<p>The Bloch representation of an <i>n</i>-qubit channel provides a way to represent quantum channels as certain affine transformations on [special characters omitted]. In higher dimensions (<i>n</i> > 1), the correspondence between quantum channels and their Bloch representations is not well-understood. Partly motivated by the ability to simplify the calculation of information theoretic quantities of a qubit channel using the Bloch representation, in this thesis we investigate the correspondence between a channel and its Bloch representation with an emphasis on <i>unital n</i>-qubit channels, in which case the Bloch representation is linear.</p><p> The thesis is divided into three main sections. First we focus our attention on qubit channels. For certain sets of quantum channels, we establish the surprising existence of a special isomorphism into the set of <i> classical channels.</i> We classify the sets of qubit channels with this property and show that information theoretic quantities are preserved by such classical representations. In a natural progression, we prove some well-known facts about SO(3), the proofs of which are either nonexistent or difficult to find in the literature. Some of this work is based on [12, 13].</p><p> In the next section, we consider the multi-qubit channels and show that every finite group can be realized as a subgroup of the quantum channels; this approach allows for the construction of a quantum representation for the free affine monoid over any finite group and gives a classical representation for it. We extend some fundamental results from [26, 28] to the multi-qubit case, including that the set of diagonal Bloch matrices is equal to the free affine monoid over the involution group [special characters omitted]. Some of this work appeared in [10]. </p><p> Lastly, we study the extreme points for the set of <i>n</i>-qubit channels. There are two types of extreme points: invertible and non-invertible; invertible channels are non-singular maps for which the inverse is also a channel. We briefly study the non-invertible extreme points and then parameterize and analyze the invertible<i>n</i>-qubit Bloch matrices, which form a compact connected Lie group. We calculate the Lie algebra and give an explicit generating set for the invertible Bloch matrices and a maximal torus.</p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:3591941 |
| Date | 04 October 2013 |
| Creators | Crowder, Tanner |
| Publisher | Howard University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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