Papers

Which quantum measurement extracts the most classical information from an ensemble? We introduce the posterior algebra, a new canonical operator algebra selected by mutual information. For faithful ensembles, an affine information bound is exact precisely when this algebra is commutative; its joint spectral measurement is then optimal, and every optimal finite POVM refines it. Binary ensembles have one generator; compactness covers singular states, giving a proof of Shor's finite-dimensional binary orthogonal-measurement conjecture. The framework also gives rigidity bounds and a certified posterior-spectral receiver, validated on 408 mixed-state instances.

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $\tau_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

We prove an upper bound on the homogenization length scale which is a fixed power of the coarse ellipticity ratio for coefficient fields that satisfy a finite range of dependence assumption.

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

We investigate the fast-advection limit ($\varepsilon \to 0$) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution $N = 128$ for four different forcing shells on both the square torus and a thin torus, with the advection parameter $\varepsilon$ varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as $\varepsilon \to 0$, with the energy-to-enstrophy ratio $E/\Omega$ and the fraction $R$ of enstrophy in the lowest modes both increasing monotonically. At finite $\varepsilon$ the degree of concentration depends on the forcing shell through the effective spectral value $B_1/B_0$, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to $\varepsilon = 0$ suggests that the limiting values may be forcing-independent, with $B_1/B_0$ governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.

Local quantum interactions generate dynamics in an exponentially large Hilbert space, yet locality and entanglement restrict the information that can spread and accumulate. Bounds on propagation and entanglement growth, together with tensor networks, exploit these restrictions to discard information unnecessary for describing the evolution. This leads to a sharper question: how much information must any low-rank representation retain? Here we determine these limits, up to logarithmic factors, for short-range interactions. For time-independent evolution, the optimal rank obeys $\log D=\widetilde O(t+\sqrt{\log(1/\epsilon)})$, with matching lower bounds fixing the accuracy exponent $1/2$; for arbitrary driving, matching fixed-time bounds instead give $2/3$. Correspondingly, the corresponding dynamical entanglement spectra exhibit distinct small-$\alpha$ Rényi laws, $\alpha^{-1}$ and $\alpha^{-2}$. In one dimension, the static limit is constructively attained by an explicit MPO algorithm, with an analogous extension to Liouvillian dynamics. Together, these results determine the irreducible information required to represent local quantum evolution and uncover distinct entanglement structures in static and driven dynamics.

We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space $V$ and $A=\en(V)$, we lift the known correspondence between double Lie algebra structures on $V$ and skew-symmetric Rota-Baxter operators on $A$ to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.

We show that the noncommutative second Chern number of a $(3+1)$D disordered axion pump remains quantized beyond global spectral gap closure at $W_g \approx 14.5$. Large-scale computation of the second Chern number and finite-size scaling demonstrate convergence of $\mathrm{Ch}_2$ toward $-1$ well into the mobility-gap regime ($W \lesssim 16.5$), while the Středa-Hall response $\partial_\phi \mathrm{Ch}_{0,3}$ departs rapidly after global spectral gap closure, serving as an internal control that distinguishes bulk mobility-gap protection from finite-size artifacts. Level statistics identify a 3D unitary Anderson transition at each pump slice and locate the mobility-edge bottleneck at $\tau=0$, with critical disorder $W_c \approx 18$.

Learning an interacting many-body Hamiltonian at the Heisenberg limit generally requires control that preserves local information for evolution times of order $1/\varepsilon$ to attain $\varepsilon$ precision. To achieve this, existing protocols often rely on trusted many-qubit operations or increasingly rapid control pulses, creating a precision barrier when gates involve unknown multi-qubit interaction or have finite duration. We show that neither resource is necessary for geometrically local Hamiltonians. Our protocol applies only static single-qubit fields, whose strength is independent of system size and grows polylogarithmically with $1/\varepsilon$. These fields generate emergent prethermal symmetries that suppress thermalization for the entire learning experiment while retaining informative symmetry-preserving dynamics. On a $d$-dimensional lattice of $n$ qubits, this enables learning every coefficient to accuracy $\varepsilon$, with failure probability at most $\delta$, using total evolution time $\mathcal{O}(\log^d(1/\varepsilon)\log^2(n/\delta)/\varepsilon).$ Thus, our learning protocol attains the information-theoretically optimal \(1/\varepsilon\) dependence up to polylogarithmic factors using product-state preparation, single-qubit measurements, and non-adaptive experiments. More broadly, our results establish emergent prethermal symmetries as a resource for quantum learning, opening a route to Heisenberg-limited characterization of many-body systems where fast or trusted many-qubit control is unavailable.

We study the fluctuation spectrum and renormalized one-loop quantum mass of the Christ–Lee kink over the full positive range of its deformation parameter. The fluctuation equation is reduced exactly to general-Heun form, yielding parity-resolved continuum solutions and implicit quantization conditions for the discrete spectrum. At the first nontrivial continuum crossing, the odd Heun solution truncates to a polynomial, giving the exact threshold \(\epsilon_1=\sqrt{\sqrt{3}-1}\). For small deformation, the kink separates into two widely spaced \(\phi^6\)-like interfaces and the fluctuation operator develops a long central cavity. This geometry explains the logarithmic growth of the bound-state count, the asymptotically geometric hierarchy of continuum-threshold crossings, and the emergence of a soft relative-translation mode. In the opposite limit, the spectrum approaches the \(\phi^4\) Pöschl–Teller problem, while its even threshold resonance becomes a shallow bound state at finite deformation. The one-loop correction is evaluated in a fixed vacuum-normal-ordering prescription using a common-regulator spectral trace and independently through an imaginary-frequency functional determinant. The two calculations agree throughout representative finite deformations and reproduce the exact \(\phi^4\) limit. In the small-deformation regime, the extended central region produces a logarithmically enhanced negative quantum correction. These results provide a unified analytic and numerical description of how the spectral reorganization of the Christ–Lee kink controls its semiclassical quantum mass.

We extend the spherical-integral approach to $\mathcal{R}$-transforms of non-Hermitian random matrices to random matrices with a fixed $K \times K$ block structure. We introduce a rank-$K$ spherical integral defining a scalar $H$-transform on $2K \times 2K$ overlap matrices and an associated matrix-valued $\mathcal{R}$-transform encoding the joint block structure. Using the replica method, we derive a conjectural subordination relation for sums of independent block matrices. This framework provides a method for determining the spectral boundaries of large non-Hermitian random matrices with a fixed block structure.

We show, analytically, that a static sine-Gordon soliton cannot exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha := (m/H)^2 < 2$, where $m$ is the mass parameter of the sine-Gordon theory and $H$ is the Hubble constant. Conversely, we also show that static sine-Gordon solitons exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha > 2$. The above threshold is explained—qualitatively and to within an O(1) factor—using a heuristic argument involving the interplay of tensile force in the Lorentzian sine-Gordon soliton and the tidal force in de Sitter spacetime. A similar heuristic argument, which remains to be confirmed analytically, also suggests the existence of a threshold, $ (m_V/H)^2 \sim O(1)$, below which the tidal forces are too strong to permit the existence of a static `t Hooft–Polyakov monopole in non-dynamical 3 + 1 de Sitter spacetime; $m_V$ is the mass of the vector boson. Linde has suggested that new inflation could have triggered secondary inflation at the core of a GUT (Grand Unified Theory) monopole even if the Hubble constant at or after the GUT phase transition was significantly smaller than the mass of the $X$ boson. We present a heuristic argument, which suggests that the $SO(3)$ `t Hooft–Polyakov monopole does not allow secondary inflation at its core when the inflationary background is weak. Based on the above, as yet analytically unconfirmed, heuristic argument for the $SO(3)$ `t Hooft–Polyakov monopole, we conjecture that secondary inflation at the core of a GUT monopole is infeasible.

Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $\chi$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$\chi(\overline{M_R})=\chi(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.

We introduce a magnetic coherent–state frame over the Landau levels of the two–dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost–orthogonal, and we use it to endow the CAR $C^*$–algebra over the one–particle Hilbert space with an asymptotic quasi–local (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet $*$–subalgebra of almost–local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost–local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete–continuum dictionary intended as the starting point for extending Lieb–Robinson–type techniques from lattice systems to interacting continuum electrons in a magnetic field.

We find a new coupling between the Airy sheet $\mathcal{S}$ and the Airy line ensemble $\{ \mathcal{A}_n \}_{n \in \mathbb{N}}$, which we call the down-the-middle isometry. In this coupling, the Airy sheet is encoded by last passage values that start at $\mathcal{A}_k(0)$ for some $k \in \mathbb{N}$ and end on the top line $\mathcal{A}_1$. Using this coupling, we give a new construction of the (extended) Airy sheet as a scaling limit of Brownian last passage percolation and study coalescence of geodesics in the directed landscape. This coupling extends to discrete polymers and last passage percolation models, as well as to the KPZ line ensemble and KPZ sheet.

We study the static matter constraint associated with a conical deficit that varies along a string near a wormhole throat. A positive scalar-dependent factor multiplying the complete Abelian-Higgs Lagrangian couples a finite-width vortex to a canonical kink. At critical coupling, we obtain the leading longitudinally varying vortex tension. The flat-kink fluctuation operator has an even translational mode and an odd shape mode of mass $\sqrt{3}/L_\chi$. Reflection symmetry excludes the translational mode from the leading sourced response. We formulate the response with nonzero dressed-string boundary conditions at the longitudinal ends and derive a finite-range transverse halo. In the overlap $w\ll\rho\ll L_\chi$, its mixed stress satisfies $2\pi C\,\delta T^{\chi\,\rho}{}_l=\partial_l\mu_v^{(0)}$ to leading order, where $2\pi C$ is the azimuthal circumference. Together with the local weak-gravity deficit relation, this supplies the leading mixed Einstein constraint of a restricted varying cone. At fixed longitudinal position the sourced halo instead decays beyond $L_\chi$. Consequently, it cannot sustain that restricted cone throughout an outer overlap unless another source contributes the missing mixed stress. This obstruction is conditional on the response of the wormhole-supporting sector and is not a no-go result for general axisymmetric metrics. A finite-width calculation checks the linear-response crossover. We specify the perturbative domain and explain why a rotating extension requires an independent solution of Gauss's law. No complete backreacted wormhole solution or stability result is claimed.

The quantum circuit complexity of an evolving many-body quantum system is believed to exhibit a sustained growth, maintained for timescales much longer than the onset of thermalization. Most previous works have focused on models which violate energy conservation, such as random unitary circuits. Here we study generic, local time-independent Hamiltonian dynamics. We unconditionally prove that for generic local Hamiltonians and typical initial product states at high effective temperature, the robust quantum circuit complexity must grow over a very long period of time, attaining an exponentially large value at late times. Our approach relies on two structural properties that we prove rigorously: (i) generic local Hamiltonians satisfy generalized spectral no-resonance conditions of arbitrary order, and (ii) typical high-temperature product states are effectively supported in exponentially many energy eigenstates. These properties have been widely assumed without proof in prior works. As corollaries of this result, we show the late-time state displays robust volume-law entanglement that is irremovable by polynomial-size circuits, and we establish a no fast-forwarding result for generic local Hamiltonians. We also lower bound the complexity of preparing sufficiently large subsystems with local quantum channels, in contrast with sufficiently small thermalizing regions which we show retain low complexity at late times.

Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the $k$-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality $k \geq 5$, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.

A seminal result of Hastings showed that entangled inputs can increase the rate of classical communication through independent uses of a quantum channel. While Hastings' result proved the existence of this phenomenon, it has remained a major open problem to find an explicit channel with this property. In this work, we resolve this question by constructing an explicit family of finite-dimensional quantum channels for which entangling inputs between pairs of channel uses yields an unbounded separation from the achievable communication rate using unentangled inputs. We describe a circuit implementation of our channel that uses only deterministic qudit Clifford unitaries and a single binary measurement with classical feedforward. The key ingredient is a careful design of the measurement and feedforward process that yields the required one-copy and two-copy output entropy bounds from moment estimates alone, without requiring strong convergence of the underlying unitary family.

A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $\Sigma\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $\Omega(n^3)$ copies, strictly exceeding the sample complexity $\Theta(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $\Sigma\ge(\frac12+\nu)I$ for any parameter $\nu>0$, we prove that single-copy tomography requires $N=\Theta\left(n^2\min(n,1+\nu^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $\nu=\Omega(1)$, the sample complexity drops to $\Theta(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.

Finite-temperature many-body theories are typically formulated in imaginary time, where they are amenable to efficient numerical treatment. However, the extraction of spectral properties from the imaginary axis is difficult and requires numerical analytic continuation, while a direct formulation on the real axis requires prohibitively expensive numerical quadrature. In this paper, we represent the objects of many-body perturbation theory using systematically improvable pole and moment expansions and develop the corresponding framework for evaluating diagrammatic contributions directly on the real axis. We introduce a controlled recompression procedure that prevents the proliferation of poles under algebraic and diagrammatic operations. We benchmark the approach on paradigmatic quantum impurity problems and apply it to self-consistent calculations of the uniform electron gas. The resulting spectral quantities converge systematically towards numerical accuracy. This work provides a practical framework for controlled and systematically improvable calculations of spectral quantities within diagrammatic many-body theories, including self-consistent conserving approaches.

We achieve linearly many parallelizable and addressable transversal logical multi-controlled-Z gates with asymptotically optimal parameters simultaneously on good quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to good qLTC constructions. To establish the nontriviality of the logical operation, we express the cup product pairing as a coefficient in a product of Moore determinants and prove that this coefficient is nonzero. Moreover, we use the symmetries of the covering spaces to construct large spaces of cocycles. By multiplying the initial cocycles by suitable elements of these spaces, we obtain linearly many independent pairings, each selected by a corresponding cycle constructed using Poincaré duality. This yields linearly many parallelizable and addressable multi-controlled-Z gates and enables asymptotically constant-overhead magic state distillation with good qLDPC codes.

The current density in a rectangle in the $D=2$ dimensional Euclidean space, where opposite point charges are placed in a source $s$ and a destination $d$, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a $q$-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.

We study the high-temperature derivation of $\Phi^4_d$ measures from grand-canonical Bose gases on two- and three-dimensional tori. Building on the upper-symbol variational approach of the author's earlier work, we use relative entropy to give a simple, unified proof of relative free-energy convergence and Hilbert–Schmidt convergence of all fixed-order rescaled reduced density matrices to their Hartree counterparts. For interaction ranges $\varepsilon=\lambda^\eta$, where $\lambda$ is the inverse temperature, these results hold for every fixed $0<\eta<1/2$ in two dimensions and $0<\eta<1/26$ in three. In two dimensions, we improve the result of Jougla and Rougerie (2026) and the local $\Phi^4_2$ derivation of Fröhlich, Knowles, Schlein and Sohinger (2025). Combined with classical approximation, the two-dimensional result gives the first local $\Phi^4_2$ derivation valid for every fixed exponent $0<\eta<1/2$, allowing polynomial exponents arbitrarily close to the diluteness threshold from below. In three dimensions, we use the identification of the local measure and the stationary SPDE estimates of Nam, R.~Zhu and X.~Zhu (2025). From these estimates we strengthen their classical correlation convergence to the Hilbert–Schmidt norm. This yields the local limit at every fixed order in the same polynomial range.

We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}$. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I ($P_{\rm I}^{2k}$) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the $P_{\rm I}^{2k}$ equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.

In this paper, it is shown that in spacetimes of maximal proper acceleration, there is a reduction of the mean lifetime of the muon. We consider several candidates for the maximal proper acceleration. For all of them, the induced corrections are small, but for the case when the maximal proper acceleration is identified with the Schwinger's acceleration, the effect is detectable in future muon collider facilities.

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

Finite monodromy provides a bridge between group representations and algebraic solutions of differential equations. We study this connection for the Katz-Long-Moody construction, which transforms representations of the semidirect product of a free group and a braid group into new representations of the same group and is related to Knizhnik-Zamolodchikov (KZ)-type equations. For a fixed finite-image input, we classify the parameter values for which the resulting representations have finite image, both on the semidirect product and on its free-group and braid-group subgroups. In particular, finiteness of the braid-group image is independent of the admissible parameter. These results give necessary and sufficient conditions for all solutions of the corresponding regular-singular KZ-type equations to be algebraic. On restriction to the free group, they also characterize finite monodromy and algebraicity of all solutions of the associated Fuchsian systems, connecting the classification to classical questions about algebraic hypergeometric functions.

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi–Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example. The proof connects simultaneous quantum smoothing with polynomial concentration on $\CP^1$. Symmetry reduces the problem to a frame inequality on $\Sym^n(\mathbb C^2)$, where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein–Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality. These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher–Fawzi and Colomer–Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.