The More You Know: Bloch Sphere

The Bloch Sphere is a useful geometrical representation of the state of a Qubit or any other two level quantum system. As discussed in this article an arbitrary pure state |\psi\rangle of a Qubit can be expressed in the computational basis as

|\psi\rangle=\alpha|0\rangle + \beta|1\rangle,

where \alpha and \beta are complex number such that |\alpha|^2+|\beta|^2=1. Since an overall phase has no observable effects, \alpha or \beta can be chosen  as real numbers and |\psi\rangle can written as

|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+\exp^{i\phi}\sin\frac{\theta}{2}|1\rangle,

where \theta and \phi \in \mathbb{R}. The parameters 0\leq\theta\leq\pi and 0\leq\phi\leq2\pi define a point on the 3D sphere called the Bloch Sphere (Fig. 1).

Bloch_Sphere

Fig. 1: Bloch Sphere representation of the State of a Qubit.

The points in the surface of the sphere represent pure states of the qubit, with |0\rangle as the North pole and |1\rangle as the South pole, and the equator states of the form |\psi\rangle=|0\rangle+\exp^{i\phi}|1\rangle. Additionally any mixed state of a qubit can be represented as a point inside the sphere, with maximally mixed states at the center of the Bloch Sphere.

This representation is useful in Quantum Computation to visualize the effect of single qubit gates.

Go to the Dictionary of Quantum Information and Quantum Computation.


All text copyright © Marco Vinicio Sebastian Cerezo de la Roca.

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The More You Know: Bloch Sphere por Marco Vinicio Sebastian Cerezo de la Roca se distribuye bajo una Licencia Creative Commons Atribución-NoComercial-CompartirIgual 4.0 Internacional.

Image credits here.

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About marcocerezo

My name is Marco Cerezo, I have a degree in physics from the Universidad Nacional de la Plata (UNLP) in Buenos Aires, Argentina, and I'm a Ph.D student. Although I study in Argentina, I was born and raised in Guatemala, so I consider myself "chapín" at heart. From an early age I discovered how exciting, interesting and fun physics can be, so I decided to devote myself to study this passionate field. This blog is dedicated to all those who want to dig a little into the mysteries and beauty of physics. And in particular, to all those who are interested in studying Quantum Computation and Quantum Information. ---------------------------------------------------------------------------------------------------------- Mi nombre es Marco Cerezo, Licenciado en Física egresado de la Universidad Nacional de la Plata (UNLP) en Buenos Aires, Argentina. Actualmente estoy haciendo mi Doctorado en física. Aunque estudio en Argentina, nací y crecí en Guatemala por lo que me considero chapín de corazón. Desde muy temprana edad descubrí lo apasionante, interesante y divertida que es la física por lo que decidí dedicarme a su estudio. Este blog es está dedicado a todos aquellos que quieren indagar un poco en los misterios y la belleza de la física; y en particular, a aquellos que quieran aprender sobre Computación e Información Cuántica.
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