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

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

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

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).

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.

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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