This work extends the local-unitary floor theorem for quantum parameter records to eigenvalue crossings and degenerate subspaces. For smooth full-rank quantum-state families with simple spectra, the fraction of a parameter record that survives every local unitary is given by the ratio of classical to quantum Fisher information, F_cl/F_Q. The present paper generalizes that result to analytic families containing transversal eigenvalue crossings. Using Rellich–Kato perturbation theory, the correct eigenvalue slopes at a degenerate point are identified as the eigenvalues of the projected derivative within the degenerate subspace. The local-unitary floor then becomes floor = F_cl^× / F_Q, where F_cl^× is the ordinary population Fisher information corrected by an explicit contribution from every crossing lying inside the observation window. For a two-branch crossing, the correction depends only on the slope difference, the crossing eigenvalue, and the fractional position of the crossing within the parameter window. This produces a universal parabolic reduction of the floor. Several consequences follow. First, the original simple-spectrum theorem is recovered when no crossing lies inside the observation window. Second, a symmetric exchange centered in the window can drive the local-unitary floor exactly to zero, providing an analytic explanation of the exact erasability previously observed numerically. Third, moving the crossing away from the center produces a parameter-free parabolic recovery of the floor, with curvature determined by the eigenvalue-slope difference and the quantum Fisher information. Fourth, corrections from multiple independent crossings are additive. Fifth, in degenerate subspaces the correct slopes are the eigenvalues of the projected derivative operator, not basis-dependent diagonal matrix elements. A naive diagonal prescription can therefore fail dramatically. All parts of the result are machine-verified on engineered analytic state families. The crossing formula agrees with direct calculations at relative errors below 10^−6, the predicted parabola and symmetric zero are reproduced, the degenerate-block prescription is confirmed, simultaneous-crossing corrections are additive, and hundreds of simple-spectrum families recover the original floor theorem. The analysis applies to analytic, full-rank families with transversal two-branch crossings. Tangential higher-order crossings, rank-changing transitions, and physical many-body realizations of the crossing formula remain open.
This work extends the local-unitary floor theorem for quantum parameter records to eigenvalue crossings and degenerate subspaces. For smooth full-rank quantum-state families with simple spectra, the fraction of a parameter record that survives every local unitary is given by the ratio of classical to quantum Fisher information, F_cl/F_Q. The present paper generalizes that result to analytic families containing transversal eigenvalue crossings. Using Rellich–Kato perturbation theory, the correct eigenvalue slopes at a degenerate point are identified as the eigenvalues of the projected derivative within the degenerate subspace. The local-unitary floor then becomes floor = F_cl^× / F_Q, where F_cl^× is the ordinary population Fisher information corrected by an explicit contribution from every crossing lying inside the observation window. For a two-branch crossing, the correction depends only on the slope difference, the crossing eigenvalue, and the fractional position of the crossing within the parameter window. This produces a universal parabolic reduction of the floor. Several consequences follow. First, the original simple-spectrum theorem is recovered when no crossing lies inside the observation window. Second, a symmetric exchange centered in the window can drive the local-unitary floor exactly to zero, providing an analytic explanation of the exact erasability previously observed numerically. Third, moving the crossing away from the center produces a parameter-free parabolic recovery of the floor, with curvature determined by the eigenvalue-slope difference and the quantum Fisher information. Fourth, corrections from multiple independent crossings are additive. Fifth, in degenerate subspaces the correct slopes are the eigenvalues of the projected derivative operator, not basis-dependent diagonal matrix elements. A naive diagonal prescription can therefore fail dramatically. All parts of the result are machine-verified on engineered analytic state families. The crossing formula agrees with direct calculations at relative errors below 10^−6, the predicted parabola and symmetric zero are reproduced, the degenerate-block prescription is confirmed, simultaneous-crossing corrections are additive, and hundreds of simple-spectrum families recover the original floor theorem. The analysis applies to analytic, full-rank families with transversal two-branch crossings. Tangential higher-order crossings, rank-changing transitions, and physical many-body realizations of the crossing formula remain open.
This work proves a quantitative theorem relating local-unitary irreversibility of quantum records to the decomposition of quantum Fisher information. For a smooth one-parameter family of full-rank quantum states with nondegenerate spectrum, consider the distinguishability between two nearby parameter values and ask how much of that distinguishability can be erased by arbitrary unitary operations acting only on the observer's subsystem. The main result is lim d→0 E_spec/E = F_cl/F_Q = 1 − F_coh/F_Q, where E is the full fidelity-based record, E_spec is the minimum record remaining after optimal local-unitary erasure, F_Q is the quantum Fisher information, F_cl is its eigenvalue-population contribution, and F_coh is the eigenvector or coherence contribution. Thus, the locally uneraseable fraction of a parameter record is exactly the fraction of its Fisher information encoded in spectral populations. The complementary coherence contribution is precisely the part that local unitary rotations can remove. The proof combines three established ingredients: the Bures expansion of quantum fidelity, the classical Bhattacharyya expansion of spectral fidelity, and the unitary-orbit theorem for maximal fidelity between density matrices. The theorem also includes a falsifiable breakdown clause. It requires the sorted eigenvalue pairing used by the unitary-orbit optimization to coincide locally with continuous eigenvalue branches. At an eigenvalue crossing this condition can fail, and the local-unitary floor can depart sharply from the naive same-branch Fisher prediction. In engineered symmetric crossings studied here, the floor collapses exactly to zero. The result is machine-verified on interacting gauge-theory record families, analytic random-state families, and engineered eigenvalue crossings. The Fisher decomposition identity is verified to approximately 10^−10 on analytic families, the previously observed gauge-theory floor fractions 0.6387 and 0.4951 are reproduced, and the inequality F_cl ≤ F_Q is confirmed across six hundred random instances. The theorem applies under smoothness, full-rank, and simple-spectrum assumptions. Degenerate subspaces, rank-changing families, and a broader operator-theoretic extension remain open.
This work develops a quantitative theory of how much information about a quantum parameter can and cannot be erased by unitary operations, and applies it to the θ = π quantum critical point of the massive lattice Schwinger model. For two nearby parameter-dependent quantum states, the distinguishability stored in eigenvalue populations cannot be changed by unitary rotations, whereas information stored in eigenbasis orientation can be moved by such operations. This motivates the erasability floor: the minimum fidelity-based distinguishability remaining after optimal unitary erasure. For smooth full-rank state families with simple spectra, the paper proves E_spec / E → F_cl / F_Q, where F_Q is the quantum Fisher information and F_cl is its eigenvalue-population contribution. Thus the locally uneraseable fraction of parameter information is exactly the population share of the quantum Fisher information, while the erasable fraction is its coherence contribution. The result is further extended to eigenvalue crossings and degenerate subspaces. A transversal crossing produces an explicit correction to the floor that is parabolic in the crossing position within the sampling window. Degenerate-point slopes are determined by the eigenvalues of the projected derivative within the degenerate subspace rather than by basis-dependent diagonal entries. Symmetric crossings can drive the erasability floor exactly to zero. The theoretical results are verified numerically on analytic and engineered state families, including crossing, degeneracy, additivity, and random-family tests. The framework is then applied to the massive lattice Schwinger model at θ = π with the Wilson-line zero mode retained as a dynamical rotor. Matrix-free GPU Krylov calculations extend the finite-size tower to N = 30 sites, corresponding to a Hilbert-space dimension of approximately 1.4 × 10^9. The global fidelity-susceptibility peak grows approximately as N^s with s = 1.87–1.89 over N = 22–30, corresponding to ν = 1.06(1), consistent with the two-dimensional Ising universality class. The pseudo-critical peak locations extrapolate to a lattice critical mass near 0.278 at a = 1/2, with a fit-form range extending to 0.296. The central value lies within 0.007 of the independently predicted value obtained by transporting the established continuum critical mass through the known additive lattice mass shift. The erasability floor reveals the information structure of the transition. Near criticality, only about 6–8% of the mass-parameter information carried by a gauge-invariant six-site region is population-borne, implying that more than 92% is coherence-borne and therefore movable by local unitary rotations. Away from the transition, the population contribution rises substantially. A regional susceptibility constructed from the same reduced states locates the critical point to within approximately 0.009 of the global susceptibility across all five large system sizes, demonstrating that the critical point remains locally detectable even as the total Hilbert space grows by more than two orders of magnitude. The study also reports unresolved finite-grid questions rather than forcing conclusions: monotonic scaling of the floor minimum and of the dip–peak separation cannot be established on the present mass grid and require finer sampling. Together, the results connect quantum Fisher information, unitary erasability, eigenvalue-crossing geometry, and quantum criticality within a single reproducible framework. The accompanying package includes theorem-verification scripts, the matrix-free lattice solver, validation gates, analysis code, figure-generation code, and raw large-N data.
This work synthesizes the gauge-theory branch of the quantum-record program and derives a quantitative relation between record irreversibility and Fisher information. The paper first corrects an interpretation from an earlier study of theta records. At a finite angular separation, the fraction of the record that could not be removed by local unitaries was approximately one half in both the massless and massive Schwinger models. The present tangent-space analysis shows that this apparent mass independence was a finite-angle coincidence. In the small-parameter-separation limit, the local-unitary floor is instead governed by a Fisher-information law: E = d²F_Q/8, E_spec = d²F_cl/8, and therefore E_spec/E = F_cl/F_Q, where F_Q is the quantum Fisher information and F_cl is its population, or diagonal, contribution in the eigenbasis of the reduced state. The relation is verified numerically to four decimal places at two mass values, two expansion centers, and two step sizes. The resulting floor fractions are approximately 0.639 in the massless theory and 0.495 in the massive theory. Operationally, the result states that the fraction of a parameter record that no local unitary can erase is precisely the fraction of its Fisher information encoded in eigenvalue populations rather than eigenvector rotations. The paper also consolidates eleven companion studies into a single gauge-record classification. Twenty-nine previously frozen numerical results spanning continuum field theory, interacting gauge theory, boundary conditions, theta physics, symmetry breaking, criticality, and non-Abelian center charge are audited against the general record principle. All twenty-nine are found to be mutually consistent. The resulting classification separates protected carriers—including Gauss flux, zero modes, energy, emergent conserved structures, and center charge—from erasable information classes such as arrangements, reference-frame orientation, and finite-size screening remnants. A final SU(3) diagnostic further localizes a previously observed quark–antiquark residual asymmetry. The effect appears only in the screened regime and only for one string convention, strongly indicating that it originates in the geometry of the screening cloud on the string side, although an analytic derivation remains open. The Fisher relation is demonstrated numerically in the present system rather than proven as a general operator theorem. Establishing that general theorem is the natural next theoretical step.
This work investigates the operational requirements for erasing a theta-dependent quantum record in the periodic Schwinger model. A companion study found that the massless theory can carry a large, energy-free, rotation-borne theta record. The present work asks whether an observer can actually undo that record using operations confined to the observer's region. The analysis begins with an exact unitary-orbit result: for two reduced states, the maximum fidelity obtainable by arbitrary local-unitary transformations depends only on their ordered spectra. This defines a spectral record, E_spec, representing the portion of distinguishability that no unitary supported inside the observer's region can remove. The initial prediction that the massless rotation-borne record would be spectrally trivial is decisively refuted. Instead, approximately one half of the theta record survives every local unitary in both the massless and massive theories. At the principal parameter point, E_spec/E ≈ 0.505, with the near-equality between the two mass regimes persisting across lattice sizes, region sizes, and theta separations. A physically motivated local chiral-current twist fails to reduce the record at all. The analysis then identifies the actual exact eraser. In the staggered lattice theory, the anomalous chiral transformation survives as a discrete one-site shift. Comparing the observer region at theta = 0 with the shifted region at theta = pi erases the record to machine precision on the mass-shifted massless line. This simultaneously provides a state-level test of the special mass-shifted line associated with the lattice chiral symmetry: the naive zero bare-mass line retains a residual record, while the massive theory does not admit the same erasure because theta + pi represents physically distinct dynamics. The results reveal a hierarchy of record reversibility. Local unitary operations cannot remove the spectral component; a discrete symmetry transformation that moves the observer's region can erase the record exactly at the chiral step; and the effect of general non-unitary dynamics remains open. The observed approximately one-half local-unitary floor is currently numerical rather than analytically derived. The study is restricted to one-dimensional staggered compact QED, finite lattices, and a single coupling regime.
This work uses the floor theorem as a quantitative probe of how information is distributed across a quantum phase transition. The system is the massive Schwinger model at θ = π, where an earlier record-susceptibility analysis located the finite-size transition. Here, neighboring ground states are analyzed through the local-unitary floor and its exact Fisher-information decomposition. Three main results are obtained. First, the floor theorem remains quantitatively valid throughout the transition. Across sixteen mass intervals, the directly measured local-unitary floor agrees with the ratio F_cl/F_Q at every point, with a maximum deviation of 0.0019. Thus the theorem functions as an accurate diagnostic even in the rapidly changing critical region. Second, the critical record is overwhelmingly coherence-borne. The population fraction of the quantum Fisher information drops to approximately 0.043 near the transition, implying that about 95.65% of the critical sensitivity is carried by eigenvector rotation rather than eigenvalue populations. Away from criticality, the information becomes progressively more population-borne, with the floor rising to approximately 0.39 by the end of the sweep. The coherence minimum does not coincide exactly with the conventional fidelity-susceptibility peak. It lies below the peak at both studied lattice sizes, with the separation decreasing as system size increases. This identifies the coherence extremum as a distinct finite-size critical locator rather than merely another representation of the susceptibility maximum. Third, the symmetry-broken order record itself is found to be almost completely local-unitary erasable. The two broken-symmetry vacua have nearly identical regional spectra, giving a unitary floor of approximately 0.014. Their distinction is therefore carried predominantly by a symmetry rotation. This does not contradict the substantial protection previously found under field-preserving channels. The same order record has different fates under different allowed channel classes: it is erasable by the broken-symmetry rotation while remaining protected when the field populations carrying the order parameter are explicitly preserved. The results provide an information-theoretic anatomy of criticality in which quantum Fisher information is separated into population and coherence components, revealing that the transition is dominated by coherence rather than population rearrangement. The analysis is limited to two lattice sizes and one coupling. The finite-size scaling of the offset between the coherence minimum and the susceptibility maximum remains open.
Quantifying properties of quantum states through the limits of their manipulation is a central goal of quantum resource theories. For symmetry breaking, the quantum geometric tensor (QGT) governs asymptotic pure-state conversion, but a complete characterization for general mixed states has remained elusive. Here we fully resolve this problem for finite-dimensional systems under compact Lie group symmetries in the i.i.d. asymptotic regime. Specifically, we establish a single-letter formula for the optimal conversion rate between arbitrary states, with vanishing trace-distance error, in the resource theory of asymmetry. The rate is determined by a one-parameter family of metric adjusted QGTs, obtained by rescaling quantum Fisher information (QFI) matrices that interpolate between the symmetric logarithmic derivative and right logarithmic derivative (RLD) QFIs. For pure states, the RLD endpoint recovers the standard QGT. No state-independent finite subset of this family suffices in general, even for U(1) symmetry, revealing a qualitative distinction from pure-state conversion. Our formula further yields an exact pure-state distillation-rate formula in terms of a single asymmetry measure, characterizes asymptotically reversible interconversion, and identifies bound asymmetry for quantum clocks. Complementarity among members of the metric adjusted QGT family also reveals an activation mechanism: a state with zero conversion rate to a mixed clock state can still enhance another input's yield under joint processing. Our proof relies on two developments of independent interest. First, we extend quantum local asymptotic normality to unitary models with arbitrary rank and spectral degeneracy. Second, we characterize convertibility between quantum Gaussian shift models in terms of the same one-parameter family of QFIs.