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Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This construction is based on flat geometry as our proposed volume is a limit of Masur-Veech volumes of moduli spaces of multi-differentials. This idea generalizes the observation in quantum gravity that the Jackiw-Teitelboim partition function is a limit of minimal string partition functions from Liouville gravity. Finally, we use the properties of these volumes to recover Mirzakhani's recursion formula for Weil-Petersson polynomials. This provides a new proof of Witten-Kontsevich's theorem.

We obtain a formal integral solution to the 3+1 D Boltzmann Equation in relaxation time approximation. The gradient series obtained from this integral solution contains exponentially decaying non-hydrodynamic terms. It is shown that this gradient expansion can have a finite radius of convergence under certain assumptions of analyticity. We then argue that, in the relaxation time model, proximity to local thermal equilibrium is not necessary for the system to be described by hydrodynamic equations., Comment: 12 pages, 0 figures

We prove a quantitative estimate on the homogenization length scale in terms of the ellipticity ratio $\Lambda/\lambda$ of the coefficient field. This upper bound is applicable to high contrast elliptic equations demonstrating near-critical behavior. Specifically, we show that, given a suitable decay of correlation, the length scale at which homogenization is observed is at most $\exp(C \log^3(1+\Lambda/\lambda))$. The proof introduces the new concept of coarse-grained ellipticity, which measures the effective ellipticity ratio of the equation -- and thus the strength of the disorder -- after integrating out smaller scales. By a direct analytic argument, we obtain an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach can be interpreted as a rigorous renormalization group argument, and provides a quantitative framework for homogenization that can be iteratively applied across an arbitrary number of length scales., Comment: 155 pages; announcement at https://www.scottnarmstrong.com/2024/05/high-contrast-homogenization/

We review the notion of symplectic duality earlier introduced in the context of topological recursion. We show that the transformation of symplectic duality can be expressed as a composition of $x-y$ dualities in a broader context of log topological recursion. As a corollary, we establish nice properties of symplectic duality: various convenient explicit formulas, invertibility, group property, compatibility with topological recursion and KP integrability. As an application of these properties, we get a new and uniform proof of topological recursion for large families of weighted double Hurwitz numbers; this encompasses and significantly extends all previously known results on this matter., Comment: 28 pages

We model an overdamped Brownian particle that is subject to resetting facilitated by a periodic ratchet potential. This asymmetric potential switches on with a constant rate, but switches off again only upon the particle's first passage to a resetting point at the minimum of the potential. Repeating this cycle sustains a non-equilibrium steady-state, as well as a directed steady-state current which can be harnessed to perform useful work. We derive exact analytic expressions for the probability densities of the free-diffusion and resetting phases, the associated currents for each phase, and an efficiency parameter that quantifies the return in current for given power input. These expressions allow us to fully characterise the system and obtain experimentally relevant results such as the optimal current and efficiency. Our results are corroborated by simulations, and have implications for experimentally viable finite-time resetting protocols., Comment: 23 pages (14 pages main), 13 figures

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schr\"odinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms., Comment: 55 pages, to appear in J. Funct. Anal

Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force., Comment: 24 pages, 4 figures

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum., Comment: 49 pages, comments welcome

This study explores the relationship between the precise asymptotics of the level-two large deviation rate function and the behavior of metastable stochastic systems. Initially identified for overdamped Langevin dynamics (Ges{\`u} et al., \textit{SIAM J Math Anal} 49(4), 3048-3072, 2017), this connection has been validated across various models, including random walks in a potential field. We extend this connection to condensing zero-range processes, a complex interacting particle system. Specifically, we investigate a certain class of zero-range processes on a fixed graph $G$ with $N > 0$ particles and interaction parameter $\alpha > 1$. On the time scale $N^2$, this process behaves like an absorbing-type diffusion and converges to a condensed state where all particles occupy a single vertex of $G$ as $N$ approaches infinity. Once condensed, on the time scale $N^{1+\alpha}$, the condensed site moves according to a Markov chain on $G$, showing metastable behavior among condensed states. The time scales $N^2$ and $N^{1+\alpha}$ are called the pre-metastable and metastable time scales. It is conjectured that this behavior is encapsulated in the level-two large deviation rate function $\mathcal{I}_N$ of the zero-range process. Specifically, it is expected that the $\Gamma$-expansion of $\mathcal{I}_N$ can be expressed as:$$\mathcal{I}_N = \frac{1}{N^2} \mathcal{K} + \frac{1}{N^{1+\alpha}} \mathcal{J},$$ where $\mathcal{K}$ and $\mathcal{J}$ are the level-two large deviation rate functions of the absorbing diffusion processes and the Markov chain on $G$. We rigorously prove this $\Gamma$-expansion by developing a methodology for $\Gamma$-convergence in the pre-metastable time scale and establishing a link between the resolvent approach to metastability (Landim et al., \textit{J Eur Math Soc}, 2023. \texttt{arXiv:2102.00998}) and the $\Gamma$-expansion in the metastable time scale., Comment: 58 pages

In this paper we introduce a simple variational model describing the ground state of a superconducting charge qubit. The model gives rise to a shape optimization problem that aims at maximizing the number of qubit states at a given gating voltage. We show that for small values of the charge optimal shapes exist and are $C^{2,\alpha}$-nearly spherical sets. In contrast, for large values of the charge the optimal shape does not exist, with the energy favoring disjoint collections of sets., Comment: 33 pages

We present a criteria affirming the existence of discrete ground states of the Dirac operators which describe the fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model. Additionally, we investigate a sufficient condition that elucidates the phenomenon of supersymmetry breaking within the supersymmetric quantum mechanics associated with this Dirac operator., Comment: 12 pages

In this paper, we consider a six parameter family of affine Segre surfaces embedded in $\mathbb C^6$. For generic values of the parameters, this family is associated to the $q$-difference sixth Painlev\'e equation. We show that different limiting forms of this family give Segre surfaces that are isomorphic as affine varieties to the the monodromy manifolds of each Painlev\'e differential equation., Comment: 66 pages, 4 figures

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed., Comment: LaTex, 80 pages

In this article, we consider Vlasov-type equations describing the evolution of single-species type plasmas, such as those composed of electrons (Vlasov-Poisson) or ions (screened Vlasov-Poisson/Vlasov-Poisson with massless electrons). We solve the final data problem on the torus $\mathbb{T}^d$, $d \geq 1$, by considering asymptotic states of regularity Gevrey-$\frac{1}{\gamma}$ with $\gamma>\frac13$, small perturbations of homogeneous equilibria satisfying the Penrose stability condition. This extends to the Gevrey perturbative case, and to higher dimension, the scattering result in analytic regularity obtained by E. Caglioti and C. Maffei in [14], and answers an open question raised by J. Bedrossian in arXiv:2211.13707., Comment: 31 pages

We study an initial-boundary-value problem (IBVP) for a system of coupled Maxwell-Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two-level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches non-vanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly-modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax Pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann-Hilbert problem is used to extract the spatio-temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium which fail to interact with a given pulse.

The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang-Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as $q$-oscillator valued vertex models with matrix elements of the $L$-operators given by generators of the $q$-oscillator algebra acting on the Fock space. Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions., Comment: 28 pages, 17 figures

We consider Dirac operators on the half-line, subject to generalised infinite-mass boundary conditions. We derive sufficient conditions which guarantee the stability of the spectrum against possibly non-self-adjoint potential perturbations and study the optimality of the obtained results. Finally, we establish a non-relativistic limit which makes a relationship of the present model to the Robin Laplacian on the half-line., Comment: 17 pages, 1 figure

We explore statistical properties of BPS q-series for 3d N=2 strongly coupled supersymmetric theories that correspond to a particular family of 3-manifolds Y. We discover that gaps between exponents in the q-series are statistically more significant at the beginning of the q-series compared to gaps that appear in higher powers of q. Our observations are obtained by calculating saliencies of q-series features used as input data for principal component analysis, which is a standard example of an explainable machine learning technique that allows for a direct calculation and a better analysis of feature saliencies., Comment: 11 pages, 4 figures, 3 tables

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.

We address the homogenization of the two-dimensional Dirac operator with position-dependent mass. The mass is piecewise constant and supported on small pairwise disjoint inclusions evenly distributed along an $\varepsilon$-periodic square lattice. Under rather general assumptions on geometry of these inclusions we prove that the corresponding family of Dirac operators converges as $\varepsilon\to 0$ in the norm resolvent sense to the Dirac operator with a constant effective mass provided the masses in the inclusions are adjusted to the scaling of the geometry. We also estimate the speed of this convergence in terms of the scaling rates., Comment: 18 pages, 2 figures

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. We obtained all the Lie symmetries admitted by the KP equation and used them to reduce the (2+1)-dimensional fractional partial differential equation with Riemann-Liouville fractional derivative to some (1+1)-dimensional fractional partial differential equations with Erd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

We have discovered analytical expressions for the probability density function (PDF) of photons that are multiply scattered in relativistic flows, under the assumption of isotropic and inelastic scattering. These expressions characterize the collective dynamics of these photons, ranging from free-streaming to diffusion regions. The PDF, defined within the light cone to ensure the preservation of causality, is expressed in a three-dimensional space at a constant time surface. This expression is achieved by summing the PDFs of photons that have been scattered $n$ times within four-dimensional spacetime. We have confirmed that this formulation accurately reproduces the results of relativistic Monte Carlo simulations.We found that the PDF in three-dimensional space at a constant time surface can be represented in a separable variable form. We demonstrate the behavior of the PDF in the laboratory frame across a wide range of Lorentz factors for the relativistic flow. When the Lorentz factor of the fluid is low, the behavior of scattered photons evolves sequentially from free propagation to diffusion, and then to dynamic diffusion, where the mean effective velocity of the photons equates to that of the fluid. On the other hand, when the Lorentz factor is large, the behavior evolves from anisotropic ballistic motion, characterized by a mean effective velocity approaching the speed of light, to dynamic diffusion., Comment: The version accepted by ApJ Letters

In this paper we introduce the $r$-equilibrium problem and discuss connections to the map $det^{S^r}$. The case $r=2$ is an application of Newton's third law of motion, while $r=3$ deals with equilibrium of torque-like forces., Comment: 11 pages, all comments are welcome

This article studies the uniqueness of the weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the investigation is provided using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness using the new approach. The second result is without the complementary conditions but is, in some sense, the "local" result investigated by another approach. In addition, here the solvability and uniqueness of the weak solutions to auxiliary problems lead out from the main problem are investigated., Comment: 24 pages

We construct the vector field associated with the GKLS generator for systems described by Gaussian states. This vector field is defined on the dual space of the algebra of operators, restricted to operators quadratic in position and momentum. It is shown that the GKLS dynamics accepts a decomposition principle, that is, this vector field can be decomposed in three parts, a conservative Hamiltonian component, a gradient-like and a Choi-Krauss vector field. The last two terms are considered a "perturbation" associated with dissipation. Examples are presented for a harmonic oscillator with different dissipation terms., Comment: 33 pages, 6 figures

We consider quantum systems with energy constraints. In general, quantum channels and continuous-time dynamics need not satisfy energy conservation. Physically meaningful channels, however, can only introduce a finite amount of energy to the system, and continuous-time dynamics may only increase the energy gradually over time. We systematically study such "energy-limited" channels and dynamics. For Markovian dynamics, energy-limitedness is equivalent to a single operator inequality in the Heisenberg picture. By tracking the output energy, we observe that the energy-constrained operator and diamond norms of Shirokov and Winter satisfy submultiplicativity estimates with respect to energy-limited channels. This makes for a powerful toolkit for quantitative analyses of dynamical problems in finite and infinite-dimensional systems. As an application, we derive state-dependent bounds for quantum speed limits and related problems that outperform the usual operator/diamond norm estimates, which have to account for fluctuations in high-energy states., Comment: Comments welcome

The probabilistic characterization of non-Markovian responses to nonlinear dynamical systems under colored excitation is an important issue, arising in many applications. Extending the Fokker-Planck-Kolmogorov equation, governing the first-order response probability density function (pdf), to this case is a complicated task calling for special treatment. In this work, a new pdf-evolution equation is derived for the response of nonlinear dynamical systems under additive colored Gaussian noise. The derivation is based on the Stochastic Liouville equation (SLE), transformed, by means of an extended version of the Novikov-Furutsu theorem, to an exact yet non-closed equation, involving averages over the history of the functional derivatives of the non-Markovian response with respect to the excitation. The latter are calculated exactly by means of the state-transition matrix of variational, time-varying systems. Subsequently, an approximation scheme is implemented, relying on a decomposition of the state-transition matrix in its instantaneous mean value and its fluctuation around it. By a current-time approximation to the latter, we obtain our final equation, in which the effect of the instantaneous mean value of the response is maintained, rendering it nonlinear and non-local in time. Numerical results for the response pdf are provided for a bistable Duffing oscillator, under Gaussian excitation. The pdfs obtained from the solution of the novel equation and a simpler small correlation time (SCT) pdf-evolution equation are compared to Monde Carlo (MC) simulations. The novel equation outperforms the SCT equation as the excitation correlation time increases, keeping good agreement with the MC simulations., Comment: 46 pages, 8 figures, 2 appendices, 103 references, Supplementary material

There exist several good reasons why one may wish to add a total derivative to an interaction in quantum field theory, e.g., in order to improve the perturbative construction. Unlike in classical field theory, adding derivatives in general changes the theory. The analysis whether and how this can be prevented, is presently limited to perturbative orders $g^n$ , $n \leq 3$. We drastically simplify it by an all-orders formula, which also allows to answer some salient structural questions. The method is part of a larger program to (re)derive interactions of particles by quantum consistency conditions, rather than a classical principle of gauge invariance., Comment: 17 pages

The renormalized spectral zeta function of the quantum Rabi Hamiltonian is considered. It is shown that the renormalized spectral zeta function converges to the Riemann zeta function as the coupling constant goes to infinity. Moreover the path measure associated with the ground state of the quantum Rabi Hamiltonian is constructed on a discontinuous path space, and several applications are shown.

Starting from the Ginzburg--Landau model in a planar simply connected domain, with a local compactly supported applied magnetic field, we derive an effective model in the strong field limit, defined on a non-simply connected domain. The effective model features oscillations in the Little-Parks and Aharonov--Bohm spirit. We discuss also a similar question for the lowest eigenvalue of the magnetic Laplacian., Comment: 12 pages

In this paper, we develop an analytical model to investigate the generation of instability waves triggered by the upstream acoustic forcing near the nozzle lip of a supersonic jet. This represents an important stage, i.e. the jet receptivity, of the screech feedback loop. The upstream acoustic forcing, resulting from the shock-instability interaction, reaches the nozzle lip and excites new shear-layer instability waves. To obtain the newly-excited instability wave, we first determine the scattered sound field due to the upstream forcing using the Wiener-Hopf technique, with the kernel function factored using asymptotic expansions and overlapping approximations. Subsequently, the unsteady Kutta condition is imposed at the nozzle lip, enabling the derivation of the dispersion relation for the newly-excited instability wave. A linear transfer function between the upstream forcing and the newly-excited instability wave is obtained. We calculate the amplitude and phase delay in this receptivity process and examine their variations against the frequency. The phase delay enables us to re-evaluate the phase condition for jet screech and propose a new frequency prediction model. The new model shows improved agreement between the predicted screech frequencies and the experimental data compared to classical models. It is hoped that this model may help in developing a full screech model., Comment: 33 pages, 11 figures

We study the dynamics of the field equations in a five-dimensional spatially flat Friedmann-Lema\^itre-Robertson-Walker metric in the context of a Gauss-Bonnet-Scalar field theory where the quintessence scalar field is coupled to the Gauss-Bonnet scalar. Contrary to the four-dimensional Gauss-Bonnet theory, where the Gauss-Bonnet term does not contribute to the field equations, in this five-dimensional Einstein-Scalar-Gauss-Bonnet model, the Gauss-Bonnet term contributes to the field equations even when the coupling function is a constant. Additionally, we consider a more general coupling described by a power-law function. For the scalar field potential, we consider the exponential function. For each choice of the coupling function, we define a set of dimensionless variables and write the field equations into a system of ordinary differential equations. We perform a detailed analysis of the dynamics for both systems and classify the stability of the equilibrium points. We determine the presence of scaling and super-collapsing solutions using the cosmological deceleration parameter. This means that our models can explain the Universe's early and late-time acceleration phases. Consequently, this model can be used to study inflation or as a dark energy candidate., Comment: 25 pages, 5 tables, 13 compound figures

In a finite non-Hermitian system, the number of zero modes does not necessarily reflect the topology of the system. This is known as the breakdown of the bulk-boundary correspondence and has lead to misconceptions about the topological protection of edge modes in such systems. Here we show why this breakdown does occur and that it typically results in hidden zero modes, extremely long-lived zero energy excitations, which are only revealed when considering the singular value instead of the eigenvalue spectrum. We point out, furthermore, that in a finite multiband non-Hermitian system with Hamiltonian $H$, one needs to consider also the reflected Hamiltonian $\tilde H$, which is in general distinct from the adjoint $H^\dagger$, to properly relate the number of protected zeroes to the winding number of $H$.

We clarify how the elliptic integrable spin chain recently found by Matushko and Zotov (MZ) relates to various other known long-range spin chains. We evaluate various limits. More precisely, we tweak the MZ chain to allow for a short-range limit, and show it is the XX model with q-deformed antiperiodic boundary conditions. Taking $q\to 1$ gives the elliptic spin chain of Sechin and Zotov (SZ), whose trigonometric case is due to Fukui and Kawakami. It, too, can be adjusted to admit a short-range limit, which we demonstrate to be the antiperiodic XX model. By identifying the translation operator of the MZ chain, which is nontrivial, we show that antiperiodicity is a persistent feature. We compare the resulting (vertex-type) landscape of the MZ chain with the (face-type) landscape containing the Heisenberg XXX and Haldane--Shastry chains. We find that the landscapes only share a single point: the rational Haldane--Shastry chain. Using wrapping we show that the SZ chain is the antiperiodic version of the Inozemtsev chain in a precise sense, and expand both chains around their nearest-neighbour limits to facilitate their interpretations as long-range deformations., Comment: 37 pages, 3 figures, 1 table

In this paper we discuss the Painlev\'e reductions of coupled KdV systems. We start by comparing the procedure with that of {\em stationary reductions}. Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the $t_2$ flow of the KdV and DWW hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical {\em structure} to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case., Comment: 11 pages

Nonclassical states are an important class of states in quantum mechanics, especially for their applications in quantum information theory. Optomechanical systems serve as an invaluable platform for exploring and harnessing these states test bed for search and application of such states. In this study, we focused on the studied the mirror-in-the-middle optomechanical system. We observed that in the absence of losses, a coherent state evolves into a entangled one. Furthermore, we demonstrate that the generation of a Schr\"odinger-cat state depends on the optomechanical coupling. We exactly solved the Gorini-Kossalokowinki-Sudarshan-Lindblad master equation, highlighting the direct influence of the reservoir on the dynamics when mechanical losses are considered. We later discussed vacuum one-photon superposition states to obtain exact entanglement dynamics using concurrence as a quantifier. Our results show that the overall entanglement of the system is attenuated by mechanical losses in the mirror., Comment: 8 pages, 5 figures

In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of $n$ for estimating $n$ parameters, this bound is not tight for large $n$. Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario., Comment: 22 pages, 9 figures, 10 appendices; presented at AIP Summer Meeting 2023

Quantum information theory is plagued by the problem of regularisations, which require the evaluation of formidable asymptotic quantities. This makes it computationally intractable to gain a precise quantitative understanding of the ultimate efficiency of key operational tasks such as entanglement manipulation. Here we consider the problem of computing the asymptotic entanglement cost of preparing noisy quantum states under quantum operations with positive partial transpose (PPT). A previously claimed solution to this problem is shown to be incorrect. We construct instead an alternative solution in the form of two hierarchies of semi-definite programs that converge to the true asymptotic value of the entanglement cost from above and from below. Our main result establishes that this convergence happens exponentially fast, thus yielding an efficient algorithm that approximates the cost up to an additive error $\varepsilon$ in time $\mathrm{poly}\big(D,\,\log(1/\varepsilon)\big)$, where $D$ is the underlying Hilbert space dimension. To our knowledge, this is the first time that an asymptotic entanglement measure is shown to be efficiently computable despite no closed-form formula being available., Comment: 7+24 pages, no figures

We utilize the topological holographic framework to characterize and gain insights into the nature of quantum critical points and gapless phases in fermionic quantum systems. Topological holography is a general framework that describes the generalized global symmetry and the symmetry charges of a local quantum system in terms of a slab of a topological order, termed as the symmetry topological field theory (SymTFT), in one higher dimension. In this work, we consider a generalization of the topological holographic picture for $(1+1)d$ fermionic quantum phases of matter. We discuss how spin structures are encoded in the SymTFT and establish the connection between the formal fermionization formula in quantum field theory and the choice of fermionic gapped boundary conditions of the SymTFT. We demonstrate the identification and the characterization of the fermionic gapped phases and phase transitions through detailed analysis of various examples, including the fermionic systems with $\mathbb{Z}_{2}^{F}$, $\mathbb{Z}_{2} \times \mathbb{Z}_{2}^{F}$, $\mathbb{Z}_{4}^{F}$, and the fermionic version of the non-invertible $\text{Rep}(S_{3})$ symmetry. Our work uncovers many exotic fermionic quantum critical points and gapless phases, including two kinds of fermionic symmetry enriched quantum critical points, a fermionic gapless symmetry protected topological (SPT) phase, and a fermionic gapless spontaneous symmetry breaking (SSB) phase that breaks the fermionic non-invertible symmetry., Comment: 33 pages, 7 figures, 6 tables

We study dimensional reduction by spherical symmetry in the context of Euclidean Chern-Simons gauge theories with gauge groups $SU(2)$ and $SL(2,\mathbb{C})$ (de Sitter gravity). We start with pure Chern-Simons theories and show that the reduced theory obtained by employing a spherically symmetric ansatz on the gauge field is similar to a fermionic theory in 1d coupled to a $U(1)$ gauge field with a 1-dimensional Chern-Simons kinetic term. Furthermore, we carry out a semiclassical computation that demonstrates that in the large $\kappa$ limit, the path integral of the 3d pure Chern-Simons theory is equivalent to the path integral of the dimensionally reduced theory, which is a 1d QFT. Moreover, we argue that the Wilson loop operators in the large $\kappa$ limit should reduce to some observables of the 1-dimensional theory. This agreement hints at the existence of a duality at the quantum level between the 3d Chern-Simons theory and the 1d reduced theory. We then study Chern-Simons Higgs theories in the Euclidean formalism. We again employ the spherically symmetric ansatz\"e on the fields, and show that at the clasical level, the action of the reduced theory is a theory with some scalar fields $\mathsf{X}_i$ coupled to fermionic fields $\Psi_i$. To get an appropriate reduced action, we define the original theory on a manifold $\mathcal{M}$ with the geometry of $\mathbb{R} \times S^2$. With a suitable choice of the potential of the Higgs field, we show that the reduced theory, which is the action that governs the spherically symmetric monopoles in the Chern-Simons Higgs theory, is equivalent to a supersymmetric quantum mechanical model. The flux of the Chern-Simons Higgs monopole is shown to be related to the fermionic number operator $F = \overline{\Psi} \Psi$, which in turn is related to the Witten index., Comment: 39 pages, no figures

The quantum dynamics of charged scalar bosons in a Bonnor-Melvin-$\Lambda$ universe is considered. In this study, the behavior of charged scalar bosons is explored within the framework of the Duffin-Kemmer-Petiau (DKP) formalism. Adopting a conical approximation ($\Lambda\ll 1$), we are considered two scenarios for the vector potential: a linear and quadratic vector potentials. In particular, the effects of this background in the equation of motion, phase shift, $S$-matrix, energy spectrum and DKP spinor are analyzed and discussed., Comment: 11 pages, 5 figures

We introduce a purely geometric formulation for two different measures addressed to quantify the entanglement between different parts of a tripartite qubit system. Our approach considers the entanglement-polytope defined by the smallest eigenvalues of the reduced density matrices of the qubit-components. The measures identify global and genuine entanglement, and are respectively associated with the projection and rejection of a given point of the polytope on the corresponding biseparable segments. Solving the so called `inverse problem', we also discuss a way to force the system to behave in a particular form, which opens the possibility of controlling and manipulating entanglement for practical purposes., Comment: 22 pages, 11 figures, 1 table

The Ornstein-Uhlenbeck (OU) process describes the dynamics of Brownian particles in a confining harmonic potential, thereby constituting the paradigmatic model of overdamped, mean-reverting Langevin dynamics. Despite its widespread applicability, this model falls short when describing physical systems where the confining potential is itself subjected to stochastic fluctuations. However, such stochastic fluctuations generically emerge in numerous situations, including in the context of colloidal manipulation by optical tweezers, leading to inherently out-of-equilibrium trapped dynamics. To explore the consequences of stochasticity at this level, we introduce a natural extension of the OU process, in which the stiffness of the harmonic potential is itself subjected to OU-like fluctuations. We call this model the OU$^2$ process. We examine its statistical, dynamic, and thermodynamic properties through a combination of analytical and numerical methods. Importantly, we show that the probability density for the particle position presents power-law tails, in contrast to the Gaussian decay of the standard OU process. In turn, this causes the trapping behavior, extreme value statistics, first passage statistics, and entropy production of the OU$^2$ process to differ qualitatively from their standard OU counterpart. Due to the wide applicability of the standard OU process and of the proposed OU$^2$ generalisation, our study sheds light on the peculiar properties of stochastic dynamics in random potentials and lays the foundation for the refined analysis of the dynamics and thermodynamics of numerous experimental systems., Comment: 32 pages, 7 figures

We propose the idea of using Kuramoto models (including their higher-dimensional generalizations) for machine learning over non-Euclidean data sets. These models are systems of matrix ODE's describing collective motions (swarming dynamics) of abstract particles (generalized oscillators) on spheres, homogeneous spaces and Lie groups. Such models have been extensively studied from the beginning of XXI century both in statistical physics and control theory. They provide a suitable framework for encoding maps between various manifolds and are capable of learning over spherical and hyperbolic geometries. In addition, they can learn coupled actions of transformation groups (such as special orthogonal, unitary and Lorentz groups). Furthermore, we overview families of probability distributions that provide appropriate statistical models for probabilistic modeling and inference in Geometric Deep Learning. We argue in favor of using statistical models which arise in different Kuramoto models in the continuum limit of particles. The most convenient families of probability distributions are those which are invariant with respect to actions of certain symmetry groups.

New one parameter family of exact solutions in General Relativity with a scalar field is found. The metric is of Liouville type which admits complete separation of variables in the geodesic Hamilton--Jacobi equation. This solution exists for the exponential potential for a scalar field and is invariant with respect to global Lorentz transformations. It describes, in particular, a black hole formation as well as a naked singularity. Solutions corresponding to the naked singularity describe accelerating expansion of the homogeneous and isotropic Universe, and can be smoothly continued along geodesics to infinite past without Big Bang., Comment: 8 pages, 2 figures

We simulated spin-spin interactions of $N$-bodies in linearized General Relativity (GR) and linearized Massive Gravity of the Fierz-Pauli type (mGR). It was noted earlier that there is a discrete difference between the spin-spin interaction potential in GR and mGR for a $2$-body system, akin to the van Dam-Veltman-Zakharov discontinuity in the static Newton's potential. Specifically, at large distances, GR favors anti-parallel spin orientation with total spin pointing along the interaction axis, while mGR favors parallel spin orientation with total spin perpendicular to the axis between the sources. For an $N$-body system, a simulation in mGR hitherto has not been done and one would like to know the total spin of the system in both theories. Here we remedy this. In the simulations of GR, we observed that the total spin tends to decrease from a random initial configuration, while for mGR with a large distance, the total spin increases., Comment: 11 pages, 6 figures

Insufficient reference data is ubiquitous in data-driven computational fluid dynamics, as it is usually too expensive to compute or impossible to observe over long enough times needed for data-driven methods to properly operate out of the sampled range. Ultimately, the lack of data can significantly compromise the fidelity of results computed with data-driven methods or make the data-driven approach inapplicable. In order to challenge this problem, we propose a probabilistic reconstruction method which enriches the hyper-parameterisation approach with ideas underlying the probabilistic-evolutionary approach. We offer to use Advection of the image point (a hyper-parameterisation method) on data sampled from the joint probability distribution of the reference dataset. The idea is to blend together the simplicity of the hyper-parameterisation (HP) method with an extra source of reference data provided through sampling from the joint probability distribution. The HP method has been tested on the sea surface temperature and surface relative vorticity computed with the global 1/4-degree and 1/12-degree resolution NEMO model. Our results show that the HP solution (the solution computed with the HP method) in the probabilistically-reconstructed and reduced (in terms of dimensionality) phase space at 1/4-degree resolution is more accurate compared to the 1/4-degree solution computed with NEMO and it is also several orders of magnitude faster to compute than the 1/4-degree run. The proposed method shows encouraging results for the NEMO model and the potential for the use in other operational ocean and ocean-atmospheric models for both deterministic and probabilistic predictions., Comment: 20 pages, 13 figures

In the context of $C^*$ dynamical systems, we consider a locally compact group $G$ acting by $^*$-automorphisms on a C$^*$ algebra $\mathfrak{U}$ of observables, and assume a state of $\mathfrak{U}$ that satisfies the clustering property with respect to a net of group elements of $G$. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity., Comment: 17 pages

We consider a semi-classical approximation to the dynamics of a point particle in a noncommutative space. In this approximation, the noncommutativity of space coordinates is described by a Poisson bracket. For linear Poisson brackets, the corresponding phase space is given by the cotangent bundle of a Lie group, with the Lie group playing the role of a curved momentum space. We show that the curvature of the momentum space may lead to rather unexpected physical phenomena such as an upper bound on the velocity of a free nonrelativistic particle, bounded motion for repulsive central force, and no-fall-into-the-centre for attractive Coulomb potential. We also present a superintegrable Hamiltonian for the Kepler problem in $3$-space with ${su}(2)$ noncommutativity., Comment: 22 pages, 3 figures

We propose a series representation for the Virasoro fusion and modular kernels at any irrational central charge. Two distinct, yet closely related formulas are needed for the cases $c\in \mathbb C \backslash (-\infty,1]$ and $c <1$. We also conjecture that the formulas have a well-defined limit as the central charge approaches rational values. Our proposal for $c <1$ agrees numerically with the fusion transformation of the four-point spherical conformal blocks, whereas our proposal for $c\in \mathbb C \backslash (-\infty,1]$ agrees numerically with Ponsot and Teschner's integral formula for the fusion kernel. The case of the modular kernel is studied as a special case of the fusion kernel., Comment: 15 pages