the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

quenched asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials this article describes the almost sure infinite volume asymptotics of the ground state energy of random schrodinger operators with scaled gibbsian potentials the random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered gibbs measure with a superstable pair interaction there is no restriction on the strength of the pair interaction it may be taken e g at a critical point the potential is scaled with the box size in a critical way i e the scale is determined by the typical size of large deviations in the gibbsian cloud the almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities the qualitative behaviour of the ground state energy asymptotics is analysed depending on the dimension and on the holder exponents of the free energy density it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy

indirect controllability of quantum systems a study of two interacting quantum bits a quantum mechanical system s is indirectly controlled when the control affects an ancillary system a and the evolution of s is modified through the interaction with a only a study of indirect controllability gives a description of the set of states that can be obtained for s with this scheme in this paper we study the indirect controllability of quantum systems in the finite dimensional case after discussing the relevant definitions we give a general necessary condition for controllability in lie algebraic terms we present a detailed treatment of the case where both systems s and a are two dimensional qubits in particular we characterize the dynamical lie algebra associated with s a extending previous results and prove that complete controllability of s a and an appropriate notion of indirect controllability are equivalent properties for this system we also prove several further indirect controllability properties for the system of two qubits and illustrate the role of the lie algebraic analysis in the study of reachable states

control of a two level quantum system in a coherent feedback scheme we consider a scheme for the control of a spin s where an auxiliary spin a plays the role of the controller and an ising interaction is assumed between the spins we demonstrate that even if there is no complete controllability on the system s a we can have full control on the system s if and only if the initial state of a is pure this provides the simplest example where the system s is indirectly controllable while the whole system s a is not completely controllable we also give an explicit algorithm for the indirect control of s

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

time optimal control of a two level quantum system via interaction with an auxiliary system we consider a two level quantum system quantum bit s which can only be controlled by modulating the difference between the two energy levels we propose a control scheme that utilizes an additional quantum bit a as an auxiliary system and obtains manipulation of s through the interaction with a for this scheme we derive the minimum time control for any unitary transformation on the target system s and the form of the optimal control law we find that the optimal time protocol involves a preliminary preparation of the state of a the control of s using interaction with a and no simultaneous control of s and a the optimal time is the same as the one for a controllable two level system

optimal control at the quantum speed limit optimal control theory is a promising candidate for a drastic improvement of the performance of quantum information tasks we explore its ultimate limit in paradigmatic cases and demonstrate that it coincides with the maximum speed limit allowed by quantum evolution

exact algebraic conditions for indirect controllability of quantum systems in several quantum control schemes a target quantum system s is put in contact with an auxiliary system a and the coherent control can directly affect only a the system s is controlled indirectly through the interaction with a the system s is said to be indirectly controllable if every unitary transformation can be performed on the state of s with this scheme the indirect controllability of s will depend on the dynamical lie algebra cal l characterizing the dynamics of the total system s a and on the initial state of the auxiliary system a in this paper we describe this characterization exactly a natural assumption is that the auxiliary system a is minimal which means that there is no part of a that is not coupled to s and we denote by n a the dimension of such a minimal a which we assume to be fully controllable we show that if n a is greater than or equal to 3 indirect controllability of s is verified if and only if complete controllability of the tot

sample average approximation method for compound stochastic optimization problems the paper studies stochastic optimization programming problems with compound functions containing expectations and extreme values of other random functions as arguments compound functions arise in various applications a typical example is a variance function of nonlinear outcomes other examples include stochastic minimax problems econometric models with latent variables and multilevel and multicriteria stochastic optimization problems as a solution technique a sample average approximation saa method also known as statistical or empirical sample mean method is used the method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems in stochastic optimization this method is widely used for optimization of standard expectation functions under constraints in this paper saa method is extended to general compound stochastic optimization problems the conditions for convergence in mean almost sure

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

constructive decomposition of the controllability lie algebra for quantum systems in this technical note we show how to use the analysis of the controllability lie algebra of a quantum mechanical system to study its dynamics and facilitate the design of controls we give algorithms to decompose the dynamics and describe an example of application to two coupled spin 1 2 s

criteria for reachability of quantum states we address the question of which quantum states can be inter converted under the action of a time dependent hamiltonian in particular we consider the problem as applied to mixed states and investigate the difference between pure and mixed state controllabilities introduced in previous work we provide a complete characterization of the eigenvalue spectrum for which the state is controllable under the action of the symplectic group we also address the problem of which states can be prepared if the dynamical lie group is not sufficiently large to allow the system to be controllable

controllability of open quantum systems with kraus map dynamics this paper presents a constructive proof of complete kinematic state controllability of finite dimensional open quantum systems whose dynamics are represented by kraus maps for any pair of states pure or mixed on the hilbert space of the system we explicitly show how to construct a kraus map that transforms one state into another moreover we prove by construction the existence of a kraus map that transforms all initial states into a predefined target state such a process may be used for example in quantum information dilution thus in sharp contrast to unitary control kraus map dynamics allows for the design of controls which are robust to variations in the initial state of the system the capabilities of non unitary control for population transfer between pure states illustrated for an example of a two level system by constructing a family of non unitary kraus maps to transform one pure state into another the problem of dynamic state controllability of open quantum systems i e controllability of state to state transformations given a set of available dynamical resources such as coherent controls incoherent interactions with the environment and measurements is also discussed

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

equivalence between indirect controllability and complete controllability for quantum systems abstract we consider a control scheme where a quantum system s is put in contact with an auxiliary quantum system a and the control can affect a only while s is the system of interest the system s is then controlled indirectly through the interaction with a complete controllability of s a means that every unitary state transformation for the system s a can be achieved with this scheme indirect controllability means that every unitary transformation on the system s can be achieved we prove in this paper under appropriate conditions and definitions that these two notions are equivalent in finite dimension we use lie algebraic methods to prove this result

degrees of controllability for quantum systems and application to atomic systems precise definitions for different degrees of controllability for quantum systems are given and necessary and sufficient conditions for each type of controllability are discussed the results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies