Combating Quantum Burst-errors with Quantum Interleaving Technique

Based on the interleaving technique, a kn-qubit code is constructed in this paper with more error-correcting ability than one n-qubit quantum error-correcting code without introducing the redundant qubits. By converting quantum bursts of errors into quantum random errors with the help of the quantum interleaving of the several states of the same quantum code, the proposed technique becomes an effective means to combat quantum bursts of errors. It is much simple and applicable for the quantum interleaving techniques to be used in the optical-fiber communications.     

Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture

In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of maximal rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank. In particular, for an estimation algebra E of maximal rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift f_i (x) and observation h_j (x). As an important corollary, we show that the number of statistics needed to compute the conditional density of the state given the observation {y(s):0?≤?s?≤?t} by the algebraic method is n where n is the dimension of the state.     

Quantum state sharing against the controller’s cheating

Most existing QSTS schemes are equivalent to the controlled teleportation, in which a designated agent (i.e., the recoverer) can recover the teleported state with the help of the controllers. However, the controller may attempt to cheat the recoverer during the phase of recovering the secret state. How can we detect this cheating? In this paper, we considered the problem of detecting the controller’s cheating in Quantum State Sharing, and further proposed an effective Quantum State Sharing scheme against the controller’s cheating. We cleverly use Quantum Secret Sharing, Multiple Quantum States Sharing and decoy-particle techniques. In our scheme, via a previously shared entanglement state Alice can teleport multiple arbitrary multi-qubit states to Bob with the help of Charlie. Furthermore, by the classical information shared previously, Alice and Bob can check whether there is any cheating of Charlie. In addition, our scheme only needs to perform Bell-state and single-particle measurements, and to apply C-NOT gate and other single-particle unitary operations. With the present techniques, it is feasible to implement these necessary measurements and operations.     

Pricing the financial Heston–Hull–White model with arbitrary correlation factors via an adaptive FDM

This paper is concerned with the pricing procedure of one of the most challenging models known as the Heston–Hull–White partial differential equation (PDE) in option pricing, at which the model is a time-dependent 3D linear PDE including three mixed derivative terms. The model comes from the fact that the price, the volatility and the interest rate are assumed to be stochastic processes. To contribute and avoid huge discretized systems, an adaptive distribution of the nodes (viz, nonuniform nodes) is taken into account with emphasis on the hot area of the solution curve. New adaptive finite difference (FD) formulas of higher orders are constructed on these meshes. Then, a set of semi-discretized equations is attained which is tackled by a time-stepping method. Several financial tests are discussed in detail to reveal the superiority of the proposed approach.     

Static state feedback triangular block decoupling for arbitrary systems: a state-space method

In this paper, the static state feedback triangular block decoupling problem for a general system described by (A, B, C, D) quadruples is studied. Necessary and sufficient conditions which are numerically verifiable are obtained and a numerically reliable decoupling method is established in the state space. Two examples are solved by our method which show the effectiveness.     

Dynamic leaders’ containment control of high-order multi-agent systems with state time-delay: an LMI approach

This paper addresses the containment control of a multi-agent system with dynamic leaders while considering the state time-delay and time-varying input disturbances. The agents dynamics are assumed to be high-order integrals and an unkown state time-delay with a known upper bound is considered. In order to deal with dynamic leaders and time-varying input disturbances, an appropriate distributed PI^p-type-based control protocol is proposed in this study. Thereafter, utilising the Lyapunov–Krasovskii stability theorem and based on LMI (Linear Matrix Inequality) approach, two theorems are proved which guarantee asymptotically stability of the multi-agent system with fixed and switching networks. In addition to stability assurance, the gains of the proposed protocol are achieved through solving the LMI. Finally, some numerical examples are given to demonstrate the validity of the theoretical results.     

Coordinate descent with arbitrary sampling I: algorithms and complexity†

We study the problem of minimizing the sum of a smooth convex function and a convex block-separable regularizer and propose a new randomized coordinate descent method, which we call ALPHA. Our method at every iteration updates a random subset of coordinates, following an arbitrary distribution. No coordinate descent methods capable to handle an arbitrary sampling have been studied in the literature before for this problem. ALPHA is a very flexible algorithm: in special cases, it reduces to deterministic and randomized methods such as gradient descent, coordinate descent, parallel coordinate descent and distributed coordinate descent—both in nonaccelerated and accelerated variants. The variants with arbitrary (or importance) sampling are new. We provide a complexity analysis of ALPHA, from which we deduce as a direct corollary complexity bounds for its many variants, all matching or improving best known bounds.     

Isomers of polycyclic conjugated hydrocarbons with arbitrary ring sizes—Part II

The numbers of C_nH_s isomers (#I) for completely condensed polycyclic conjugated hydrocarbons with arbitrary ring sizes are studied. Solutions for #I in the cases r=4, n_i=2 and r=4, n_i=0 were achieved. Here r and n_i are used to denote the number of rings and number of internal carbons, respectively. Combinatorial methods were applied, in combination with computer programming. The present work, in supplement to the preceding part of this article series, completes the analysis of #I for all cases where r ⪕ 4.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

Non-linear uncertain feedback systems with initial state values†

A single input-output non-linear plant w is considered which, due to uncertainty, is known only to be a member of a set . At t = 0, the plant has initial state values given by a vector Y ₀, member of a defined set ₀. At this instant it is switched into a feedback structure with compensation function G, which has zero initial state values. For certain classes of  it is shown how G may be designed to achieve arbitrary fast attenuation of the plant output for all wε Y ₀ε₀. The design philosophy is to convert this problem into one of disturbance attenuation in a linear time-invariant plant P with zero initial states ; Pε𝒫 a set of linear time-invariant plants and the disturbance dε a set of disturbances. ,𝒫 are determined by ₀,  and the performance specifications. Schauder's fixed point theorem is used to rigorously justify this technique. The design procedure may be readily used by engineers who are quite ignorant of the mathematical theory of non-linear operators. Ordinary, familiar frequency-response concepts are used in design execution, A design example is included.     

Quantum algorithm to solve function inversion with time–space trade-off

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time–space trade-off in this paper. The details are as follows. Let function \(f(x)=y\) have k solutions, where \(x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}\) for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability \(1-\left( 1-\frac{k}{2^{n}}\right) ^{L}\) for \(L\in Z^{+}\). The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time–space trade-off strategy, we improve the successful probability of algorithm.     

H ∞ control of networked control systems with state quantisation

This article addresses the problem of controller design for networked control systems over digital communication. The systems under consideration are stabilised via state feedback, where the effects of sampled signal, state quantisation, network-induced delay and packet dropout are considered. The proposed delay-dependent stability criteria are formulated in the form of a linear matrix inequality, which ensure asymptotic stability and a prescribed H _∞ performance level for networked control systems with admissible uncertainties. Maximum allowable delay bound of networked control systems is obtained by solving a convex optimisation problem. Furthermore, a numerical example is given to illustrate the effectiveness of the main result.     

Is the cutoff radius in DPD simulations with a fluid of constant density arbitrary?

The DPD (Dissipative Particle Dynamics) method was used to simulate the flow of a fluid occupying the space inside a rotating infinite long cylinder. The fluid was initially at rest and rigid body rotation was expected to be reached. However, as the simulation proceeded, the DP?s (Dissipative Particles) began accumulating near the wall, leaving the cylinder?s core empty of DP?s, obviously, an aphysical solution. To solve this problem it was suggested to set the cutoff radius as a function of the DP?s number density. A conceptual explanation is suggested.     

Extending Hoare Logic with an Infinite While-Rule

In the paper we generalize the while-rule in Hoare calculus to an infinite one and then present a sufficient condition much weaker than the expressiveness for Cook‘2 relative completeness theorem with respect to our new axiomatic system.Using the extended Hoare calculus we can derive true Hoare formulas which contain while statements free of loop invariants.It is also pointed out that the weak condition is a first order property and therefore provides a possible approach to the characterization of relative completeness which is also a first order property.     

Quantum measurement and the Aharonov–Bohm effect with superposed magnetic fluxes

We consider the magnetic flux in a quantum mechanical superposition of two values and find that the Aharonov–Bohm effect interference pattern contains information about the nature of the superposition, allowing information about the state of the flux to be extracted without disturbance. The information is obtained without transfer of energy or momentum and by accumulated nonlocal interactions of the vector potential $\varvec{A}$ with many charged particles forming the interference pattern, rather than with a single particle. We suggest an experimental test using already experimentally realized superposed currents in a superconducting ring and discuss broader implications.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

Robust H ∞ filtering for networked systems with multiple state delays

In this paper, a new robust H _∞ filter design problem is studied for a class of networked systems with multiple state-delays. Two kinds of incomplete measurements, namely, measurements with random delays and measurements with stochastic missing phenomenon, are simultaneously considered. Such incomplete measurements are induced by the limited bandwidth of communication networks, and are modelled as a linear function of a certain set of indicator functions that depend on the same stochastic variable. Attention is focused on the analysis and design problems of a full-order robust H _∞ filter such that, for all admissible parameter uncertainties and all possible incomplete measurements, the filtering error dynamics is exponentially mean-square stable and a prescribed H _∞ attenuation level is guaranteed. Some recently reported methodologies, such as delay-dependent and parameter-dependent stability analysis approaches, are employed to obtain less conservative results. Sufficient conditions, which are dependent on the occurrence probability of both the random sensor delay and missing measurement, are established for the existence of the desired filters in terms of certain linear matrix inequalities (LMIs). When these LMIs are feasible, the explicit expression of the desired filter can also be characterized. Finally, numerical examples are given to illustrate the effectiveness and applicability of the proposed design method.     

Linear control with incomplete state feedback and known initial-state statistics†

A method is proposed for the design of a linear, time-invariant state feedback control for a linear, time-invariant, finite-dimensional system with infinite duration of control (regulator problem), which makes use of only those state variables which are available for measurement, providing that these are sufficient to render the system stable. The expected value vector and the covariance matrix for the initial state are presumed to be known.

The cost function is quadratic and is expressed in terms of the initial state statistics and the cost-weighting matrix. The necessary conditions are derived for the minimization of the expected value of the cost. The minimization results in a set of m simultaneous polynomial equations in m unknowns where m is the product of the number of the available state variables and the number of control signals formed out of these. The theory is illustrated by a simple example of a third-order system.     

Delay-dependent robust stability of uncertain networked control systems with multiple state time-delays

In this paper, delay-dependent robust stability for a class of uncertain networked control systems （NCSs） with multiple state time-delays is investigated. Modeling of multi-input and multi-output （MIMO） NCSs with networkinduced delays and uncertainties through new methods are proposed. Some new stability criteria in terms of LMIs are derived by using Lyapunov stability theory combined with linear matrix inequalities （LMIs） techniques. We analyze the delay-dependent asymptotic stability and obtain maximum allowable delay bound （MADB） for the NCSs with the proposed methods. Compared with the reported results, the proposed results obtain a much less conservative MADB which are more general. Numerical example and simulation is used to illustrate the effectiveness of the proposed methods.