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


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).


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.

About marcocerezo

I'm Marco Cerezo, I have a Ph.D in Physics and I'm currently a Postdoctoral Research Associate at Los Alamos National Laboratory in New Mexico, USA. My main fields of study are Quantum Information, Quantum Computing and Condensed Matter. Currently I'm working to develop novel quantum algorithms which can be useful in near-term quantum devices.
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