optimal large scale quantum state tomography with pauli measurements quantum state tomography aims to determine the state of a quantum system as represented by a density matrix it is a fundamental task in modern scientific studies involving quantum systems in this paper we study estimation of high dimensional density matrices based on pauli measurements in particular under appropriate notion of sparsity we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and frobenius norm losses and show how these rates can be achieved by a common thresholding approach numerical performance of the proposed estimator is also investigated

efficient method for computing the maximum likelihood quantum state from measurements with additive gaussian noise we provide an efficient method for computing the maximum likelihood mixed quantum state with density matrix given a set of measurement outcomes in a complete orthonormal operator basis subject to gaussian noise our method works by first changing basis yielding a candidate density matrix which may have nonphysical negative eigenvalues and then finding the nearest physical state under the 2 norm our algorithm takes at worst o d 4 for the basis change plus o d 3 for finding where d is the dimension of the quantum state in the special case where the measurement basis is strings of pauli operators the basis change takes only o d 3 as well the workhorse of the algorithm is a new linear time method for finding the closest probability distribution in euclidean distance to a set of real numbers summing to one

quantum tomography by regularized linear regressions abstract in this paper we study extended linear regression approaches for quantum state tomography based on regularization techniques for unknown quantum states represented by density matrices performing measurements under certain basis yields random outcomes from which a classical linear regression model can be established first of all for complete or over complete measurement bases we show that the empirical data can be utilized for the construction of a weighted least squares estimate lse for quantum tomography taking into consideration the trace one condition a constrained weighted lse can be explicitly computed being the optimal unbiased estimation among all linear estimators next for general measurement bases we show that l 2 regularization with proper regularization gain provides even a lower mean square error under a cost in bias the optimal regularization parameter is defined in terms of a risk characterization for any finite sample size and a resulting implementable estimator is proposed finally a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model which proves that the proposed implementable tuning estimator is asymptotically optimal as the number of copies grows to infinity additionally several numerical examples are provided to validate the established results

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

optimal quantum tomography of states measurements and transformations we present the first complete optimization of quantum tomography for states positive operator value measures and various classes of transformations for arbitrary prior ensemble and arbitrary representation giving corresponding feasible experimental schemes in terms of random bell measurements

quantum gate identification error analysis numerical results and optical experiment the identification of an unknown quantum gate is a significant issue in quantum technology in this paper we propose a quantum gate identification method within the framework of quantum process tomography in this method a series of pure states are applied to the gate and then a fast state tomography on the output states is performed and the data are used to reconstruct the quantum gate the algorithm has computational complexity o d3 with the system dimension d the identification approach is compared with the maximum likelihood estimation method for the running time which shows an efficiency advantage of our method an error upper bound is established for the identification algorithm and the robustness of the algorithm against impurities in the input states is also tested we perform a quantum optical experiment on a single qubit hadamard gate to verify the effectiveness of the identification algorithm

necessary optimality conditions for infinite dimensional state constrained control problems abstract 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

a non stationary multi scale oscillating free boundary for the laplace and heat equations the paper is concerned with the homogenization problem for the poisson equation in a domain a part of whose boundary varies in time and is rapidly oscillating in space at each time such a problem arises in growing semiconductor film taking into account the fact that the surface of the film is changing in time by chemical vapor deposition or by etching we prove existence and uniqueness of the solution and establish its homogenizated approximation we obtain an l estimate on the error between the free boundary and the homogenized boundary

hamiltonian identification through enhanced observability utilizing quantum control this note considers hamiltonian identification for a controllable quantum system with nondegenerate transitions and a known initial state we assume to have at our disposal a single scalar control input and the population measure of only one state at an arbitrarily large final time t we prove that the quantum dipole moment matrix is locally observable in the following sense for any two close but distinct dipole moment matrices we construct discriminating controls giving two different measurements this result suggests that what may appear at first to be very restrictive measurements are actually rich for identification when combined with well designed discriminating controls to uniquely identify the complete dipole moment of such systems

an invitation to quantum tomography we describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems the state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the wigner function we present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its wigner function we illustrate the performance of the estimators on simulated data an em algorithm is proposed for practical implementation there remain many open problems e g rates of convergence adaptation and studying other estimators a main purpose of the paper is to bring these to the attention of the statistical community

observer based hamiltonian identification for quantum systems a symmetry preserving observer based parameter identification algorithm for quantum systems is proposed starting with a 2 level quantum system qubit where the unknown parameters consist of the atom laser frequency detuning and coupling constant we prove an exponential convergence result the analysis is inspired by lyapunov and adaptive control techniques and is based on averaging theory the observer is then extended to the multi level case where all the atom laser coupling constants are unknown the extension of the convergence analysis is discussed through some heuristic arguments the relevance and the robustness with respect to various noises are tested through numerical simulations

pontryagin maximum principle for semilinear and quasilinearparabolic equations with pointwise state constraints this paper studies the first order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial differential equations with pointwise state constraints a pontryagin type maximum principle is obtained

minimax control for evolutionary variational bilateral problem this paper deals with a minimax control problem for semilinear evolutionary variational inequalities associated with bilateral constraints the control domain is not necessarily convex the cost functional which is to be minimized is the sup norm of some function of the state and the control the major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation variational inequality and the nonsmooth cost functional the sup norm in this paper the existence and the pontryagin type necessary condition for optimal controls are established

hedged maximum likelihood quantum state estimation this letter proposes and analyzes a new method for quantum state estimation called hedged maximum likelihood hmle hmle is a quantum version of lidstone s law also known as the add rule a straightforward modification of maximum likelihood estimation mle it can be used as a plug in replacement for mle the hmle estimate is a strictly positive density matrix slightly less likely than the ml estimate but with much better behavior for predictive tasks single qubit numerics indicate that hmle beats mle according to several metrics for nearly all true states for nearly pure states mle does slightly better but neither method is optimal

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

state constrained optimal control problems governed by coupled nonlinear wave equations with memory this work considers the optimal control problems governed by coupled nonlinear wave equations with memory where the state constraint is a type of integral first we investigate the existence of the optimal solution for the optimal control problem and then we deduce the maximum principle for the optimal solution with state constraint moreover we give a concrete example which is covered by the main result

asymptotic equivalence of quantum state tomography and noisy matrix completion matrix completion and quantum tomography are two unrelated research areas with great current interest in many modern scientiflc studies this paper investigates the statistical relationship between trace regression in matrix completion and quantum state tomography in quantum physics and quantum information science as quantum state tomography and trace regression share the common goal of recovering an unknown matrix it is nature to put them in the le cam paradigm for statistical comparison regarding the two types of matrix inference problems as two statistical experiments we establish their asymptotic equivalence in terms of deflciency distance the equivalence study motivates us to introduce a new trace regression model the asymptotic equivalence provides a sound statistical foundation for applying matrix completion methods to quantum state tomography we investigate the asymptotic equivalence for sparse density matrices and low rank density matrices and demonstrate that sparsity and low rank are not necessarily helpful for achieving the asymptotic equivalence of quantum state tomography and trace regression in particular we show that popular pauli measurements are bad for establishing the asymptotic equivalence for sparse density matrices and low rank density matrices

backwards uniqueness of the c_0 semigroup associated with a parabolic hyperbolic stokes lame partial differential equation system in this paper the backward uniqueness property is ascertained for a two or three dimensional fluid structure interactive partial differential equation pde system for which an explicit co semigroup formulation was recently given by avalos and triggiani 2007 on the natural finite energy space h see also their contemporary mathematics article of 2007 for a preliminary simplified canonical model this system of coupled pdes comprises the parabolic stokes equations and the hyperbolic lame system of dynamic elasticity each dynamic evolves within its respective domain while being coupled on the boundary interface between fluid and structure in terms of said fluid structure semigroup e at posed on the associated finite energy space h the backward uniqueness property can be stated in this way if for given initial data y 0 h e at y 0 0 for some t 0 then necessarily y 0 0 the proof of this property hinges on establishing necessary pde estimates for a certain static fluid structure equation in order to invoke the abstract backward uniqueness resolvent based criterion by lasiecka renardy and triggiani 2001 the backward uniqueness property for the coupled stokes lame pde is motivated by and has positive implications to the problem of exact controllability in the hyperbolic state variables w w t and simultaneously approximate controllability in the parabolic state variable u of the present coupled pde model under boundary control a similar situation occurred for thermoelastic models as shown in papers by m eller v isakov i lasiecka m renardy and r triggiani

a quantum hamiltonian identification algorithm computational complexity and error analysis quantum hamiltonian identification qhi is important for characterizing the dynamics of quantum systems calibrating quantum devices and achieving precise quantum control in this paper an effective two step optimization tso qhi algorithm is developed within the framework of quantum process tomography in the identification method different probe states are input into quantum systems and the output states are estimated using the quantum state tomography protocol via linear regression estimation the time independent system hamiltonian is reconstructed based on the experimental data for the output states the hamiltonian identification method has computational complexity o d 6 where d is the dimension of the system hamiltonian an error upper bound o frac d 3 sqrt n is also established where n is the resource number for the tomography of each output state and several numerical examples demonstrate the effectiveness of the proposed tso hamiltonian identification method

quantum state reconstruction by maximizing likelihood and entropy quantum state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements as a rule does not yield a unique result we derive a reconstruction scheme where both the likelihood and the von neumann entropy functionals are maximized in order to systematically select the most likely estimator with the largest entropy that is the least bias estimator consistent with a given set of measurement data this is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity an interesting structure of such estimators will also be explored