Infinite State Research Articles

Necessary optimality conditions for infinite dimensional state constrained control problems

This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed.

On compact vector formats in the solution of the chemical master equation with backward differentiation

SummaryA stochastic chemical system with multiple types of molecules interacting through reaction channels can be modeled as a continuous‐time Markov chain with a countably infinite multidimensional state space. Starting from an initial probability distribution, the time evolution of the probability distribution associated with this continuous‐time Markov chain is described by a system of ordinary differential equations, known as the chemical master equation (CME). This paper shows how one can solve the CME using backward differentiation. In doing this, a novel approach to truncate the state space at each time step using a prediction vector is proposed. The infinitesimal generator matrix associated with the truncated state space is represented compactly, and exactly, using a sum of Kronecker products of matrices associated with molecules. This exact representation is already compact and does not require a low‐rank approximation in the hierarchical Tucker decomposition (HTD) format. During transient analysis, compact solution vectors in HTD format are employed with the exact, compact, and truncated generated matrices in Kronecker form, and the linear systems are solved with the Jacobi method using fixed or adaptive rank control strategies on the compact vectors. Results of simulation on benchmark models are compared with those of the proposed solver and another version, which works with compact vectors and highly accurate low‐rank approximations of the truncated generator matrices in quantized tensor train format and solves the linear systems with the density matrix renormalization group method. Results indicate that there is a reason to solve the CME numerically, and adaptive rank control strategies on compact vectors in HTD format improve time and memory requirements significantly.

Security proof of continuous-variable quantum key distribution using three coherent states

We introduce a ternary quantum key distribution (QKD) protocol and asymptotic security proof based on three coherent states and homodyne detection. Previous work had considered the binary case of two coherent states and here we nontrivially extend this to three. Our motivation is to leverage the practical benefits of both discrete and continuous (Gaussian) encoding schemes creating a best-of-both-worlds approach; namely, the postprocessing of discrete encodings and the hardware benefits of continuous ones. We present a thorough and detailed security proof in the limit of infinite signal states which allows us to lower bound the secret key rate. We calculate this is in the context of collective eavesdropping attacks and reverse reconciliation postprocessing. Finally, we compare the ternary coherent state protocol to other well-known QKD schemes (and fundamental repeaterless limits) in terms of secret key rates and loss.

Lean Reachability Tree for Unbounded Petri Nets

Elaborate efforts have been made to eliminate fake markings and refine ${\omega }$ -markings in the existing modified or improved Karp–Miller trees for various classes of unbounded Petri nets since the late 1980s. The main issues fundamentally are incurred due to the generation manners of the trees that prematurely introduce some potentially unbounded markings with ${\omega }$ symbols and keep their growth into new ones. Aiming at addressing them, this work presents a non-Karp–Miller tree called a lean reachability tree (LRT). First, a sufficient and necessary condition of the unbounded places and some reachability properties are established to reveal the features of unbounded nets. Then, we present an LRT generation algorithm with a sufficiently enabling condition (SEC). When generating a tree, SEC requires that the components of a covering node are not replaced by ${\omega }$ symbols, but continue to grow until any transition on an output path of an unbounded place has been branch-enabled at least once. In return, no fake marking is produced and no legal marking is lost during the tree generation. We prove that LRT can faithfully express by folding, instead of equivalently representing, the reachability set of an unbounded net. Also, some properties of LRT are examined and a sufficient condition of deadlock existence based on it is given. The case studies show that LRT outperforms the latest modified Karp–Miller trees in terms of size, expressiveness, and applicability. It can be applied to the analysis of the emerging discrete event systems with infinite states.

Combinatorial Lévy processes

Combinatorial Lévy processes evolve on general state spaces of combinatorial structures, of which standard examples include processes on sets, graphs and $n$-ary relations and more general possibilities are given by processes on graphs with community structure and multilayer networks. In this setting, the usual Lévy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. The main theorems characterize both finite and infinite state space combinatorial Lévy processes by a unique $\sigma$-finite measure. Under the additional assumption of exchangeability, I prove a more explicit characterization by which every exchangeable combinatorial Lévy process corresponds to a Poisson point process on the same state space. Associated behavior of the projection into a space of limiting objects reflects certain structural features of the covering process.

Roughness as classicality indicator of a quantum state

We define a new quantifier of classicality for a quantum state, the Roughness, which is given by the L2(R2) distance between Wigner and Husimi functions. We show that the Roughness is bounded and therefore it is a useful tool for comparison between different quantum states for single bosonic systems. The state classification via the Roughness is not binary, but rather it is continuous in the interval [0,1], being the state more classic as the Roughness approaches to zero, and more quantum when it is closer to the unity. The Roughness is maximum for Fock states when its number of photons is arbitrarily large, and also for squeezed states at the maximum compression limit. On the other hand, the Roughness approaches its minimum value for thermal states at infinite temperature and, more generally, for infinite entropy states. The Roughness of a coherent state is slightly below one half, so we may say that it is more a classical state than a quantum one. Another important result is that the Roughness performs well for discriminating both pure and mixed states. Since the Roughness measures the inherent quantumness of a state, we propose another function, the Dynamic Distance Measure (DDM), which is suitable for measure how much quantum is a dynamics. Using DDM, we studied the quartic oscillator, and we observed that there is a certain complementarity between dynamics and state, i.e. when dynamics becomes more quantum, the Roughness of the state decreases, while the Roughness grows as the dynamics becomes less quantum.

On the Decay of Correlations in the Random Field Ising Model

In a celebrated 1990 paper, Aizenman and Wehr proved that the two-dimensional random field Ising model has a unique infinite volume Gibbs state at any temperature. The proof is ergodic-theoretic in nature and does not provide any quantitative information. This article proves the first quantitative version of the Aizenman-Wehr theorem. The proof introduces a new method for proving decay of correlations that may be interesting in its own right. A fairly detailed sketch of the main ideas behind the proof is also included.

Robust Control of a Cable From a Hyperbolic Partial Differential Equation Model

This article presents a detailed study of the robust control of a cable’s vibrations, with emphasis on considering a model of infinite dimension. Indeed, using a partial differential equation model of the vibrations of an inclined cable with sag, we are interested in studying the application of $\mathcal H_{\infty }$ -robust feedback control to this infinite dimensional system. The approach relies on Riccati equations to stabilize the system under measurement feedback when it is subjected to external disturbances. Henceforth, this article focuses on the construction of a standard linear infinite dimensional state space description of the cable under consideration before writing its approximation of finite dimension and studying the $\mathcal H_{\infty }$ feedback control of vibrations with partial observation of the state in both cases. The closed-loop system is numerically simulated to illustrate the effectiveness of the resulting control law.

Equivalence of Initialized Fractional Integrals and the Diffusive Model

Fractional calculus is viewed as a novel and powerful tool to describe the stress and strain relations in viscoelastic materials. Consequently, the motions of engineering structures incorporated with viscoelastic dampers can be described by fractional-order differential equations. To deal with the fractional differential equations, initialization for fractional derivatives and integrals is considered to be a fundamental and unavoidable problem. However, this issue has been an open problem for a long time and controversy persists. The initialization function approach and the infinite state approach are two effective ways in initialization for fractional derivatives and integrals. By comparing the above two methods, this technical brief presents equivalence and unification of the Riemann–Liouville fractional integrals and the diffusive representation. First, the equivalence is proved in zero initialization case where both of the initialization function and the distributed initial condition are zero. Then, by means of initialized fractional integration, equivalence and unification in the case of arbitrary initialization are addressed. Connections between the initialization function and the distributed initial condition are derived. Besides, the infinite dimensional distributed initial condition is determined by means of input function during historic period.

Control Space Reduction and Real-Time Accurate Modeling of Continuum Manipulators Using Ritz and Ritz–Galerkin Methods

To address the challenges with real-time accurate modeling of multisegment continuum manipulators in the presence of significant external and body loads, we introduce a novel series solution for variable-curvature Cosserat rod static and Lagrangian dynamic methods. By combining a modified Lagrange polynomial series solution, based on experimental observations, with Ritz and Ritz–Galerkin methods, the infinite modeling state space of a continuum manipulator is minimized to geometrical position of a handful of physical points (in our case two). As a result, a unified easy to implement vector formalism is proposed for the nonlinear impedance and configuration control. We showed that by considering the mechanical effects of highly elastic axial deformation, the model accuracy is increased up to 6%. The proposed model predicts experimental results with 6%–8% (4–6 mm) mean error for the Ritz–Galerkin method in static cases and 16%–20% (12–14 mm) mean error for the Ritz method in dynamic cases, in planar and general three-dimensional motions. Comparing to five different models in the literature, our approximate solution is shown to be more accurate with the smallest possible number of modeling states and suitable for real-time modeling, observation, and control applications.

Topological orders and quantum phase transitions in a one-dimensional extended quantum compass model

By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.

Progress of detecting orbital angular momentum states of optical vortices through diffraction gratings

Optical vortices are a new kind of laser beam and receiving more and more attention currently.The complex amplitudes of optical vortices comprise a helical term exp (il),with l being the topological charge and the azimuthal angle.Each photon in optical vortices carries the orbital angular momentum (OAM) with a value of lħ,where ħ is the Planck's constant divided by 2.The topological charge l is the eigenvalue of optical vortices,and determines the helical wavefront distribution,thus also known as OAM state.Moreover,such an OAM state can be an infinite integer state. And vortices with various OAM states are orthogonal to each other,making it possible to be employed in high capacity data-transmission system.In addition,the above unique features contribute to their widely applications in lots of areas such as optical tweezers and spanners,rotation detection,quantum entanglement,etc.In these applications,detecting OAM states is basic,and greatly significant.Recently researchers have developed lots of approaches to detecting the OAM states,including the methods of interference,diffraction gratings,metasurface,etc.Of such approaches,the scheme of diffraction gratings is the simplest and most widely used,where one or more diffraction gratings are employed. When optical vortices propagate through such gratings,the OAM states are acquired immediately through capturing and analyzing the distinct OAM-related diffraction patterns.In this review,we focus on the techniques of detecting OAM states through diffraction gratings,which have been demonstrated by our group and other researchers.Some of the main detection gratings,including double-slit,triangle aperture and slit,angular slit,cylindrical lens,graduallychanging-period grating,annular grating,are introduced.In addition,schemes like composite fork grating,Dammann vortex grating and integrated Dammann grating,are presented to detect the OAM state for coaxial multiplexed vortices. Besides diagnosing OAM state,measuring the intensity proportion of each OAM channel,known as OAM spectrum, in multiplexed vortices is also necessary in some cases.Therefore we also introduce the ways to measure the OAM spectrum,e.g.the OAM mode sorter,the gray-scale algorithm.

On Generating A Variety of Unsafe Counterexamples for Linear Dynamical Systems

Counterexamples encountered in formal verification are typically used as evidence for violation of specification. They also play a crucial role in CEGAR based techniques, where the counterexample guides the refinements to be performed on the abstractions. While several scalable techniques for verification have been developed for safety verification of hybrid systems, less attention has been paid to extracting the various types of counterexamples for safety violations. Since these systems are infinite state systems, the number of counterexamples for safety violations are potentially infinite and hence searching for the right counterexample becomes a challenging task. In this paper, we present a technique for providing various types of counterexamples for a safety violation of the linear dynamical system. More specifically, we develop algorithms to extract the longest counterexample — the execution that stays in the unsafe set for most time, and deepest counterexample — the execution that ventures the most into the unsafe set in a specific direction provided by the user.

Probing the anomalous dynamical phase in long-range quantum spin chains through Fisher-zero lines.

Using the framework of infinite matrix product states, the existence of an anomalous dynamical phase for the transverse-field Ising chain with sufficiently long-range interactions was first reported in J. C. Halimeh and V. Zauner-Stauber [Phys. Rev. B 96, 134427 (2017)10.1103/PhysRevB.96.134427], where it was shown that anomalous cusps arise in the Loschmidt-echo return rate for sufficiently small quenches within the ferromagnetic phase. In this work we further probe the nature of the anomalous phase through calculating the corresponding Fisher-zero lines in the complex time plane. We find that these Fisher-zero lines exhibit a qualitative difference in their behavior, where, unlike in the case of the regular phase, some of them terminate before intersecting the imaginary axis, indicating the existence of smooth peaks in the return rate preceding the cusps. Additionally, we discuss in detail the infinite matrix product state time-evolution method used to calculate Fisher zeros and the Loschmidt-echo return rate using the matrix product state transfer matrix. Our work sheds further light on the nature of the anomalous phase in the long-range transverse-field Ising chain, while the numerical treatment presented can be applied to more general quantum spin chains.

Probability distributions for Markov chain based quantum walks

We analyze the probability distributions of the quantum walks induced from Markov chains by Szegedy (2004). The first part of this paper is devoted to the quantum walks induced from finite state Markov chains. It is shown that the probability distribution on the states of the underlying Markov chain is always convergent in the Cesaro sense. In particular, we deduce that the limiting distribution is uniform if the transition matrix is symmetric. In the case of a non-symmetric Markov chain, we exemplify that the limiting distribution of the quantum walk is not necessarily identical with the stationary distribution of the underlying irreducible Markov chain. The Szegedy scheme can be extended to infinite state Markov chains (random walks). In the second part, we formulate the quantum walk induced from a lazy random walk on the line. We then obtain the weak limit of the quantum walk. It is noted that the current quantum walk appears to spread faster than its counterpart-quantum walk on the line driven by the Grover coin discussed in literature. The paper closes with an outlook on possible future directions.

Comment on “Many-body localization in Ising models with random long-range interactions”

This comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power law interactions, $r^{-\alpha}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has been earlier argued [1, 2] that this model obeys the dimensional constraint suggesting the delocalization of all finite temperature states in thermodynamic limit for $\alpha \leq 2d$ in a $d$-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model in Ref. [3]. To resolve this controversy we reexamine the model of Ref. [3] and demonstrate that the infinite temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization [2] is extended to small exponents $0 \leq \alpha \leq d$. Disagreements between two works are explained by the non-standard selection of investigated states in the ordered phase and misinterpretation of the localization-delocalization transition in Ref. [3].

Rényi entropies after releasing the Néel state in the XXZ spin-chain

We study the Rényi entropies in the spin- anisotropic Heisenberg chain after a quantum quench starting from the Néel state. The quench action method allows us to obtain the stationary Rényi entropies for arbitrary values of the index α as generalised free energies evaluated over a calculable thermodynamic macrostate depending on α. We work out this macrostate for several values of α and of the anisotropy Δ by solving the thermodynamic Bethe ansatz equations. By varying α different regions of the Hamiltonian spectrum are accessed. The two extremes are for which the thermodynamic macrostate is either the ground state or a low-lying excited state (depending on Δ) and when the macrostate is the infinite temperature state. The Rényi entropies are easily obtained from the macrostate as function of α and a few interesting limits are analytically characterised. We provide robust numerical evidence to confirm our results using exact diagonalisation and a stochastic numerical implementation of Bethe ansatz. Finally, using tDMRG we calculate the time evolution of the Rényi entanglement entropies. For large subsystems and for any α, their density turns out to be compatible with that of the thermodynamic Rényi entropies.

Verifying chemical reaction network implementations: A pathway decomposition approach

The emerging fields of genetic engineering, synthetic biology, DNA computing, DNA nanotechnology, and molecular programming herald the birth of a new information technology that acquires information by directly sensing molecules within a chemical environment, stores information in molecules such as DNA, RNA, and proteins, processes that information by means of chemical and biochemical transformations, and uses that information to direct the manipulation of matter at the nanometer scale. To scale up beyond current proof-of-principle demonstrations, new methods for managing the complexity of designed molecular systems will need to be developed. Here we focus on the challenge of verifying the correctness of molecular implementations of abstract chemical reaction networks, where operation in a well-mixed “soup” of molecules is stochastic, asynchronous, concurrent, and often involves multiple intermediate steps in the implementation, parallel pathways, and side reactions. This problem relates to the verification of Petri nets, but existing approaches are not sufficient for providing a single guarantee covering an infinite set of possible initial states (molecule counts) and an infinite state space potentially explored by the system given any initial state. We address these issues by formulating a new theory of pathway decomposition that provides an elegant formal basis for comparing chemical reaction network implementations, and we present an algorithm that computes this basis. Our theory naturally handles certain situations that commonly arise in molecular implementations, such as what we call “delayed choice,” that are not easily accommodated by other approaches. We further show how pathway decomposition can be combined with weak bisimulation to handle a wider class that includes most currently known enzyme-free DNA implementation techniques. We anticipate that our notion of logical equivalence between chemical reaction network implementations will be valuable for other molecular implementations such as biochemical enzyme systems, and perhaps even more broadly in concurrency theory.

Dynamical phase diagram of quantum spin chains with long-range interactions

Using an infinite Matrix Product State (iMPS) technique based on the time-dependent variational principle (TDVP), we study two major types of dynamical phase transitions (DPT) in the one-dimensional transverse-field Ising model (TFIM) with long-range power-law ($\propto1/r^{\alpha}$ with $r$ inter-spin distance) interactions out of equilibrium in the thermodynamic limit -- \textit{DPT-I}: based on an order parameter in a (quasi-)steady state, and \textit{DPT-II}: based on non-analyticities (cusps) in the Loschmidt-echo return rate. We construct the corresponding rich dynamical phase diagram, whilst considering different quench initial conditions. We find a nontrivial connection between both types of DPT based on their critical lines. Moreover, and very interestingly, we detect a new DPT-II dynamical phase in a certain range of interaction exponent $\alpha$, characterized by what we call \textit{anomalous cusps} that are distinct from the \textit{regular cusps} usually associated with DPT-II. Our results provide the characterization of experimentally accessible signatures of the dynamical phases studied in this work.

Attitude stabilization of rigid spacecraft implemented in backstepping control with input delay

A backstepping method is used for nonlinear spacecraft attitude stabilization in the presence of external disturbances and time delay induced by the actuator. The kinematic model is established based on modified Rodrigues parameters (MRPs). Firstly, we get the desired angular velocity virtually drives the attitude parameters to origin, and then backstep it to the desired control torque required for stabilization. Considering the time delay induced by the actuator, the control torque functions only after the delayed time, therefore time compensation is needed in the controller. Stability analysis of the close-loop system is given afterwards. The infinite dimensional actuator state is modeled with a first-order hyperbolic partial differential equation (PDE), the L 2 norm of the system state is constructed and is proved to be exponentially stable. An inverse optimality theorem is also employed during controller design. Simulation results illustrate the efficiency of the proposed control law and it is robust to bounded external disturbances and time delay mismatch.