[PAST EVENT] Diane Pelejo, Applied Science - Ph.D. Dissertation Defense

November 9, 2016
3pm - 5pm

ABSTRACT: Several matrix-related problems and results motivated by quantum information theory is presented in this dissertation.
In the first problem we look at 2^n-by-2^n unitary matrices, which describe operations on closed a n-qubit system.
We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost.
We then present a recurrence scheme to decompose a general 2^n-by-2^nunitary matrix to the product of no more than
2^(n-1)(2^n-1) single qubit gates with small number of controls.
In the second problem
Next, we address the problem of finding a specific element P among a given set of quantum channels S that will produce
the optimal value of a scalar function D(r_1,P(r_2)), on two fixed quantum states r_1 and r_2. Some of the  functions
we considered for D(-,-) are the trace distance,  quantum fidelity and quantum relative entropy. We discuss the optimal
solution when S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum
channels, and the set of all quantum channels.
In the third problem, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit
subsystem. More specifically, given positive numbers a_1>=...>=a_{2n}>=0 we want to determine if there exist a 2n-by-2n
density matrix r having eigenvalues a_1,...,a_{2n} and satisfying tr_1(r)=I_n/n. This problem is a special case of the
more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a_1,...,a_{2n} for n<=6
and state some observations on general values of n.
Next, we discuss projection methods and illustrate its usefulness in:
(a) constructing a quantum channel P, if it exists, such that P(r_1)=s_1,..., P(r_k)=s_k for given n-by-n density matrices
r_1,...,r_k and m-by-m density matrices s_1,...,s_k; (b) constructing a multipartite state r having a prescribed set of
reduced states r_1,...r_s on s of its subsystems; (c) constructing a multipartite state r having prescribed reduced states and
additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining
if a square matrix A can be written as a product of two positive semidefinite contractions.
Lastly, we examine the shape of the Minkowski product of  convex subsets K_1,K_2 of the complex plane given by
K_1K_2={ab : a is in K_1, b is in K_2} which has applications in the study of the product numerical range and quantum error-correction.
In 2011, Karol et al,  conjectured that K_1K_2 is star-shaped when K_1 and K_2 are convex. We give counterexamples to show that
this conjecture does not hold in general but we show that the set K_1K_2 is star-shaped if K_1 is a line segment or a circular disk.

BIO: Diane Pelejo is an international student from the town of Montalban, Rizal in the Philippines.
She obtained her Bachelor's and Master's degree in Mathematics from the University of the Philippines Diliman (UPD).
She joined the PhD Applied Science program at William & Mary in the Spring of 2013. She worked on
matrix-related problems in quantum information theory under the guidance of Dr. Chi-Kwong Li of the Mathematics department.
After graduation, Diane will be returning to the Philippines, where an assistant professor position is waiting for her in UPD.
She aims to contribute to research and development in Mathematics in her country.