#### Date of Award

Spring 1-1-2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### First Advisor

Martin Walter

#### Second Advisor

Markus J. Pflaum

#### Third Advisor

Graeme Smith

#### Fourth Advisor

Judith Packer

#### Fifth Advisor

Arlan Ramsay

#### Abstract

We construct a classical code, called a *Heisenberg code*, which is not uniquely decipherable in order to mimic the quantum behavior of uncertainty. We classify this code according to two properties and determine the possible codeword lengths for a Heisenberg code. We suggest a possible example of a physical system which utilizes Heisenberg codes. We define a channel for Heisenberg codes, called a *Heisenberg channel*, which is a composite of a *sender state* and a *receiver state* which are matrices of probability amplitudes. We demonstrate that Heisenberg channels have partial trace properties similar to density matrices for quantum states. Next, we show that certain Heisenberg channels can be associated to the correlations between different partite systems of a quantum states, and define *Heisenberg states* and *Heisenberg density matrices* which are sender states and Heisenberg channels with complex entries, respectively. We prove that a Heisenberg state exists for any quantum state and that a Heisenberg density matrix relating to an *n*-qubit quantum state is itself a density matrix for a (2*n* − 1)-qubit quantum state.

#### Recommended Citation

Ledbetter, Sion Nicolas, "Heisenberg Codes and Channels" (2018). *Mathematics Graduate Theses & Dissertations*. 57.

https://scholar.colorado.edu/math_gradetds/57