Papers

The fundamental task of quantum hypothesis testing, which consists in discriminating quantum objects, has been extensively studied for quantum states and channels. Here, we investigate the fundamental limits of discriminating an arbitrary pair of quantum Markovian dynamics generated by Lindbladians, focusing on the roles of total exposure time and temporal resolution. We allow adaptive choices of evolution durations and arbitrary instantaneous controls between them. We characterise the optimal type-II error exponent (Stein exponent) as the largest instantaneous increase of quantum relative entropy, and give an algebraic criterion for its finiteness in terms of the Lindbladians. Remarkably, an infinite exponent is equivalent to a vanishing type-II error probability at sufficiently large finite total time, for every positive type-I tolerance. While some dynamics admit a finite protocol with zero type-II error, others require increasingly fine temporal control. For the latter, the error vanishes as the minimum allowed evolution duration, or resolution time, tends to zero at fixed sufficiently large total time, but remains bounded away from zero at every fixed positive resolution time. We achieve this using quantum Zeno protocols, establishing temporal resolution as a resource for discrimination. When the null hypothesis is the identity semigroup, an infinite asymptotic error exponent has a stronger consequence: at a fixed finite total exposure time, the type-II error can be made arbitrarily small while keeping the type-I error exactly zero. This is impossible in the corresponding classical setting. Discriminating the identity semigroup from generalised amplitude damping on a qubit provides examples of all three regimes: a finite asymptotic error exponent, zero type-II error attainable at positive resolution time, and vanishing type-II error requiring the resolution time to tend to zero.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini–Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini–Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

The susceptibility tensor of a homogeneous collisionless magnetized plasma possesses parity-time (PT) symmetry. For a stationary gyrotropic equilibrium, with the wave vector $\mathbf{k}$ and the equilibrium magnetic field $\mathbf{B}_0$ in the $x$–$z$ plane, the kinetic response satisfies $\chi(\omega^*,\mathbf{k})=R\chi(\omega,\mathbf{k})^*R$, where $R=\operatorname{diag}(1,-1,1)$. We show that the susceptibility PT-symmetry is a reduced-response-space manifestation of PT symmetry of the underpinning kinetic system. We identify parity $P$ and time reversal $T$ on the electromagnetic kinetic eigenmode space and show that the Hamiltonian operator $\mathcal{H}_{\mathrm{VM}}$ of the linearized Vlasov–Maxwell system is PT-symmetric, satisfying $PT\mathcal{H}_{\mathrm{VM}}(PT)^{-1}=\mathcal{H}_{\mathrm{VM}}$. The PT symmetry of the homogeneous magnetized Vlasov–Poisson eigenproblem is also established. Many kinetic instabilities associated with nonmonotonic distributions, rings, loss cones, and resonant energetic-particle populations are fundamentally spontaneous PT-symmetry breaking, which is achieved through and only through Krein collisions. Three examples demonstrate distinct utilities of this analysis. Definite-sign cyclotron residues protect thermal Bernstein modes from spontaneous PT-symmetry breaking. The nonmonotonic Dory–Guest–Harris distribution permits sign-indefinite residues and Krein collisions, for which we derive an analytical instability threshold and growth rate. The recently discovered mode-pole resonance instability provides a distinct realization of spontaneous PT-symmetry breaking and illustrates how PT symmetry can guide the discovery of kinetic instabilities with applications to fusion energy. These results establish PT symmetry as a structural principle of plasma waves and instabilities and open new directions for studying spectral topology in kinetic plasmas.

For the coupled three-dimensional double well $H=\tfrac12|p|^2+2\sum_i(q_i^2-1)^2-\sum_{i<j}q_i^2q_j^2$ we exhibit, at the first excited energy level (and also at the ground state), a linear functional on polynomials of degree $\le 6$ that satisfies every constraint of the standard level-3 eigenstate bootstrap, has a strictly positive definite moment matrix, and nevertheless assigns a negative value to a nonnegative polynomial. It is therefore an interior point of the level-3 relaxation that is realized by no quantum state. The construction reflects a fact that holds at every level: the eigenstate rows never reach the top-degree momentum moments. At level 3, where strictly feasible points exist, this means that the corresponding block of the moment matrix carries no information about which energies admit one. Every algebraic ingredient is an exact rational certificate, and the only numerical ingredients (spectral enclosures of an eigenfunction) are carried out in rigorous ball arithmetic.

Dynkin classified the maximal subalgebras of the simple finite-dimensional Lie algebras. The super version of his problem is solved, for the matrix Lie superalgebras, by Shchepochkina, and, for the simple vectorial ones with polynomial coefficients taken with their Weisfeiler gradings, in the companion paper by Leites and Shchepochkina — with three exceptions left open: the three simple finite-dimensional Lie superalgebras $\mathfrak{osp}_a(4|2)$, $\mathfrak{ag}(2)$ and $\mathfrak{ab}(3)$, which are vectorial in more than one way and are not covered by the general argument. Here we settle these three cases for graded subalgebras. For each of the three Lie superalgebras, we list all the $\mathbb{Z}$-gradings to which the method of that paper applies — two for $\mathfrak{osp}_a(4|2)$, five for $\mathfrak{ag}(2)$, eight for $\mathfrak{ab}(3)$ — and describe, for every one of them, all the maximal graded subalgebras, the semi-simple ones explicitly. A simple criterion upgrades most of the answers from “maximal graded\rq\rq\ to “maximal\rq\rq. Every answer is computed over $\mathbb{Q}$. Two by-products: the classical list of five cases into which the graded subalgebras of a vectorial Lie superalgebra split is incomplete for ambients of depth $>1$ — a sixth case has to be added, and it is non-empty precisely here; and the parameter $a$ of the copies of $\mathfrak{osp}_a(4|2)$ sitting inside $\mathfrak{ag}(2)$, resp. $\mathfrak{ab}(3)$, is computed intrinsically.

We consider the interaction-round-a-face version of the nineteen-vertex model and derive a new quantum spin-$1$ chain. In the isotropic limit, the associated spin chain can be seen as three coupled Babujian-Taktahjan spin chains with different boundary twists. We compute the ground state short-distance correlation functions of the isotropic model in the thermodynamic limit by exploiting the shared integrable structure between the models.

S. Lie was the first to try to classify maximal subalgebras of simple vectorial (i.e., consisting of vector fields) Lie algebras with polynomial coefficients. In Appendix it is shown that, as stated, the problem is wild: describing all subalgebras contains the classification of pairs of matrices up to simultaneous conjugation, so there are too many such subalgebras. Several researchers distinguished various classes of maximal subalgebras important in applications. Here, in simple (or in Cartan prolongations close to simple) vectorial Lie superalgebras with polynomial coefficients considered with gradings associated with Weisfeiler filtrations, we describe maximal simple (and close to simple) graded subalgebras — a superization of what Sophus Lie tackled. Together with I. Shchepochkina's description of the maximal subalgebras of matrix Lie superalgebras, see arXiv:hep-th/9702122, this paper solves the super and infinite-dimensional version of Dynkin's problem — the description of the maximal subalgebras of simple finite-dimensional Lie algebras — except for the 3 exceptional ambients considered elsewhere.

Uncertainty quantification (UQ) for random partial differential equations (PDEs) is ubiquitous in computational science and engineering. However, classical spectral solvers for this class of problems face the curse of dimensionality, and existing neural solvers often ignore the stochastic structure that makes moments and calibration tractable. We introduce a stochastic separable physics-informed neural network, dubbed S$^{2}$-PINN, that represents the solution $u(t,\mathbf{x},\mathbf{Z})$ of a random PDE with a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos (gPC) stochastic basis, coupled by a low-rank Canonical Polyadic (CP) tensor decomposition core. The method is trained with a hybrid strong-form and gPC-projected residual loss. Our theoretical analysis establishes that the separable class is dense in $L^2$ under mild conditions, and the projected residual corresponds exactly to a stochastic Galerkin constraint. Furthermore, we show that mini-batch projection coefficients are logarithmically dependent on the number of gPC modes, and that the orthogonality penalty controls the conditioning of the learned spatial dictionary. Using four manufactured random PDE benchmarks, we show that S$^{2}$-PINN outperforms nine baselines in terms of mean and variance accuracy, as well as calibration, while using significantly fewer parameters. Further evaluations on non-manufactured Poisson and Darcy problems, a stochastic Navier–Stokes problem, a diffusion scaling study of higher random dimensions, and two stochastic inverse problems reveal the generalization capabilities of the proposed structure. Together, these results support stochastic separability as an effective design principle for physics-informed neural UQ. The code for the experiments can be found in https://github.com/DMax1314/s2pinn

We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $\alpha'$. Auxiliary fields for the inverse coframe, spinor derivative, flux and induced tangent connection give a shifted-cotangent Hamiltonian of degree at most four. Eliminating the auxiliary fields recovers the physical functional. On the flat seven-torus, this functional has a nonzero quintic Taylor coefficient in local physical spinor coordinates. For $Y=S^1\times X$, with $X$ a Calabi–Yau threefold, we restrict to the product locus with horizontal gauge and gerbe data. The functional reduces to $ \frac L4\operatorname{Im}\int_X(H+i\,d\omega)\wedge\Omega, $ where $L$ is the circle length and $\Omega=e^{-2\Phi}\Psi$ is the weighted complex volume. The induced $SU(3)$ structure selects a Hermitian tangent Courant subalgebroid. Transverse isotropic lifts in its complexification encode the holomorphic connection and string equations, together with the induced tangent connection, weighted canonical line and gerbe hierarchy. Their Taylor brackets form an $L_3$ algebra whose string part agrees with the known holomorphic heterotic algebra. We then show that this holomorphic theory is obtained by characteristic Hamiltonian reduction of a finite contractible stabilization of the product variational theory, in every cochain degree. While the quotient captures the holomorphic equations, conformal balance, gauge primitivity and leading-flux admissibility are additional conditions on a real representative.

We derive long-time asymptotics in different space-time regions for the Sawada-Kotera (SK) equation with admissible data involving continuous spectrum and finite discrete spectrum. Our result proves the soliton resolution conjecture, yields the asymptotics governed by the F-XVIII Painlevé transcendent, and establishes the asymptotic stability of \(N\)-soliton solutions. More importantly, this extends the long-time asymptotic theory to the coexistence of solitons and radiation, whereas previous results were restricted to the purely continuous spectrum. One key ingredient in our argument is a new meromorphic Riemann-Hilbert framework that incorporates the discrete spectrum through complete six-point pole orbits while preserving cofactor analyticity and the singular structure at the spectral origin. A second ingredient is the combination of an orbitwise, region-dependent pole reduction mechanism and the \(\bar{\partial}\)-nonlinear steepest descent method, which allows us to select the solitons contributing in each region and separate them from the continuous radiation. Our framework also resolves the coalescing stationary points at the singular spectral origin through the regular modified SK problem and the Miura map.

We develop a cancellation mechanism for linear second-order kinetic equations in non-divergence form with measurable uniformly elliptic coefficients. We decompose the dynamics into spatial waves and follow two families of velocity paths along which the waves acquire nearly opposite phases. The parabolic Krylov–Safonov theory gives a common lower bound for the two velocity marginals. The corresponding contributions cancel up to a small phase error, producing a contraction \textità la Döblin}. Iterating this contraction yields exponential Fourier decay estimates with enhanced dissipation. We then present two applications. The first contribution is a kinetic Aleksandrov estimate: a maximum principle in which the source is measured in an $L^p$ norm. For rough coefficients $A(t,v)$ independent of position, we obtain the estimate for every $p>2n+1$, where $n$ is the dimension of position and velocity. The Fourier decay also gives spatial smoothness, and parabolic regularity gives Hölder continuity in time and velocity. For autonomous coefficients $a(x,v)$ in dimension one, position serves as time away from zero velocity. A Harnack comparison controls returns to small velocity intervals and yields the estimate for every $p>4$. Both thresholds are optimal among those valid for all ellipticity ratios. The second contribution concerns position on the torus and velocity on the sphere, with measurable uniformly elliptic diffusion coefficients $A(t,v)$ independent of position. We prove enhanced dissipation with the optimal square-root frequency power and Gevrey regularity in position. For autonomous coefficients $A=A(v)$, we also obtain exponential convergence to equilibrium in $L^2$ weighted by the invariant velocity measure, and a quantitative spectral gap. To our knowledge, these are the first results on decay established for laws of kinetic equations with rough coefficients.

In Rational Extended Thermodynamics, the monatomic limit of polyatomic gases is often singular. For the internal-state density $\phi(I)=I^a$, with $a=(D-5)/2$, we prove that the normalized equilibrium distribution of the molecular internal energy $I$ concentrates at $I=0$ as $D\to3$, providing a rigorous justification of the Dirac prescription and its first-order correction. In classical ET14, the internal-mode energy differs from equilibrium by $-3\Pi/2$; positivity yields $-p<\Pi<(D-3)p/3$, while the exact entropy develops a boundary corner at $\Pi=0$. Negative dynamic pressures remain admissible, but fixed negative values do not approach the monatomic state with the prescribed total energy. This explains the need for compatible initial data. In the relativistic case, the internal energy is conditionally gamma distributed at fixed momentum; the constitutive integrals follow directly, and we derive an exact energy identity and the first correction to the Synge energy.

An extension with temporal modulation is presented of the canonical 2 + 1-dimensional solitonic Dym equation as originally introduced by Konopelchenko and Dubrovsky. Application is made of admitted Painleve' II symmetry reduction to derive exact solution to a class of associated 2 + 1-dimensional nonlinear moving boundary problems of generalized Stefan-type.

We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra–Itzykson–Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices $A,B$ with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for $(BA)^*(BA)\otimes\mathbf{1}_N$ has two limiting descriptions at speed $N^2$ for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for $(A^*A)\otimes(B^*B)$. For every fixed number of factors $k\geq3$, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number $k$ of factors, global Haar vectors at speed $N^k$ and product-state vectors at speed $N$ yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the $S$-transform.

The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system. While it has been asserted in the literature on quantum information theory that decomposing this state space into equirank pieces yields a Whitney stratification, rigorous proofs have remained absent. Using methods from invariant theory, this work provides a proof that the rank decomposition of the matrix algebra constitutes a (b)-regular stratification in the sense of Whitney. Furthermore, we demonstrate how the Uhlmann connection - which describes the adiabatic holonomy of mixed states and reduces to the Berry connection on pure states - can be geometrically treated as a stratified connection on a stratified vector bundle. Finally, our appendix provides a self-contained exposition of stratified fiber bundles and the new concept of stratified connections, which may be of independent interest.

We develop a rigorous justification for a parameter estimation algorithm which couples continuous data assimilation to generic optimization. For finite dimensional systems, we provide a rigorous justification of an asymptotic approximation of the sensitivity for the underlying modeled dynamical system, prove that the $L^2$ loss function satisfies an approximate Polyak-Lojasiewicz inequality, and use that result to justify convergence of gradient descent for the proposed algorithm. Numerical examples are provided that demonstrate the precision of the rigorous results.

Inferring an Allen-Cahn constitutive force from images requires more than an accurate optimizer: nonlinear observation noise alters the mean of force features, shared noise couples instruments to the response, and a temporal balance borrowed from another solver can remain biased even on clean data. We develop observation-aware, noise-adjusted moment equations that address these effects before inversion. The surface-tension-calibrated constitutive transformer (SCCT) couples the resulting moments to a positive Bernstein inverse and can adapt its prior and precision from images and moments. A conditional stability bound separates moment residuals from prior error. Synthetic tests show that correcting the estimating equation matters more than increasing regularizer flexibility; the benefit of learned adaptation is smaller and depends on the training comparison. Forces inferred from planar movies predict unseen planar and three-dimensional evolutions. A ring-cage breakup shows why diffuse-interface accuracy distinguishes forces even when their topological predictions agree, while external surface tension sets the energy scale separately.

We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.

We investigate whether the geometry of a curved conductor–dielectric interface can modify the effective mass and propagation of scalar surface plasmons (SSPs). In the hydrodynamic limit, we describe SSPs as density perturbations in a semiclassical jellium electron gas. Using a thin–layer dimensional reduction procedure, we derive a massive Klein–Gordon equation for SSPs on the curved interface with a geometric potential. We find that under specific conditions, the geometric potential allows SSP propagation with a vanishing effective mass. We then apply the formalism to a well known analogue black hole configuration, determining the corresponding quasinormal mode frequencies. Our results show that interface curvature provides a geometric handle on plasmon propagation and localization, suggesting new possibilities for plasmon transport and analogue gravity experiments with surface plasmons.

The convergence of hydrodynamics is commonly investigated by Taylor-expanding the gapless dispersion relations $\omega(k)$ around $k=0$. We show that this diagnostic can fail arbitrarily close to the hydrodynamic limit: sound modes can collide at arbitrarily small wave number, rendering $\omega(k)$ nonanalytic, while the relationship between densities and fluxes remains analytic there. We instead formulate the gradient expansion off-shell (i.e. in the presence of external forces), directly at the level of the constitutive relations. For causal theories with relaxational spectra, we establish convergence for $|i\omega|+w|ik|<1/\tau_g$, where $w$ is the maximal information speed and $1/\tau_g$ the nonhydrodynamic gap. On the naturally Lorentzian $\{i\omega,ik\}$ plane, this region is a causal diamond, whose boundary marks the first possible encounter with nonhydrodynamic modes. Beyond it, analytic continuation reveals the nonhydrodynamic spectrum through singularities of the resummed constitutive relations. Thus, hydrodynamics is aware of the degrees of freedom beyond hydrodynamics, but this information becomes fully accessible only off-shell.

Dissipative dynamics can stabilize a quantum target without providing a reliable sequence of finite control operations. We derive checkable conditions for implementing such stabilization with imperfect controls, with bounds on convergence, errors during operations, and physical cost. The analysis includes pulses, idle intervals, and all simultaneously acting background dynamics. Using a common Lyapunov measure of deviation from the target, we require a contraction bound that remains uniformly below one over the prescribed uncertainty set after accounting for implementation errors. If all allowed paths preserve the target subspace throughout each operation, repeated cycles converge to that subspace from every initial state, even when errors vary between cycles. Comparison with these certified target-preserving paths gives an explicit long-time error bound under bounded leakage. Two single-qubit counterexamples show why a stabilizing average or a stabilizing generator obtained by unitary conjugation does not ensure these path conditions. The certification approach also supports a finite-library search. For a specified four-level cascade model with finite pulses, calibration and timing errors, dephasing, and bounded pumping, this search yields a sequence with an explicit preparation-time bound. For every initial state, it guarantees trace distance at most $0.1$ from the target at the end of preparation and throughout subsequent cycles.

We establish lower bounds on the number of resonances for three-dimensional semiclassical Dirac operators with mass-type potentials. For a pair $\Dl_j=\Dl_0+\mbf{\beta}v_j$, $j=1,2$, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution $\mu$ to the corresponding mass functions. We prove that if $E_0>1$ is a boundary point of $\supp\mu$, then every complex neighbourhood of $E_0$, and likewise of $-E_0$, contains in total at least $C\hbar^{-3}$ resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of $\mu$ to the relative classical density of states and combines this with Khochman's local trace formula and a relative semiclassical trace formula. As a consequence, if $v$ is a single bounded mass-type potential, analytic with power decay in an exterior sector, with $\inf(1+v)>0$ and $\sup v>0$, then $\Dl=\Dl_0+\mbf{\beta}v$ has at least $C\hbar^{-3}$ resonances in every complex neighbourhood of $E_0=1+\sup v$ and of $-E_0$, for all sufficiently small $\hbar$. The exponent agrees with Khochman's upper bound, the two resonance families are related by spectral symmetry, and $E_0$ is necessarily a trapping energy. We also prove the relative semiclassical trace formula needed in the argument for general Hermitian matrix-valued perturbations.

We construct spectral-parameter solutions of the quantum Yang–Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ – which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang–Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang–Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.

Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus–Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.

We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any $[[n,k]]$ CSS quantum low-density parity-check (QLDPC) code can be transformed into an $[[nm,k]]$ CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in $O(\log m)$ parallel time. Below a constant threshold $p_{\rm RG}>0$ for joint local stochastic physical and measurement noise, the logical failure probability over $T$ correction rounds is bounded by \begin{equation*} O(T n)\exp[-\Omega(m^\alpha)] \end{equation*} for a constant $\alpha>0$; we say that the threshold is $p_{\rm RG}$ and error suppression is $\Omega(m^{\alpha})$. Moreover, if the input code is equipped with a single-shot decoder with threshold $p_c>0$ and error suppression $\Omega(n^{\beta})$, then the output code enhances the error suppression to $\Omega(m^{\alpha} n^{\beta})$, while maintaining the decoder-independent threshold $p_{\rm RG}$, and thus enhanced if $p_c < p_{\rm RG}$.

We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local $\Oc(h^{-3})$ upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy–Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations $\mbf{\beta}V(x)$ in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time $T_\chi$ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed $M$ below the resonance-free depth constant, derive the explicit local-energy decay threshold $T_N>T_\chi+(N+1)/M$. The same analysis gives a rigorous microlocal obstruction to a uniform $[h\log(1/h)]^{-1}$ bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.

Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on $n$ $d$-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error $\epsilon$ in diamond distance using $O(n\min\{n,d\}^{3}\log^2(n/\epsilon))$ random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the $\mathrm{SU}(d)$-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.

Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the $\mathbb{Z}_2$ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the $\mathbb{Z}_N$ toric code, we find that an order-two Clifford realization exists for odd $N$, whereas it is impossible for even $N$. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.