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Matrix Results and Techniques in Quantum Information Science and Related Topics
Pelejo, Diane Christine
Pelejo, Diane Christine
Abstract
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at $2^n\times 2^n$ unitary matrices, which describe operations on a closed $n$-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 $2^n\times 2^n$ unitary matrix to the product of no more than $2^{n-1}(2^n-1)$ single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element $\Phi$ among a given set of quantum channels $\mathcal{S}$ that will produce the optimal value of a scalar function $D(\rho_1,\Phi(\rho_2))$, on two fixed quantum states $\rho_1$ and $\rho_2$. Some of the functions we considered for $D(\cdot,\cdot)$ are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when $\mathcal{S}$ 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 chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers $a_1\geq\ldots\geq a_{2n}\geq 0$, we want to determine if there exist a $2n\times 2n$ density matrix $\rho$ having eigenvalues $a_1,\ldots,a_{2n}$ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on $a_1,\ldots,a_{2n}$ for $n\leq 6$ and state some observations on general values of $n$. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that $\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}$ for given $\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_n$ and $\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_m$, (b) constructing a multipartite state $\rho$ having a prescribed set of reduced states $\rho_1,\ldots, \rho_r$ on $r$ of its subsystems, (c) constructing a multipartite state$\rho$ 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. In chapter 6, we examine the shape of the Minkowski product of convex subsets $K_1$ and $K_2$ of $\IC$ given by $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was 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.
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2016-10-21
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Applied Science
DOI
http://doi.org/10.21220/S2CQ13