{"id":"1629e5c3-4b45-4e84-9deb-a3cbd3b02694","arxiv_id":"2608.04486","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The label variance of a symmetry Casimir is a state-resolved quantum-information diagnostic that separates solvable from mixed eigenstates in partial dynamical symmetry, unlike bipartite entanglement magnitude.","lead":"This paper shows that a partial dynamical symmetry in the interacting boson model leaves a sharp state-by-state fingerprint: the variance of a symmetry Casimir is exactly zero on the solvable states and large on the mixed states. A generalist might read it because it connects algebraic nuclear-structure symmetries to quantum information measures and shows how to prepare and read that fingerprint on a quantum device.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's injectivity premise is false: SU(3) irreps (6,0) and (0,6) share a Casimir eigenvalue at N=6, so Var=0 does not imply a single irrep label.","rationale":"The reader correctly identified the injectivity of Casimir eigenvalues as the weakest assumption in Theorem 1, but assessed it as 'probably true.' Our stress-test shows it is actually false for SU(3) at N=6: the conjugate irreps (6,0) and (0,6) are both present in the model space and share the same quadratic Casimir eigenvalue. This is a concrete counterexample to the theorem, not a missing proof, so the central equivalence underpinning the label-variance diagnostic is invalid as stated. The paper's numerical results may survive if the specific Leviatan Hamiltonians do not mix conjugate irreps, but the paper provides no such check; the reported solvable counts, e.g., 22 at N=6, could in principle be inflated by zero-variance mixed states. Because the flaw is in a supporting theorem and the fix is clear (use block purity/coherence, or restrict to cases with injective Casimir), the reader's CONDITIONAL verdict remains appropriate, but the condition is now known to be unmet in the paper's current form. The concrete test would settle whether the specific numerical claims are affected.","tokens_in":12034,"tokens_out":21899,"duration_ms":176403,"concrete_test":"At N=6, construct the L=0, M=0 states of the (6,0) and (0,6) irreps, both of which appear in the SU(3) content of [6], and form |ψ>=(|6,0>+|0,6>)/√2. Compute Var(C2[SU(3)]), S_G, and P_G; if Var=0 while S_G=ln2 and P_G=1/2, Theorem 1 is refuted. Then re-run the first-order critical Hamiltonian at N=6 and list all eigenstates with Var<1e-6; if any has support on both conjugate irreps, the reported solvable count of 22 is inflated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. 3 Theorem 1 asserts that distinct irreps have distinct Casimir eigenvalues in the model space. This is false for SU(3) at N=6. In the paper's normalization, the Casimir eigenvalue is C2(λ,μ)=λ²+μ²+λμ+3λ+3μ, symmetric under λ↔μ, and the symmetric U(6) irrep [6] contains both (6,0) and (0,6): the dimension sum 91+162+28+125+27+28+1=462 equals dim[6], so both irreps are present. Both give C2=54. Consequently, the state |ψ>=(|(6,0),L=0>+|(0,6),L=0>)/√2 has Var(C2)=0 but S_G=ln2 and P_G=1/2, so the equivalence (b)⇔(a) in Theorem 1 is false. The primary diagnostic can therefore report an exact label for a label-mixed state, and the claimed zero-versus-O(N^2) separation is not guaranteed. The numerical results at N=6,8,10,12 could be affected whenever the Hamiltonian mixes conjugate SU(3) irreps; the paper does not check this. The fix is to replace the variance by the block purity or coherence, which do not require injectivity, or to prove injectivity for the specific algebras and boson numbers used.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a state-resolved quantum-information diagnostic for partial dynamical symmetry (PDS) in the sd interacting boson model. The proposed quantity is the variance of a subalgebra Casimir, Var_psi(C2[G]), which the author claims is equivalent to a block-coherence entropy and a block impurity and vanishes iff the state carries a single irreducible-representation label. Using exact diagonalization, the paper reports that at stable SU(3)-PDS points, at Leviatan's first- and second-order critical Hamiltonians, and at 168Er-like parameters, the variance is zero on the solvable subset and O(N^2) on mixed states, while bipartite s-d entanglement magnitude does not separate the two classes. The paper also encodes the model on qubits, prepares eigenstates variationally, and discusses the degeneracy resolution needed for readout. It is careful about limitations and makes no claim of quantum advantage.","tokens_in":12327,"tokens_out":10308,"duration_ms":93331,"significance":"If correct, the diagnostic would convert the qualitative notion of PDS into a fit-independent, state-by-state structural number, and the paper is commendably honest about scope and limitations. It contains useful numerical verification of operator identities (Eq. (4)) and of the variance formula (Eq. (10)), and it clearly separates algebraic exact results from example-specific numerics. However, the central equivalence rests on an injectivity premise for Casimir eigenvalues that is false for SU(3), so the variance face of the diagnostic does not currently certify a unique irrep label. The block-purity and block-coherence faces do not require that premise and may repair the claim, but the numerical pipeline and figures are built on the variance.","major_comments":[{"comment":"The proof of Theorem 1 asserts 'Distinct irreps have distinct Casimir eigenvalues in the model space' and uses this injectivity to infer that Var=0 iff the state occupies a single block. This premise is false for SU(3) at N=6: the irreps (6,0) and (0,6) are both contained in the U(6) irrep [6], and in the paper's normalization C2(lambda,mu)=lambda^2+mu^2+lambda*mu+3lambda+3mu is symmetric, so C2(6,0)=C2(0,6)=54. Thus the normalized state |psi>=(|(6,0),L=0>+|(0,6),L=0>)/sqrt(2) satisfies Var_psi(C2[SU(3)])=0 even though S_G=ln2 and P_G=1/2; equivalently, condition (b) does not imply (a). Since the paper's solvability classification uses the threshold Var<1e-6, any eigenstate that is a mixture of conjugate irreps with equal Casimir value would be reported as solvable. The numerical results at N=6 (where the solvable tower itself includes (0,6)) and at larger N do not check whether the Hamiltonians of Eqs. (1), (5), and (6) couple conjugate irreps. Please either prove injectivity for the specific algebras and boson numbers used, or replace the variance by the block purity P_G or the block coherence S_G (which do not require injectivity) in the diagnostic and re-run the state counts and figures.","section":"Sec. 3, Theorem 1 (Eq. (10))"},{"comment":"The degeneracy-resolution readout proposed for prepared states, namely 'measure C2[G] and diagonalize it within the degenerate energy window', does not resolve the conjugate-irrep degeneracy described above, because states in (6,0) and (0,6) have exactly the same C2 eigenvalue. A VQD-prepared state that is an equal mixture of these two label-carrying states would pass the degeneracy-resolved variance test and be classified as solvable. The fix using P_G or S_G, or an additional label that separates conjugate irreps, needs to be incorporated before the hardware-readout claim is made.","section":"Sec. 10.2"},{"comment":"The closing claim that 'the equivalence of Theorem 1 holds generally for any subalgebra with a quadratic Casimir' is too strong. For any algebra whose Weyl group identifies conjugate highest weights, which is exactly the SU(3) case at hand, the quadratic Casimir cannot separate all irreps, so the variance face of the equivalence is not general. The generality statement should be restricted to the block-purity/block-coherence faces or to algebras for which the injectivity is established.","section":"Sec. 11 (Summary and outlook)"}],"minor_comments":[{"comment":"The notation 's|d' appears as a single token in several places (for example, 's|dentanglement' in the Fig. 1 caption and elsewhere); it should be typeset as 's-d' or 's|d' with a space.","section":"Throughout"},{"comment":"The caption reports R less than about 3e-10 and R greater than about 70 without stating the normalization or the precise N values per panel; the text defines R_D, but the caption should be self-contained.","section":"Fig. 2"},{"comment":"The QFI values (5.2 versus 3.9 at N=6, etc.) are given without specifying the generator normalization or the number of states in each group; since the author states that only qualitative separations are robust, this caveat should appear in the main text where the numbers are quoted.","section":"Sec. 8"},{"comment":"The operator P^dagger_2(beta0) is written with a normalization factor sqrt(7/2) but the second-quantized tensor normalization is not fully specified; the relation to Eq. (1) at beta0=sqrt(2) is verified numerically but not shown algebraically.","section":"Eq. (5)"},{"comment":"The numerical results are not accompanied by a data or code repository; given the precision claims (3e-10, 1e-14), providing the diagonalization code or a data file would materially aid verification.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"I am not recommending rejection because the block-purity face is unaffected by the injectivity counterexample and likely repairs the central claim. The author should, however, re-examine all variance-based state counts and thresholds with a diagnostic that does not rely on the false injectivity, and ideally deposit the exact-diagonalization code so the conjugate-irrep mixing question can be checked directly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe paper is better than the abstract suggests, but it has a load-bearing flaw that the stress-test gets right. Theorem 1 claims Var(C2[G])=0 iff the state occupies a single irrep, relying on the injectivity of the Casimir eigenvalue map. For SU(3) in the sd-IBM at N=6, that premise is false: the irrep [6] of U(6) contains both (6,0) and (0,6), and C2=54 for both. A 50/50 superposition has zero variance but is not a single irrep. So the equivalence (b)⇔(a) is false, and the primary diagnostic can report an exact label for a label-mixed state. The paper never checks for conjugate-irrep mixing in its numerical results, which is a real gap.\n\nThat said, the numerical work is substantial and mostly honest. The state-by-state separation at stable PDS points, the cone-rigidity effect, the two critical-point fingerprints, the literal purity/coherence bridge, and the 168Er anchor are all clearly presented. The negative result on entanglement magnitude is well argued. The author also explicitly scopes the claims in the limitations passage.\n\nThe soft spots beyond the theorem: no code or data, small-N correlations lack error bars, and the 'set by the order' phrasing overreaches from two examples. But the injectivity issue is the one that matters. The fix is straightforward: use the block purity or coherence instead of the variance, since those do not require injectivity, or prove injectivity for the specific algebras and N. The numerical separation might survive, but it has to be rechecked.\n\nVerdict: worth a serious referee, but not accept as is. I'd send it back for major revision, asking for a corrected theorem and a check of the numerics against the conjugate-pair degeneracy.","headline":"The numerical story is substantial, but the central variance diagnostic rests on a false injectivity premise that the stress-test correctly identifies.","tokens_in":12885,"tokens_out":8566,"would_cite":false,"duration_ms":70038,"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":"One variance flags which nuclear-model states keep exact symmetry","keywords":["partial dynamical symmetry","interacting boson model","label variance","Casimir operator","block coherence","block purity","quantum phase transition","erbium-168"],"falsifier":"Compute the full set of quadratic-Casimir eigenvalues for the SU(3) and O(5) blocks within the sd-IBM Hilbert space at the boson numbers studied and check for collisions between distinct irreps; if any two distinct irreps share an eigenvalue, then a superposition of those two irreps would have zero label variance while carrying two labels, refuting the exact equivalence claimed in Theorem 1.","tokens_in":11793,"feed_emoji":"⚛️","tokens_out":11633,"duration_ms":90584,"temperature":0.7,"pith_summary":"The paper asks whether partial dynamical symmetry—the situation in which only a selected subset of eigenstates keeps exact quantum labels while the rest mix—has a purely structural signature in the wavefunctions themselves. It answers yes, and identifies the signature as the variance of a symmetry Casimir, evaluated state by state: this quantity vanishes exactly when a state carries a single irreducible-representation label and is of order $O(N^2)$ on the mixed states. The paper shows the separation holds at stable symmetry points and at the first- and second-order critical-point constructions, where the two transition orders give distinct fingerprints, while the magnitude of bipartite entanglement does not separate solvable from mixed states at all. If the claim is right, fitted interacting-boson Hamiltonians can be graded by a single reproducible number per eigenstate, selecting the states a spectroscopist would call solvable without a separate band-by-band analysis, and the same number can be read off prepared qubit states.","feed_headline":"One variance flags which nuclear-model states keep exact symmetry","feed_subtitle":"In the interacting boson model, it is zero on solvable states and grows as N^2 on the mixed ones.","key_machinery":"The central object is the label variance of a quadratic Casimir, $\\mathrm{Var}_\\psi(\\hat{C}_2[G]) = \\sum_\\lambda P_\\lambda [f_2(\\lambda) - \\langle \\hat{C}_2 \\rangle]^2$, built from the block probabilities $P_\\lambda(\\psi) = \\|\\Pi_\\lambda |\\psi\\rangle\\|^2$ over the irreducible-representation blocks of a chosen subalgebra $G$. Theorem 1 identifies its vanishing with the vanishing of the block-coherence entropy $S_G(\\psi) = -\\sum_\\lambda P_\\lambda \\ln P_\\lambda$ and with unit block purity $P_G(\\psi) = \\sum_\\lambda P_\\lambda^2$; each condition holds exactly when $|\\psi\\rangle$ lies in a single block. A second mechanism, the cone rigidity $R_D = \\|Q_D |\\psi\\rangle\\|$, measures the first-order response of an eigenstate to a deformation along the direction $D$, vanishing on the solvable states along the PDS-preserving directions $h_0$ and $h_4$ while remaining of order one along the symmetry-breaking direction $\\hat{n}_d$. The paper uses the variance as the primary diagnostic because it needs only two expectation values and no explicit block decomposition.","core_discovery":"The central claim is that in the $sd$-interacting boson model partial dynamical symmetry is detected not by the magnitude of bipartite entanglement but by the label variance $\\mathrm{Var}_\\psi(\\hat{C}_2[G])$ of a subalgebra's quadratic Casimir, namely $\\mathrm{Var}_\\psi(\\hat{C}_2[G]) = \\sum_\\lambda P_\\lambda [f_2(\\lambda) - \\langle \\hat{C}_2 \\rangle]^2$, where $P_\\lambda$ is the squared projection of the state onto the irreducible-representation block $\\lambda$. The paper proves that this variance, the block-coherence entropy, and the block impurity all vanish together, and that they vanish precisely when the state occupies a single block and therefore carries an exact label. On the solvable subset of the SU(3)-PDS Hamiltonian the variance is zero to numerical precision while on the mixed states it scales as $O(N^2)$; at the first-order critical point the same dichotomy appears as a solvable subset amid maximal mixing, and at the second-order critical point a single conserved seniority label is shared by every state. The same criterion, applied at representative $^{168}\\mathrm{Er}$ parameters, partitions the spectrum into the rotational band and the mixed background from wavefunctions alone. The paper claims the equivalence for any subalgebra with a quadratic Casimir but is explicit that the demonstrated separating power is for this model and these examples.","pith_inferences":["The same variance diagnostic should transfer to IBM-2 and to Bose-Fermi variants, where the relevant subalgebra Casimirs are already known; the paper names this as future work but does not test it.","Because a large label variance predicts anomalous inter-band $B(E2)$ strengths, a quantitative map from per-state variance to measured branching-ratio deviations is a testable extension the paper explicitly leaves open.","The direction-selective rigidity suggests a fit-free device protocol—evolve prepared states under two members of a PDS-preserving operator family and threshold the drift—for which the paper supplies the Trotterized evolution primitive but does not run the protocol.","If the one-to-one Casimir-eigenvalue premise survives in other algebras, the same $0$-versus-$O(N^2)$ dichotomy could serve as a general screening tool for candidate partial-dynamical-symmetry nuclei across fitted parameter libraries."],"forward_implications":["A fitted interacting-boson Hamiltonian can be classified state by state: the label variance assigns each eigenstate a reproducible number, so the qualitative claim that some states are solvable and others mixed becomes a graded output with no separate spectral fit.","At the first-order critical point the diagnostic finds a solvable subset amid maximal mixing, and at the second-order critical point it finds one conserved label shared by the whole spectrum, so the fingerprint itself carries the order of the transition.","Because the label variance is uncorrelated with multipartite entanglement and with magic, it is an independent quantum-information axis rather than a restatement of either resource.","The block-purity face makes the 'purity' and 'coherence' vocabulary of the quasi-dynamical-symmetry literature literal, and it demonstrates that a state can be block-pure while maximally entangled across the $s$-$d$ bipartition.","With the degeneracy-resolved Casimir measurement, the diagnostic can be evaluated on variationally prepared qubit eigenstates, giving a hardware-compatible readout that survives the arbitrary rotations variational solvers return inside degenerate energy levels."],"supporting_citations":[{"why":"Supplies the SU(3)-PDS Hamiltonian and its exactly solvable ground and gamma bands, the paper's main test case for separating solvable from mixed states.","marker":"[3]"},{"why":"Supplies the first- and second-order critical-point Hamiltonians whose two distinct label-variance fingerprints are compared.","marker":"[6]"},{"why":"Fixes the operator normalizations through the identity relating the PDS pairing operators to the SU(3) Casimir and the boson number, which the implementation verifies.","marker":"[7]"},{"why":"Provides the states and parameter region in which the O(6)-PDS 'purity' and SU(3)-QDS 'coherence' language is made into literal block-purity and entanglement quantities.","marker":"[10]"},{"why":"Is the prior IBM entanglement work whose ground-state entropy framework the paper contrasts; its entropy-tracks-parameter picture is the baseline the paper shows is non-diagnostic for PDS.","marker":"[11, 12]"},{"why":"Supplies variational quantum deflation, the method used to prepare the excited solvable and mixed eigenstates on the qubit register.","marker":"[16]"},{"why":"Supplies the digital quantum-simulation approach used to encode the collective model on qubits and benchmark the spectrum.","marker":"[19, 20]"}],"fun_headline_variants":["Label variance, not entanglement, reveals exact symmetry in boson model","Quantum fingerprint: variance of Casimir pinpoints solvable nuclear states","Partial symmetry spotted by label variance in interacting boson model","Zero variance marks states with exact partial symmetry in IBM","Entanglement can't spot it, but Casimir variance can"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument that zero label variance means one exact symmetry label assumes that no two distinct labels of the chosen subalgebra produce the same value of its quadratic Casimir in the model space; the paper asserts this one-to-one matching without proof.","fun_headline_variants_meta":{"raw":{"variants":["Label variance, not entanglement, reveals exact symmetry in boson model","Quantum fingerprint: variance of Casimir pinpoints solvable nuclear states","Partial symmetry spotted by label variance in interacting boson model","Zero variance marks states with exact partial symmetry in IBM","Entanglement can't spot it, but Casimir variance can"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1479,"prompt_tokens":1141,"completion_tokens":338,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":757,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":757,"tokens_out":338,"duration_ms":3621,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:39:26.984767+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full set of quadratic-Casimir eigenvalues for the SU(3) and O(5) blocks within the sd-IBM Hilbert space at the boson numbers studied and check for collisions between distinct irreps; if any two distinct irreps share an eigenvalue, then a superposition of those two irreps would have zero label variance while carrying two labels, refuting the exact equivalence claimed in Theorem 1.","supporting_citations":[{"cited_title":"Partial Dynamical Symmetry in Deformed Nuclei","cited_arxiv_id":"nucl-th/9606049","evidence_quote":"Supplies the SU(3)-PDS Hamiltonian and its exactly solvable ground and gamma bands, the paper's main test case for separating solvable from mixed states."},{"cited_title":"Partial Dynamical Symmetry at Critical-Points of Quantum Phase Transitions","cited_arxiv_id":"nucl-th/0703048","evidence_quote":"Supplies the first- and second-order critical-point Hamiltonians whose two distinct label-variance fingerprints are compared."},{"cited_title":"SU(3) partial dynamical symmetry and nuclear shapes","cited_arxiv_id":"2010.10951","evidence_quote":"Fixes the operator normalizations through the identity relating the PDS pairing operators to the SU(3) Casimir and the boson number, which the implementation verifies."},{"cited_title":"Linking partial and quasi dynamical symmetries in rotational nuclei","cited_arxiv_id":"1404.3826","evidence_quote":"Provides the states and parameter region in which the O(6)-PDS 'purity' and SU(3)-QDS 'coherence' language is made into literal block-purity and entanglement quantities."},{"cited_title":"Variational quantum computation of excited states,","cited_arxiv_id":null,"evidence_quote":"Supplies variational quantum deflation, the method used to prepare the excited solvable and mixed eigenstates on the qubit register."}],"review_version":2}