Bloch sphere

The Bloch sphere is used in quantum mechanics to ( for example, a qubit ) represent the state of a two- state system graphically. It was named after the physicist Felix Bloch, who developed these clear illustration for superpositions of states. This is a geometric representation by means of which the state of a two state system is characterized as a point on the surface of the Bloch sphere.

Tableau

The vectors pointing to the poles of the Bloch sphere, the vectors of the base are predetermined. Points which lie on the equator of the Bloch sphere, correspond those states which consist of equal proportions of the two base states. The points that lie on the upper hemisphere, sit down for the greater part of the basic state of the upper base vector along and points on the lower hemisphere come together to form a larger part of the lower base state. The right figure shows the standard basis vectors are ( for spin systems one chooses usually ) and the so-called Bloch vector drawn, wherein the vector is defined as follows:

The vector corresponding to the eigenvector of the spin operator in direction with a unit vector in any direction ( in real space of intuition ) is, which is determined by the angle and

The spin operator vector. This eigenvector is not a vector in the space of intuition in the example the direction of lives. Instead, it is the eigenvector of the room element is spanned by the eigen vectors of the operator.

Related to the Riemann sphere

The linear combination of the two poles of the associated state vectors (hereinafter referred to and designated by ), since it does not depend on a quantum state of the phase and the magnitude of the result is normalized to one, are represented by a single complex number as follows:

Note that the counter of this fraction is a vector, the denominator only a number necessary for normalization.

The Bloch sphere is now the Riemann sphere of the complex number.

Pure and mixed states

The Pauli matrices are Hermitian and, together with the unit matrix is ​​a basis of the vector space of complex matrices. The density matrix of a qubit can always relative to a fixed base as

Are shown. Summing up as a vector in, then is always positive semi-definite, ie an admissible density matrix, when in the closed unit ball of lies. The vector is called the Bloch vector. The condition is then purely exactly when the Bloch vector has length one, ie, is on the ball surface.

Two pure states are orthogonal if their Bloch vectors are located at diametrically opposite points on the Bloch sphere. In the middle of the Bloch sphere is the fully mixed condition, the Bloch vector is the null vector.

Forming a mixture of a portion of the state of Bloch vector and a component of the state vector with Bloch, then the mixture is described by Bloch vector. So you can write all the states as a convex combination of pure states, and the Bloch sphere also shows that the state space is a convex set whose extreme points are the pure states.

Geometric interpretation

And are spin states to the spin quantum number of 1/2, substantially parallel position, and anti- parallel orientation of an electron in the magnetic field, showing the superposition state of the expected value of the ( vectorial ) spin operator in the direction which indicates the assigned point on the Bloch sphere.