Abstract We study the quantum backflow problem of a relativistic charged Dirac fermion constrained to move on a ring of radius R. Using the relativistic current operator we compute the probability flux through a generic time interval to show emergence of quantum backflow. We also discuss the limiting case when the particle moves along a line.
Abstract An important problem in quantum technologies is to generate a target quantum evolution of an open quantum system. As a core element, it requires the ability to determine whether the actual evolution of the system coincides with the target evolution. For generation of unitary quantum channels in open quantum systems, it was shown by Goerz et al (2014 New J. Phys. 16 055012) that for determining whether the actual evolution coincides with a target unitary it is sufficient to compare their action on three special density matrices. In this work, we consider controlled generation and unique determination of non-unitary quantum channels in open quantum systems with particular emphasis on rank-two single-qubit quantum channels. We prove that for a unique determination of such quantum channels it is sufficient to consider their action on only three special density matrices. Based on this theoretical result, we numerically investigate generation of various target non-unitary single-qubit quantum channels in open quantum systems using coherent and incoherent controls.
Abstract We study scattering for continuous-time quantum walks on finite graphs with attached leads. The focus of our study is perfect transmission, which is essential for a given graph to be considered as a scattering gadget. To this end, we introduce three real parameters: the terminal-responses µ1 and µ2, and a terminal-to-terminal coupling ν, defined in terms of characteristic polynomials of the graph and its vertex deleted subgraphs. Up to a momentum-dependent transformation, these quantities are the independent entries of an admittance matrix, and each is additive under parallel composition of graphs. In these variables the perfect transmission at a fixed momentum is characterized by a hyperbola in (µ1, µ2, ν)-space, the points of which determine the transmission phase. This turns the search for graphs with prescribed transmission properties into a geometric vector-sum problem for smaller building blocks.
In cosmological fluctuations around a space of constant negative 3-curvature the relevant basis modes are associated Legendre conical functions. To facilitate the evaluation of the cosmic microwave background temperature anisotropy in the negative 3-curvature case we present some new properties of associated Legendre conical functions of the first and second kind,In particular we show that with the τwhere n ranges from -ℓ to ℓ in unit steps when K is a non-negative integer ℓ, and from -∞ to ∞ in unit steps otherwise. Also we can set, where n ranges from 0 to 2ℓ in unit steps when K is a non-negative integer ℓ, and from 0 to ∞ in unit steps otherwise. With these forms isolating the entire τ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over τ of associated Legendre conical functions and their products. The QWe show how to use the divergence of this integral outside of this range in order to characterize the complex τ plane pole structure of Q -1/2-K -1/2+iτ (χ). We present a new treatment of the Borwein integral. We present both a new derivation of and a new generalization of the Nyquist-Shannon sampling theorem that is used in signal processing. We apply this analysis to the cosmic microwave background temperature anisotropy, to thus uncover an unexpected connection between signal processing and cosmological perturbation theory.
Abstract Entropic uncertainty relations (EURs) bound the joint unpredictability of incompatible quantum measurements, but standard formulations such as the Maassen–Uffink (MU) bound compress measurement incompatibility into a single maximal overlap and can therefore be loose for mixed states with strongly nonuniform overlap structure. We develop a calibrated, rank-resolved framework based on the characteristic decomposition of a density operator into maximally mixed states on nested leading eigenspaces. This induces spectral staircase weights and rank-layer measurement statistics which, combined with rank-resolved overlap envelopes, yield analytic state-adaptive EURs. In the regime where one measurement basis diagonalizes the state, our bounds recover the MU form as a relaxation and can be tighter when large overlaps are confined to low-rank subspaces carrying little spectral weight. We further obtain a calibrated direct-sum majorization relation, closed-form calibrated certificates for guessing probability and min-entropy, and a quantum-memory extension with a Holevo-information penalty.
Abstract We investigate the thermodynamic performance of a quantum Otto heat engine modulated by stochastic resetting, utilizing a discrete-time framework of a qubit coupled to a hierarchical environment via a collision model. Our results reveal that stochastic resetting functions as a potent mechanism for modulating information flow, effectively driving the system toward the Markovian regime by erasing system-environment correlations. We identify a fundamental trade-off in the relaxation dynamics: resetting speeds up the convergence to steady state in non-Markovian regimes but hinders it in Markovian cases due to perturbative effects. By implementing an optimal resetting protocol-resetting to the excited and ground states during the heating and cooling strokes, respectively-we show that work extraction increases monotonically with the resetting rate. Notably, the effective efficiency exhibits a regime-dependent behavior, where the suppression of heat absorption leads to significant efficiency gains specifically under strong memory effects. This work provides a theoretical framework for leveraging resetting protocols to optimize nonequilibrium quantum thermal machines in complex environments.
Abstract Evolution of sphalerons in a class of quartic Klein–Gordon models are studied under a growing perturbation. Sphalerons are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation increases in time and asymptotically approaches the speed of light. To explain this behaviour analytically, a nonlinear collective coordinate method is developed which has three dynamical parameters and leads to an explicit asymptotic solution using a power series expansion. The solution describes the emergence of a spreading tabletop profile whose height approaches the true vacuum while its flanks steepen and accelerate outward. In addition, the energy density is shown to concentrate at the flanks, indicating the onset of a gradient blow-up at large times. These results provide a detailed description of the long-time dynamics of positively perturbed sphalerons, and reveal a universal mechanism for the formation of relativistically expanding structures in nonlinear field theories.
Abstract Local operator entanglement (LOE) has emerged an indicator of quantum chaos in many-body systems. Numerical studies have shown that, in chaotic systems, LOE grows linearly in time and displays a volume-law behavior at late times, scaling proportionally with the number of local degrees of freedom. Despite extensive numerical evidence, complemented by analytical studies in integrable systems, a fully analytical understanding of the emergence of the volume law remains incomplete. In this paper, we contribute toward this goal by deriving a late-time expression for LOE in chaotic systems that exhibits volume-law scaling. Our derivation proceeds by expressing the late-time LOE in the Liouville eigenstate basis and relies on three main assumptions: a higher-order non-resonance condition for the Hamiltonian eigenenergies, the eigenstate thermalization hypothesis ansatz for the matrix elements of the initial local operator, and the replacement of Hamiltonian eigenstates with random states in the final expression for LOE. Under these assumptions, we obtain an explicit formula displaying volume-law scaling. Finally, we complement our analytical derivation with numerical simulations of the 1D mixed-field Ising model, testing the resulting formula and exploring the regime of validity of our assumptions.
Abstract We discuss the Kontsevich–Segal–Witten (KSW) criterion for the allowability of complex metrics, in the context of the gravitational path integral that calculates the supersymmetric index. We focus on the saddle points that capture the contribution of supersymmetric black holes in AdS 5 space. We show that, for such black holes with two independent angular momenta, the conditions imposed on the corresponding saddle point by the KSW criterion are equivalent to the ones arising from the convergence of the microscopic trace form of the supersymmetric index. This result adds to previous results establishing such an equivalence in other, simpler examples of the gravitational index in AdS space and flat space. Along the way, we give a practical algorithm for implementing the KSW criterion in terms of eigenvalues of certain matrices.
Abstract Stabilizer convolution channels in odd-prime dimensions have a natural description in finite-field phase space. We analyze these channels using Clifford symmetry and show that, for a fixed environmental state, the maximal single-letter coherent information is unchanged under Clifford-equivalent choices of the stabilizer convolution matrix. This reduces the apparently quadratic family of admissible stabilizer convolutions, at the one-use level, to sign-orbit classes of a single finite-field parameter. We also isolate quadratic cases in which the induced Clifford symmetry becomes either the identity or phase-space inversion; these cases give symmetry-protected families with zero maximal single-letter coherent information. To complement this symmetry analysis, we introduce a minimal weighted magic entropy and prove that it sharpens the magic-relative-entropy upper bound for positive convolution channels at the single-letter level. Finally, we define the single-letter coherent-information power of a convolution and show that the optimizing environmental state may be taken to be pure. Small-dimensional numerical examples illustrate the sign-orbit classification and show that coherent-information enhancement is governed by the finite-field convolution parameter rather than by dimension alone.
The min-plus process is a stochastic coagulation-annihilation-type process on the binary tree, of interest in mathematics, physics, and computer science as a tractable instance of max-type recursive distributional equations. We carry out large Monte Carlo simulations at effective tree depths up to N=60 that provide finite-depth corroboration of the Beta(2,1) stretched-exponential limit for its root value X_N at p=1/2, on the asymmetric √(N) side of the random-homogeneous-systems classification recently introduced by Chen, Duquesne, and Shi and by Morfe. Off criticality, our simulations confirm the sub-critical closed form ℙ(X_∞=1)=(1-2p)/(1-p) within Monte Carlo error and document a super-critical mean growth exceeding the elementary (2p)^N lower bound at the depths we reach. For a Bernoulli(q)-initial-condition variant, we identify an elementary closed-form identity at p=1/2 that pins down the order parameter ℙ(X_N=0)=q exactly, locates the absorbing-state phase transition at p_c=1/2 in the operator-mixing probability rather than in the initial-zero density, and shows that the conditional law on positives deforms substantially with q. Our simulations use a level-wise recursion and an FFT-based precomputed leaf table which reduce the effective simulation depth while preserving the recursive tree law and may be useful for the simulation of related recursive equations on large trees.
Abstract We analyze a variant of the Hopfield model that incorporates an unlearning mechanism based on spin correlations in the high-temperature regime. In particular, we focus on the effective interaction obtained from a high-temperature approximation. In the large system limit where extensively many patterns are stored, we employ the replica method under the replica-symmetric (RS) ansatz to characterize the model analytically. Our analysis provides a systematic and self-consistent framework that yields order-parameter equations and stability conditions at finite temperatures for this effective model. Within the approximation, the resulting theory provides RS estimates of the behavior of the signal-to-noise ratio, the memory capacity, and the criteria for selecting optimal hyperparameters, in qualitative agreement with the findings of Nokura (1996 J. Phys. A: Math. Gen. 29 3871). Moreover, the theoretical predictions are compared with synchronous dynamics simulations initialized from the target stored pattern, suggesting that unlearning suppresses spurious memories and thereby improves retrieval performance, consistent with the RS analysis.
Abstract By combining stability and the analysis of topological solitons in ( 1 + 1 ) -dimensional scalar field theories with the Darboux transformation technique, we create models featuring kink-like solutions whose perturbations are all bounded. The stability analysis of extended classical solutions in scalar field theories is determined from the spectrum of quantum mechanical Schrödinger operators whose eigenfunctions serve to expand quantum perturbations to the classical soliton. Our strategy is to invert the procedure: we start from the Schrödinger operator for a single harmonic oscillator and ask whether there is a field theory admitting kink solutions. Having identified the parent exotic field theory to the harmonic oscillator, the Darboux transformation allows for constructing new exotic but solvable Schrödinger equations. This framework relates the quantum harmonic oscillator and its rational deformations to exotic scalar theories featuring non-trivial potentials. Depending on the structure of the spectrum of perturbation frequencies, these potentials may have various local maxima, minima, and inflexion points. The stationary solutions take the form of a definite integral over a finite interval of a function times the Gaussian bell distribution, including the error and the Owen T functions. In such models, the spectrum of the Schrödinger operator governing kink fluctuations is purely discrete. Therefore, Riemann − ζ function regularization allows us to achieve finite one-loop quantum corrections to the classical mass.
Abstract We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.
Abstract Quantum affine algebras are a fundamental family of infinite-dimensional quantum groups with profound connections to integrable systems, solvable lattice models, and conformal and quantum field theories. In recent years, scattering amplitudes in gauge theory, particularly in planar N = 4 supersymmetric Yang–Mills theory, quantum chromodynamics, and related models, have revealed unexpected mathematical structures, including cluster algebras from Grassmannians and partial flag varieties. Meanwhile, the Grothendieck rings of finite-dimensional representations of quantum affine algebras have been shown to carry canonical cluster algebra structures closely related to those arising from Grassmannians and partial flag varieties. These connections open a pathway for transferring methods and results from the theory of quantum affine algebras to the study of scattering amplitudes. In this survey, we review recent developments in this emerging interface, highlighting the representation-theoretic and cluster-algebraic structures that play an increasingly important role in the modern study of scattering amplitudes.
The maximum clique problem (MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an enhanced quantum solution method that introduces a pre-detection phase based on a hybrid heuristic algorithm and a vertex pruning strategy, encoding the prior constraints on clique size into quantum registers as global information. It encodes the prior constraints on clique size into quantum registers as global information. In contrast, state-of-the-art Grover-based methods for n-vertex graphs require O(n2n) iterations and O(n) measurements. By leveraging a global information encoding mechanism, the proposed method compresses the number of iterations to O(2n) and only needs a constant number of measurements to identify all maximum cliques. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
Abstract We investigate the extendibility problem for Brauer states, focusing on the symmetric two-sided extendibility and the de Finetti extendibility. By employing the representation theory of the unitary and orthogonal groups, we provide a general recipe for determining the set of (n, m)-extendible and n-de Finetti-extendible Brauer states. From the concrete form of the commutant of the diagonal action of the orthogonal group, we explicitly determine the set of parameters for which the Brauer states are (1, 2)-, (1, 3)- and (2, 2)-extendible in any dimension d and find that Brauer states extend with a non-trivial trade-off in n and m. Using the same recipe we also provide an estimate of the set of (1, m)-extendible Brauer states for any m and dimension d. Finally, using the branching rules from U(d) to O(d), we obtain the set of n-de Finetti-extendible Brauer states in any dimension, and also analytically describe the n → ∞ limiting shape which turns out not to be a polygon for odd dimensions.
Abstract Theorem 2.5 gives a proof of the GLE and 2FDT for the Mori projection. We note that the same result is obtained by a variation of constants, which greatly simplifies the semigroup approach presented in section 2. We therefore add a short proof of the GLE by means of the variation of constants formula for strongly continuous semigroups.
We present a constructive framework for deriving Poisson-algebraic structures in classical two-dimensional superintegrable Hamiltonian systems, characterized by three functionally independent integrals of motion H, L, and A. Under explicit geometric hypotheses-namely, that one can select an integral L whose Hamiltonian flow is complete and 2 pi-periodic on each connected regular region-we obtain globally defined generators H, L, A, and B={L,A} closing to form a four-generated Poisson algebra. The construction is fully explicit and does not require the integrals to be polynomial in the momenta, of bounded differential order, or associated with separable coordinates. Polynomial closure follows whenever the corresponding structure function G(H,L) is polynomial in H and L. This provides a general mechanism for constructing polynomial Poisson algebras well beyond the standard polynomial-momentum setting, including systems with higher-order and non-polynomial integrals of motion. We illustrate the method in several relevant examples, including the Kepler, Holt, Smorodinsky-Winternitz, Fokas-Lagerstrom, Hamiltonian trigonometric in the momenta, non-separable Post-Winternitz, and curved-space oscillator systems. In several cases, the same set of integrals also allows one to reconstruct the trajectories in configuration space by algebraic elimination of the momenta, without direct integration of Hamilton's equations. We also discuss distinguished invariant sectors associated with special values of the integrals of motion.
Abstract Establishing a concrete link between quantum information measures and thermodynamic quantities remains a significant challenge in contemporary research. This paper presents novel contributions exploring three key aspects. First, we investigate the relationship between quantum Fisher information (QFI) and work and heat exchange during a feedback control measurement cycle, unveiling a quantitative interplay between energy and information. Second, we demonstrate the behavior of heat exchange with the environment at the point of maximum estimation precision, and a direct relation that links the von Neumann entropy in systems with a large number of microstates with the precision of the estimation through QFI, shedding light on the thermodynamic signature of optimal quantum parameter estimation, bridging quantum metrology, and statistical thermodynamics in the context of high-dimensional stochastic systems. Finally, we propose a method for detecting entanglement based on thermodynamic work, introducing a framework to identify entanglement through energy considerations in a quantum system. Together, these findings advance our understanding of the intersection between quantum information theory and quantum thermodynamics, providing an additional foundation for future research.
