The most general state of a quantum two-level system can be written in the form

where α and β are complex numbers, so it might seem like one has four real parameters to
work with. However, the state has to be normalized, so ∣α∣^{2} + ∣β∣^{2} = 1, and an overall phase
makes no difference, so either α or β can be chosen to be real. This leads to the “canonical”
parametrization

| (1) |

in terms of only two real numbers θ and φ, with natural ranges 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state (1) as a point on a unit sphere called the Bloch sphere.

In the notation of (1), the state is represented by the North pole, and the state by the South pole, whereas the states on the equator correspond to superpositions of and with equal weights (θ = π∕2), and different phases, parametrized by the azimuthal angle φ (the “longitude”). An alternative notation is and . Given any state on the sphere, the diametrically opposite point always represents an orthogonal state.

All the pictures of Bloch spheres generated by the applets on these pages can be freely rotated with the mouse or using the sliders provided. We have tried to indicate depth by graying out the parts of the sphere or the axes that are farthest from the viewer.

The vector V drawn from the center of the sphere to a point representing a state on the surface is called the Bloch vector, and in some cases it has a direct counterpart in “real” 3-dimensional space: in particular, for a spin-1/2 system, such as a proton, it represents the direction in which the particle’s magnetic moment is pointing, at least to the extent that it is possible to determine the direction of such an object in quantum mechanics.

Although a “pure state” such as (1) must lie on the surface of the sphere, the applets on these pages can also exhibit examples in which the state is represented by a point inside the sphere. Such states are called “mixed states” and arise as a result of the coupling of the two-level system to other quantum-mechanical objects. When this happens, one typically loses the ability to assign to the two-level system a pure state such as (1).

In practice, the difference between between pure and mixed states could be summarized as follows. For any pure state such as (1), one is always (in principle) able to predict with certainty the outcome of at least one possible measurement on the system; for a mixed state, on the other hand, the farther it is from the surface of the sphere, the less able the theory is to predict the outcome of any measurement. For example, if the system was a spin-1/2 particle in the maximally mixed state represented by the point at the center of the sphere, that would mean that we had absolutely no idea of what direction the magnetic moment was pointing in.

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