Characterization of quantum states in predicative logic

We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations. We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.Comment: 14 pages, Boston, IQSA10, to appea

A stability with optimality analysis of consensus-based distributed filters for discrete-time linear systems

In this paper we investigate how stability and optimality of consensus-based distributed filters depend on the number of consensus steps in a discrete-time setting for both directed and undirected graphs. By introducing two new algorithms, a simpler one based on dynamic averaging of the estimates and a more complex version where local error covariance matrices are exchanged as well, we are able to derive a complete theoretical analysis. In particular we show that dynamic averaging alone suffices to approximate the optimal centralized estimate if the number of consensus steps is large enough and that the number of consensus steps needed for stability can be computed in a distributed way. These results shed light on the advantages as well as the fundamental limitations shared by all the existing proposals for this class of algorithms in the basic case of linear time-invariant systems, that are relevant for the analysis of more complex situations

Basic Logic and Quantum Entanglement

As it is well known, quantum entanglement is one of the most important features of quantum computing, as it leads to massive quantum parallelism, hence to exponential computational speed-up. In a sense, quantum entanglement is considered as an implicit property of quantum computation itself. But...can it be made explicit? In other words, is it possible to find the connective "entanglement" in a logical sequent calculus for the machine language? And also, is it possible to "teach" the quantum computer to "mimic" the EPR "paradox"? The answer is in the affirmative, if the logical sequent calculus is that of the weakest possible logic, namely Basic logic. A weak logic has few structural rules. But in logic, a weak structure leaves more room for connectives (for example the connective "entanglement"). Furthermore, the absence in Basic logic of the two structural rules of contraction and weakening corresponds to the validity of the no-cloning and no-erase theorems, respectively, in quantum computing.Comment: 10 pages, 1 figure,LaTeX. Shorter version for proceedings requirements. Contributed paper at DICE2006, Piombino, Ital

LQ non-Gaussian Control with I/O packet losses

The paper concerns the Linear Quadratic non-Gaussian (LQnG) sub-optimal control problem when the input and output signals travel through an unreliable network, namely Gilbert-Elliot channels. In particular, the input/output packet losses are modeled by Bernoulli sequences, and we assume that the moments of the non-Gaussian noises up to the fourth order are known. By mean of a suitable rewriting of the system through an intermittent output injection term, and by considering an augmented system with the second-order Kronecker power of the measurements, a simple solution is provided by substituting the Kalman predictor with intermittent observations of the LQG control law with a quadratic optimal predictor. Numerical simulations show the effectiveness of the proposed method

A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic

Delay-State Dynamics to Filtering Gaussian Systems with Markovian Delayed Measurements

In this paper we propose a solution to the problems of detecting a stochastic output delay sequence characterized by a Markov chain and estimating the state of a linear system driven by Gaussian noise through an augmented delaystate dynamics. This is the model for uncertain observations resulting from losses in the propagation channel due to fading phenomena or packet dropouts that is common in wireless sensor networks, networked control systems, or remote sensing applications. The solution we propose consists of two parallel stages: a nonlinear detector, which identifies at each time instant the delay and a filtering stage. Numerical simulations show the performance of the proposed method

Distributed estimation of nonlinear systems

In a classical distributed framework, we present a novel distributed observer for genuinely nonlinear continuous-time plants. A network of sensors monitors a multiple-outputs plant. Each sensor measures only a portion of the plant’s outputs, and the sensing capability is different from sensor to sensor. The assumption of strongly connected digraph on the underlying sensor network ensures robustness and direct communication paths between nodes. Moreover, incremental homogeneity assumptions on the plant embrace a very large class of nonlinear systems for which a distributed observer can be designed. The distributed observer consists of local observers associated with each sensor, asymptotically estimating the entire state of the plant only by using the local sensing capability and information exchanged through the communication network. Numerical simulations on a network of interconnected Van Der Pol oscillators confirms theoretical results. Robustness and switch

Stochastic output delay identification and filtering of discrete-time gaussian systems

In this paper we propose a solution to the problems of detecting a generally correlated stochastic output delay sequence of a linear system driven by Gaussian noise. This is the model for uncertain observations resulting from losses in the propagation channel due to fading phenomena or packet dropouts that is common in wireless sensor networks, networked control systems, or remote sensing applications. The solution we propose consists of a nonlinear detector which identifies online the stochastic delay sequence. The solution provided is optimal in the sense that minimizes the probability of error of the delay detector. Finally, a filtering stage fed with the information given by the detector can follow to estimate the state of the system. Numerical simulations show good performance of the proposed method

Distributed Infinite-Horizon Optimal Control of Discrete-Time Linear Systems over Networks

In this paper we consider the distributed infinite-horizon Linear-Quadratic-Gaussian optimal control problem for discrete-time systems over networks. In particular, the feedback controller is composed of local control stations, which receives some measurement data from the plant process and regulates a portion of the input signal. We provide a solution when the nodes have information on the structural data of the whole network but takes local actions, and also when only local information on the network are available to the nodes. The proposed solution is arbitrarily close to the optimal centralized one (in terms of cost index) when the intermediate consensus steps are sufficiently large