Time: Tuesday, October 20, 2015

Speaker: Dr. Radhakrishnan Balu (US Army Research Laboratory)

Title: An Introduction to Quantum Stochastics

Abstract: Quantum probability (QP) is a generalization of axiomatic probability theory and quantum mechanics. The classical ideas of random variables and measures are extended to operators (Hermitian) and trace operations enabling the rich theory of operators to be exploited. In QP probability amplitudes play a prominent role producing counter intuitive interference effects not possible in classical situations. Quantum analogues of stochastic processes can be defined as operator processes that can be used to model noise in quantum systems interacting with an environment. We will start with a review of notions in classical probability space and define the corresponding notions in QP. To make the ideas concrete we will consider a discrete time quantum processes to model classical Markov chains to express a bias removal algorithm. We will also discuss examples of continuous time versions in modeling quantum networks. As a second application we will discuss a quantum mechanics based logic programming language to express probability distributions and statistical correlations, by extending the notion of propositions to operators. We will discuss examples from quantum communication protocols in this new language and outline how to prove properties about them. Finally, we will discuss issues in designing efficient algorithms in solving the stochastic partial differential equations derived as part of modeling quantum noises.

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