Papers

Every two-qubit measurement basis composed of maximally entangled eigenstates can be transformed into the Bell basis via local unitary operations. For higher dimensions, in contrast, there exist inequivalent bases composed of maximally entangled eigenstates. Here, we provide a single-parameter family of two-qutrit maximally entangled measurement bases, and demonstrate that none of the bases are equivalent to each other under local unitaries. These bases are constructed from the continuous family of symmetric informationally complete sets of states in dimension three. By studying the local unitary bases that generate the family of two-qutrit maximally entangled measurement bases, we construct the first examples of wild error bases in the smallest dimension where these can exist. Finally, we discuss how distinct measurements in the family lead to differences in performance in several scenarios relevant in quantum information.

For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

In the damage variant of Cops and Robber, the damage number \(\dmg(G)\) is the number of distinct vertices damaged by the robber under optimal play, with one cop minimizing and the robber maximizing this number. We introduce the zombie damage number \(\zdmg(G)\), obtained by requiring the pursuer to move at every turn along a shortest path toward the survivor. The parameter therefore measures the cost of geodesic pursuit when the objective is containment rather than capture alone. We prove that \(\dmg(G)\leq\zdmg(G)\) and characterize the graphs with \(\zdmg(G)=0\). Geodesic pursuit incurs no additional damage on trees, and we determine the parameter exactly for paths, cycles, split graphs, and complete multipartite graphs. In particular, \(\zdmg(C_n)=n\) for \(n\geq5\), while \(\zdmg(K_{n_1,\ldots,n_k})=n_1+n_2-2\) when \(n_i\geq2\) for every \(i\). We also develop a nonbacktracking trace argument for sparse graphs. If \(G\) is connected, \(\delta(G)\geq2\), and \(g(G)\geq5\), then every vertex is damaged, and hence \(\zdmg(G)=n(G)\). The same argument gives the sharp bound \(\zdmg(G)\geq g(G)\) for every connected graph of finite girth at least five. It follows that fully subdividing each edge of a connected graph of minimum degree at least two produces a graph with maximum zombie damage. These results yield an unbounded separation from the ordinary damage number. Most of the questions leading to these results, as well as the open conjectures, arose through Theo-Conjecture, an advisor-supervised discovery loop combining automated conjecturing, exact computation, language-model-assisted exploration, and human mathematical judgment.

Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Sym$^+_n$ equipped with the Bures–Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum $\Pi=\tfrac12 L_\Sigma[\dot\Sigma]$, where $L_\Sigma$ is the Lyapunov operator, and explicit Hamiltonian $\mathcal{H}(\Sigma,\Pi) = 2{\rm tr}(\Pi\Sigma\Pi)+V(\Sigma)$. Adding Rayleigh dissipation recovers the Bures–Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as $(s-\nu)^{-2}$ as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures–Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.

The magnitude of a metric space constitutes an expressive invariant that subsumes numerous different geometrical-topological invariants. Building on recent advances in magnitude homology, i.e., a bigraded homology theory that recovers the magnitude, we develop a novel local measure of the centrality or importance of nodes in a graph. Our measure is inspired by the concept of relative homology as it considers the change in magnitude homology when removing a vertex. We show that our proposed measure satisfies several properties a centrality measure is reasonably expected to respect and demonstrate that we introduce a new perspective on centrality by comparing to several established centrality measures.

For centered bosonic Gaussian states, the covariance pullback of the Bures cometric differs from the lift-normalized classical covariance Fisher–Rao cometric by a state-independent bilinear form. Under the canonical identification of symmetric covariance covectors with $\mathfrak{sp}(2N,\mathbb{R})$, this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with $m$ pure modes has an $m^2$-dimensional cometric kernel isomorphic to $\mathfrak u(m)$, while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction $(\nu-\hbar/2)/(\nu+\hbar/2)$ of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.

In this paper, the directed extended zero-divisor graph $\overrightarrow{\Gamma}_E(S)$ of a non-commutative semigroup $S$ is introduced and the relationships between $\overrightarrow{\Gamma}_E(S)$ and the zero-divisor graph $\overrightarrow{\Gamma}(S)$ of $S$ are examined. The fact that $\overrightarrow{\Gamma}(S)$ is a spanning subgraph of $\overrightarrow{\Gamma}_E(S)$ is proved. Necessary and sufficient conditions for the equality $\overrightarrow{\Gamma}_E(S)=\overrightarrow{\Gamma}(S)$ are obtained. Moreover, the conditions under which $\overrightarrow{\Gamma}_E(S)$ contains a cycle were established. In addition, conditions under which the diameter of $\overrightarrow{\Gamma}_E(S)$ is strictly smaller than that of $\overrightarrow{\Gamma}(S)$ are characterized. The vertices that are adjacent to or adjacent from all other vertices are investigated. The results are established by using several algebraic notions such as nilpotent elements, their nilpotency indices, and annihilator sets of elements.

We give a negative answer to a question due to Kim, Pearcy and Shields, namely, we prove that every nonscalar bounded linear operator on a Hilberts space $\mathcal{H}$ belongs to the class $\Delta(\mathcal{H})$ defined in [56]. This gives an affirmative solution to the (hyper) invariant subspace problem in $\mathcal{H}$. Our method of proof is based on Berezin symbol technique in reproducing kernel Hilbert spaces.

We introduce a filtration on the Chow ring of an abelian surface $A$, inspired by O'Grady's filtration on K3 surfaces. We give a geometric description of the filtration, and we prove that it is deeply linked with the rational orbit of points in the generalized Kummer variety of $A$. We propose a conjecture on the second Chern class of sheaves on this abelian surface, and we provide some evidence for the conjecture and prove some of its applications.

Over-the-air computation (AirComp) exploits multiple-access superposition to compute functions of distributed data without separately decoding all terminal messages. We study a reusable two-input AirComp gate for the exp-minus-log (EML) operation $\eml(u,v)=\exp(u)-\log(v)$, $v>0$. Thus, all internal nodes of a prescribed real-admissible EML tree reuse one gate type, avoiding node-specific nonlinear gate designs. Given an explicit EML tree whose intermediate logarithm arguments remain positive on a given compact domain, we derive additive white Gaussian noise (AWGN) and coherent flat fading implementations under peak-power constraints. We then characterize the number of gate evaluations, the dependency depth, evaluation latency, node-wise feasibility, deterministic error propagation, positivity preservation, and a high-probability AWGN error bound for the complete tree. A four-terminal two-hop example gives explicit positivity and end-to-end error conditions, and a digital interface propagates quantization and gate errors across the tree.

We develop an approach for computing the universal, in Vogel's sense, quantum dimensions. In a previous work a method for calculating the $\gamma$-independent part of these dimensions for universal multiplets with nonzero associated members for classical algebras was proposed. In the present paper, this approach is extended to the $\gamma$-dependent factors. In particular, we derive these contributions for the $E$ universal multiplet, which appears in the universal decomposition of the fourth power of the adjoint representation, thereby obtaining its universal quantum dimension for the first time. The analysis requires knowledge of the quantum dimensions of the classical and $E_8$ members of the multiplet; for the remaining exceptional algebras, the corresponding quantum dimensions are recovered automatically, providing an additional consistency check of the approach. We also partially extend a previously conjectured relation between the $sl$ and $so, sp$ members - formulated in terms of vertical and horizontal sums of Young diagrams - to the exceptional case. Finally, we present an explicit algorithm implementing the proposed method, which enables the derivation of universal formulae for higher-dimensional representations. The universal quantum dimensions provide, in some cases, a unified description of Chern-Simons observables, including knot invariants and partition functions, in a form valid simultaneously for all gauge algebras.

The cycle double cover conjecture states that every bridgeless graph has a list of cycles such that every edge is in exactly two of them. In July 2026, OpenAI announced a proof. This exposition presents the proof with slight modifications intended to make it more accessible.

We give a uniform structure for the first five tensor powers of the adjoint representation of an exceptional simple Lie algebra.

We prove an asymptotic formula for the density of parameters $c\in \mathbb{Z}_p$ for which the polynomial $x^d+c$ admits a $\mathbb{Q}_p$-rational $n$-cycle. In the cases where $d=2$ and $p\neq 2,3$, we compute these densities explicitly for $n=1,2,$ and $3$.

We study the problem of optimal control for dynamical systems near a pitchfork bifurcation, motivated by the role of external cues in guiding symmetry-breaking transitions in cell-fate selection and other natural processes. Using an asymptotic expansion of the optimality conditions obtained from the Pontryagin maximum principle, the leading-order optimal control law for a general n-dimensional system is examined across three dynamical regimes distinguished by scaling of control strength with respect to the distance from criticality. While in the strong control limit the results reduce to known approximations from linear-quadratic control, we derive generalized amplitude equations for the co-evolution of state and costate variables describing the optimized trajectory in the weak and intermediate control regimes. These control normal forms are validated against numerical solutions of the full optimal control problem for a canonical model of a bistable biochemical switch. The bifurcation structure of the optimal control problem is analyzed in the weak control regime. Finally, we demonstrate the construction of asymptotic solutions in the long time limit in this regime using boundary-layer methods.

We study $(2+1)$-dimensional oscillatory integral operators of the form \[ T_\lambda f(x,y)=\int_{\mathbb{R}}e^{i\lambda P(x,y)t^k}\psi(x,y,t)f(t)dt,\qquad k\geq 1, \] where the phase $P$ is a real-analytic function with a critical point at the origin. We establish the sharp $L^2\to L^2$ decay rate of $\frac12\min\{1/h_{P}, 1/k\}$, where $h_{P}$ denotes Varchenko's Newton height of $P$. The two terms in the minimum reflect a natural competition between the spatial degeneracy of $P$ and the temporal degeneracy of $t^k$; their optimality is confirmed by a Knapp-type and a focusing example, respectively. A $TT^{*}$ reduction transforms the $L^2$ estimate into a scalar oscillatory integral, allowing Varchenko's theorem to apply directly. Building on this foundation, complex interpolation yields the sharp $L^2\to L^{2k+2}$ bound. Finally, in the regime $h_{P}\geq k$, we obtain sharp $L^2\to L^p$ decay estimates for all $p$.

In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.

Reinforcement learning (RL) has recently emerged as a promising feedback control strategy for nonlinear and complex dynamical systems. However, RL algorithms are sample inefficient and require a large number of interaction with the environment to synthesize optimal control strategies. Consequently, applications of RL are typically limited to sparse sensors and actuators due to the curse of dimensionality entailed by the exploration-exploitation dilemma in high-dimensional spaces. In this work, we bridge RL and traditional optimal control for dynamical system with a novel Physics-EnhAnced Reinforcement Learning (PEARL) paradigm tailored to the control of high-dimensional and parametric dynamical systems, exploiting the differentibility of their dynamics. Specifically, PEARL employs an actor-adjoint algorithm that leverages automatic differentiation to compute policy gradients over short horizons and adjoint-based sensitivities of future returns approximated via neural networks, significantly reducing the number of environment interactions, while mitigating long-term gradient instabilities. Through two challenging parametric navigation problems in unsteady flows, we show that PEARL (i) effectively exploits differentiable environments to outperform state-of-the-art RL algorithms, (ii) is sample efficient, thanks to the physics-guided policy learning, (iii) generalizes across multiple scenarios, which is crucial when dealing with parametric systems, and (iv) enables scaling RL to high-dimensional state and action spaces, without requiring low-dimensional state representations or multi-agent strategies.

We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provide a machine-checked Lean 4 formalization of the main result.

We propose a definition of the reduction type of a curve over a discretely valued field in terms of the special fibre of an arbitrary regular model. We show that under this definition, the reduction type in terms of the minimal regular model determines and reduction type in terms of the minimal regular normal crossings model and vice versa. We also show that our definition is compatible with earlier classification results in low genus, namely the Kodaira-Néron classification for elliptic curves and the classification result of Namikawa-Ueno in genus 2. The essential new element in our definition is a suitable invariant of the singularities on the special fibre, building on the notion of equisingularity introduced by Zariski.

Computation of zigzag persistence has progressed in recent years, with results showing that complexities of many problems closely align with those in the non-zigzag setting. The major efficiency gap now lies in the updating of zigzag representatives. In this paper, we propose efficient algorithms for updating zigzag representatives based on a recent algorithm for extracting zigzag representatives from a $R=DV$ decomposition of a constructed non-zigzag. The main difficulty for designing our update algorithms lies in the adjacency change occurring in two operations that lengthen or shorten a filtration. Despite the adjacency change, we find that the update can still be done efficiently in quadratic time.

This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $\Gamma$ be the fundamental group of a closed orientable surface. We prove that if an element $g\in\Gamma$ has the same possible images as a given simple closed curve $\gamma\in \Gamma$ under epimorphisms from $\Gamma$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(\Gamma)$-orbit of $\gamma$, i.e. $g$ is itself a simple closed curve with the same topological type as $\gamma$. Consequently, the set of simple closed curves in $\Gamma$ is closed in the profinite topology of $\Gamma$; and we obtain a new algorithm to decide whether a given element in $\Gamma$ can be represented by a simple closed curve. Proper powers of simple closed curves and the pro-$p$ cases are also discussed.

A random sequence sampled from a known continuous distribution is observed with the objective to choose an item with the overall rank one. A rejected item cannot be recalled and is immediately erased from the memory. Under this memory constraint, the choice problem is not amenable to recursive methods of optimal stopping and becomes a global optimisation task. We focus on a heavy-traffic form of the problem with infinitely many choice opportunities, which we state in terms of a planar Poisson process (PPP). Symmetries of the PPP are used to derive basic structural properties of the optimal stopping rule, including the balance at the boundary equation, and two key integral identities. Throughout, we make throrough comparison to the classic full-information counterpart of the problem, revisiting both discrete- and continuous-time models. The optimal value, stopping rule and other characteristics of the problem are determined analytically and approximated numerically with high precision.

We construct stationary solutions of the three-dimensional incompressible Euler equations with helical symmetry and support in a neighborhood of a helix. The solutions are piecewise smooth and arise from a nonlinear overdetermined elliptic boundary value problem associated with a stream-function formulation. A distinguishing feature is that the vortex cross-sections are intrinsically anisotropic: after rescaling, the leading-order shape is elliptic rather than radial, and the boundary exhibits a nontrivial third Fourier mode reflecting helical effects absent in previous axisymmetric constructions. A key step in the proof is the analysis of a genuinely anisotropic overdetermined elliptic problem with prescribed Dirichlet and nonconstant Neumann conditions.

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $\delta$; the critical regime $\varepsilon = \delta$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (\delta)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier–Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (\delta^{1 + \iota})$ for some $\iota > 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = \Theta( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.

While quantum computers promise to solve classically intractable problems, identifying the point at which fault-tolerant quantum computation outperforms the best classical algorithms for practical applications remains an outstanding challenge. Here we establish a concrete quantum-classical crossover for quantum many-body dynamics under realistic hardware conditions. We introduce a scalable fault-tolerant framework that combines coherent observable estimation with a space-time-efficient implementation of non-Clifford rotations, suppressing the residual logical errors that limit existing partially fault-tolerant approaches. A benchmark against state-of-the-art tensor-network and variational Monte Carlo algorithms reveals a concrete crossover for mixed-field Ising dynamics at modest system sizes. For a physical error rate of $p=10^{-3}$, fault-tolerant simulation requires approximately 2 hours and $3.7 \times 10^5$ physical qubits for a 100-site 1D system, whereas tensor network approaches would require about 100 years. For 2D models, where rapid entanglement growth limits the classical evolution time, we project quantum runtimes within minutes. A physical error rate of $p=10^{-4}$ leads to at least an order of magnitude reduction in qubit count ($3.1 \times 10^4$ physical qubits) and runtime (minutes for 1D and seconds for 2D). The reduction in quantum runtime arises from our improved rotation-state injection and co-design of quantum error correction and observable-estimation protocols, which jointly suppress logical-error accumulation and reduce sampling overhead. Our results establish a scalable route towards practical quantum advantage and identify quantitative engineering targets for future fault-tolerant architectures.

It is well-known that combinatorial circuits are modeled mathematically by string diagrams in monoidal categories. Given a gate set $\Sigma$, the circuits over $\Sigma$ can be thought of as string diagrams in the free monoidal category generated by $\Sigma$. In this model, circuit semantics are then given by monoidal functors out of this free category. For quantum circuits, this functor is often valued in the category of unitary matrices. This model suffices for concrete quantum circuits, but fails to describe parameterized families of quantum circuits, such as those which arise in the analysis of ansatz circuits. In this paper, we introduce an approach to parameterized circuit semantics, which is based on enriched category theory. We first introduce an abstract categorical construction, and use this to gain new insights on controlled operations and quantum communication. We then study the special cases of Cartesian monoidal parameters and monoidal closed parameters, both endowing the parameterized semantics with useful constructions. We conclude by showing that the monoidal closed case can be used to unify two perspectives on quantum control.

We prove $L^p\rightarrow L^4$ Fourier adjoint restriction estimates for three-dimensional quadratic surfaces in $\mathbb{R}^5$. These estimates are sharp up to the endpoint, and improve results by Oberlin and Guo–Oh.

We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most \[ \frac{69}{25}(n-2) \] edges, for every $n\ge 8$. This improves the best known bound \[ \frac{323}{108}n-6 \qquad \text{for every } n\ge 27. \]