# Research

I am interested in mathematical physics research that lies in the intersection of probability theory and dynamical systems. Specifically, I am researching on probabilistic methods to model classical noise and disorder on repeated interaction dynamics, repeated measurements and random repeated applications of quantum operations.

### Papers

Thesis (a version of the thesis) Limiting Theorems for Stationary and Ergodic Quantum Processes - pdf

L. Pathirana, J. Schenker, "Law of Large Numbers and Central Limit Theorem for Ergodic Quantum Processes" - arXiv:2303.08992 - J.Math.Phys 64(8):082201(2023)

L. Pathirana, J. Schenker, "An Ergoidc Theorem for Continuous Parameter quantum Processes - L., J. Schenker" - In preparation

O. Ekblad, E. Moreno-Nadales, L. Pathirana, J. Schenker "Asymptotic Purification of Disordered Quantum Trajectories" - arXiv:2404.03168

## Recent (and planed) Talks:

University of Copenhagen, QLunch - November 22nd 2023

University of Illinois Urbana-Champaign Probability Seminar - May2nd 2023

AMS Spring Southeastern Sectional Meeting - Special Session on Disordered and Periodic Quantum Systems - March 18th 2023 - University of Atlanta

QInfo at INRIA - January 18th

Mathematical Physics Seminar - November 22nd 2022 - University of Kentucky

MSU Probability Seminar (Mathematics and Statistics) - November 9th 2022, Michigan State University

Mathematical Physics and Operator Algebras Seminar (MPOT) - October 18th 2022, Michigan State University

6th Great Lakes Mathematical Physics Meeting (GLaMP) - June 11th 2022, Michigan State University

11th Ohio River Analysis Meeting, University of Kentucky (ORAM11), April 2nd 2022

Student analysis seminar, Michigan State University, February 11th 2022

Student analysis seminar, Michigan State University, December 7th 2021

### Research Introduction:

I am interested in mathematics research that uses probability theory, complex analysis and dynamical systems as main tools. I am currently working with my advisor Prof. Jeffrey Schenker on problems that are related to mathematical physics. In a broad sense I am interested in introducing dynamical systems aspects into existing areas in mathematical physics.

Such applications of dynamics in existing math concepts can be used as a tool to model classical noise alongside quantum noise and in a more pure-mathematics aspect can be used to study certain products (compositions) of random maps that generalizes products (compositions) of IID random maps. Regarding these probabilistic and mathematical physics aspects, my recent and ongoing research can be vaguely classified into three categories:

Asymptotic behavior of quantum processes:

Discrete time:

The net change in an initial state resulting from a quantum process is obtained from composition of the quantum operations acting on the initial state. If this composition is generalized so that it is taken along a ergodic sequence of quantum operations which is defined in the following sense: Consider a discrete, ergodic, dynamical system (N, W, t) (N is the set of natural numbers, W is the probability space and t is the ergodic measure preserving transformation on W) and a map f that takes points (n,w) in (N x W) from the dynamical system to space of quantum operations. Starting from any point w (in W), we can obtain an ergodic sequence of quantum operations by evaluating the map f along the trajectory starting at w. Study of these quantum processes were developed by Movasaagh and Schenker (2021,2022) as means to study thermodynamic limits of matrix product states (MPS). I like to study limiting behavior (law of large numbers, central limit theorem, stables laws, etc) for discrete parameter stationary and ergodic quantum processes.

Continuous Time:

In continuous time parameter, a quantum processes is a double indexed family f_(s,t) of positive map valued random variables that satisfies for r<s<t, f_(r,t) = f_(s,t) o f_(r,s). This is a generalization of the discrete parameter processes. Along with the asymptotic behavior of theses processes one can also study properties of the Lindblad operator(also related to the ones above) that generates such positivity-improving maps as described above.

Study product (composition) of random linear maps:

Composition of random linear maps is a generalization of the discrete parameter quantum processes and an extension of the study of product of random matrices. Similar to some existing results in the study of product of random matrices under "irreducibility" conditions we are capable of obtaining asymptotic limits of theses compositions. I have obtained a LLN and a CLT that describe asymptotic behavior of such compositions that are governed by the (maximal) Lyapunove exponent of the sequence.

Repeated indirect measurements with classical noise:

Quantum trajectories are mathematically studied by the pioneers Maassen and Kümmerer in 2005. Quantum trajectory forms a Markov chain and it is proved that the paths of the Markov chain purifies with probability 1 if the associated Kraus operators are not proportional to an isometry in restriction to the range of P for any projection of dimension at least two. In the case where the quantum trajectories are modified to also include classical noise we expect the quantum trajectory to purify with probability 1 in the probability space [W x A^∞] where W is the probability space that models classical noise and A is the outcome measure space. We assume that the classical noise introduce a distribution to each of the Kraus operators V_i in the usual trajectory thus randomizing V_i in an ergodic and stationary manner.

From the article: An Ergodic Theorem for Quantum Processes with Applications to Matrix Product States by Ramis Movassagh , Jeffrey Schenker in Commun. Math. Phys. 395, 1174-1196 (2022)