13.5: Detail- Qubit and Applications

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Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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Sections 13.5 to 13.11 are based on notes written by Luis Pérez-Breva May 4, 2005.

The previous sections have introduced the basic features of quantum information in terms of the wave function. We first introduced the wave function in the context of physical systems in Chapter 10. The wave function is a controversial object, that has given rise to different schools of thought about the physical interpretation of quantum mechanics. Nevertheless, it is extremely useful to derive probabilities for the location of a particle at a given time, and it allowed us to introduce the multi-state model in Chapter 10 as a direct consequence of the linearity of Schrödinger equation.

In the first chapters of these notes, we introduced the bit as a binary (two-state) quantity to study classical information science. In quantum information science we are also interested in two-state systems. However, unlike the classical bit, the quantum mechanical bit may also be in a state of superposition. For example we could be interested in superpositions of the first two energy levels of the infinite potential well. The type of mathematics required for addressing the dynamics of two-state systems, possibly including superpositions, is linear algebra (that we reviewed in Chapter 2 in the context of the discrete cosine transformation.)

To emphasize that we are interested in the dynamics of two-state systems, and not in the dynamics of each state, it is best to abstract the wave function and introduce a new notation, the bracket notation. In the following sections, as we introduce the bracket notation, we appreciate the first important differences with the classical domain: the no-cloning theorem and entanglement. Then we give an overview of the applications of quantum mechanics to communication (teleportation and cryptography), algorithms (Grover fast search, Deutsch-Josza) and information science (error correcting codes).