]"},{"why":"Gives the generalized geometric measure used to quantify genuine multipartite entanglement in the definitions of $E_N$ and $D_N$.","marker":"[62]"},{"why":"Defines multipartite entangling power and its hierarchy, giving the optimization-based framework the paper adopts.","marker":"[72]"},{"why":"Extends the entangling-power framework to imperfect gates, supporting the numerical maximization convention used here.","marker":"[73]"},{"why":"Defines the geometric measure of entanglement on which the GGM computation is based.","marker":"[84]"},{"why":"Introduces the Dzyaloshinskii-Moriya interaction used in the even-$N$ Hamiltonian simulation.","marker":"[75]"},{"why":"Formulates the DM interaction Hamiltonian used alongside Heisenberg coupling for the odd-$N$ simulation.","marker":"[76]"}],"fun_headline_variants":["Even qubits unlock one-sided entanglement power of unitaries","Which unitaries entangle more than they disentangle? Parity decides","Even-number qubits: entangling power exceeds disentangling for some gates","Unequal entangling vs disentangling: even-odd qubit parity is key","Even or odd: the key to a gate's entanglement asymmetry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim rests on the assumption that the numerical maximization of the generalized geometric measure over all fully separable input states returns the true global maximum for both $U$ and $U^\\dagger$; the paper gives no analytical proof of the inequalities for the non-diagonal families, so a missed better input state for either quantity could erase the reported disparity.","fun_headline_variants_meta":{"raw":{"variants":["Even qubits unlock one-sided entanglement power of unitaries","Which unitaries entangle more than they disentangle? Parity decides","Even-number qubits: entangling power exceeds disentangling for some gates","Unequal entangling vs disentangling: even-odd qubit parity is key","Even or odd: the key to a gate's entanglement asymmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000957,"raw_usage":{"total_tokens":4124,"prompt_tokens":1035,"completion_tokens":3089,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":2997}},"tokens_in":651,"tokens_out":3089,"duration_ms":20309,"temperature":1.0,"reasoning_tokens":2997,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:29:14.955223+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $E_4(U_{ND}(\\lambda))$ and $D_4(U_{ND}(\\lambda))$ at $\\lambda=\\pi/3$ using a dense grid over all four single-qubit Bloch angles followed by local refinement; if the two maxima agree within numerical precision, or if any fully separable state beats the reported value for either quantity, the parity-dependent disparity reported for even $N$ is an artifact of the optimization rather than a property of the unitary.","supporting_citations":[{"cited_title":"Zanardi, Entanglement of quantum evolutions, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the original definition of entangling power that the paper generalizes to the multipartite setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines multipartite entangling power and its hierarchy, giving the optimization-based framework the paper adopts."},{"cited_title":"Cleve, I","cited_arxiv_id":null,"evidence_quote":"Extends the entangling-power framework to imperfect gates, supporting the numerical maximization convention used here."},{"cited_title":"Vazirani and A","cited_arxiv_id":null,"evidence_quote":"Introduces the Dzyaloshinskii-Moriya interaction used in the even-$N$ Hamiltonian simulation."},{"cited_title":"Gottesman and I","cited_arxiv_id":null,"evidence_quote":"Formulates the DM interaction Hamiltonian used alongside Heisenberg coupling for the odd-$N$ simulation."}],"review_version":1}