**Time: ** Tuesday, October 20, 2015

2:00pm

MTH1311

**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.