Bloch sphere

The Bloch sphere (after its developer Felix Bloch ) is a graphical-geometric representation in quantum mechanics . It represents the superposition of the states of a two-state system (e.g. a qubit ) as points on a spherical surface.

Illustrative representation

Bloch sphere

The vectors pointing to the poles of the Bloch sphere are the vectors of the given base . Points that lie on the equator of the Bloch sphere correspond to those states that consist of both base states in equal proportions. The points that lie on the upper hemisphere are composed for the greater part of the basic state of the upper basis vector, and points on the lower hemisphere are composed for the greater part of the lower basic state.

The figure on the right shows:

• the standard basis vectors (for spin systems one usually chooses ) ${\displaystyle |0\rangle ,|1\rangle }$ ${\displaystyle |0\rangle =|z+\rangle ,|1\rangle =|z-\rangle }$
• the Bloch vector , which is defined as follows: ${\displaystyle |\psi \rangle }$

${\displaystyle {\begin{aligned}|\psi \rangle &=\cos {\frac {\theta }{2}}|0\rangle \,+\,e^{i\varphi }\sin {\frac {\theta }{2}}|1\rangle \\&=\cos {\frac {\theta }{2}}|0\rangle \,+\,(\cos \varphi +i\sin \varphi )\,\sin {\frac {\theta }{2}}|1\rangle \end{aligned}}}$

With and you get all the states where the squares of the coefficients can be interpreted as probabilities with the sum one. The coefficient at is restricted to real values ​​in order to eliminate the physically non-existent degree of freedom of a common complex phase of both components. ${\displaystyle 0\leq \theta \leq \pi }$ ${\displaystyle 0\leq \varphi <2\pi }$ ${\displaystyle |0\rangle }$

The Bloch vector corresponds to the eigenvector of the spin operator in the direction, where ${\displaystyle |n+\rangle }$ ${\displaystyle {\hat {S}}_{\vec {n}}={\vec {n}}\cdot {\hat {\vec {s}}}{\overset {.}{=}}{\frac {\hbar }{2}}({\vec {n}}\cdot {\vec {\sigma }})}$ ${\displaystyle {\vec {n}}}$

• the direction in the real visual space is given by the angle and ${\displaystyle {\vec {n}}={\begin{pmatrix}\sin \theta \cdot \cos \varphi \\\sin \theta \cdot \sin \varphi \\\cos \theta \end{pmatrix}}}$ ${\displaystyle \theta ,\varphi }$
• ${\displaystyle {\hat {\vec {s}}}\;{\overset {.}{=}}\;{\frac {\hbar }{2}}\cdot {\vec {\sigma }}=\!\sum _{i\in \{x,y,z\}}\!\!{\vec {e}}_{i}{\frac {\hbar }{2}}\cdot \sigma _{i}}$ is the spin operator vector.

The eigenvector is not a vector in the visual space in which z. B. the direction lives. Instead, it is an element of the space that is spanned by the eigenvectors of the operator . ${\displaystyle |n+\rangle }$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\hat {S}}_{\vec {n}}}$

Connections

With the Riemann number ball

The linear combination of the state vectors assigned to the two poles (hereinafter referred to by and ) can be represented with a single complex number because the phase is not important in a quantum state and the magnitude of the result is normalized to one : ${\displaystyle \left|\uparrow \right\rangle }$ ${\displaystyle \left|\downarrow \right\rangle }$ ${\displaystyle c}$

${\displaystyle \left|\Psi \right\rangle ={\frac {\left|\uparrow \right\rangle +c\left|\downarrow \right\rangle }{\sqrt {1+\left|c\right|^{2}}}}}$

Note that the numerator of this fraction is a vector, but the denominator is only a number required for normalization .

The Bloch sphere is now the Riemann number sphere for the complex number . ${\displaystyle c}$

With the Poincaré ball

Closely related to the Bloch sphere is the Poincaré sphere , which is used to represent the polarization of transverse waves (e.g. light) and for the mean-field description of larger spin systems.

Pure and mixed states

The Pauli matrices are Hermitian and together with the identity matrix form a basis of the vector space of the complex matrices. The density matrix of a qubit can always be represented on a fixed basis as ${\displaystyle E}$ ${\displaystyle 2\times 2}$

${\displaystyle \rho ={\frac {1}{2}}\left(E+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}\right)={\frac {1}{2}}{\begin{pmatrix}1+z&x-\mathrm {i} y\\x+\mathrm {i} y&1-z\end{pmatrix}}}$

If im is understood as a vector , then it is always positive semidefinite , i.e. a permissible density matrix, if the is located in the closed unit sphere . The vector is called the Bloch vector . The state is pure if and only if the Bloch vector has length one, i.e. lies on the surface of the sphere. ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \rho }$ ${\displaystyle (x,y,z)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle (x,y,z)}$

Two pure states are orthogonal if their Bloch vectors are at exactly opposite points on the Bloch sphere. In the middle of the Bloch sphere lies the completely mixed state, whose Bloch vector is the zero vector .

If one forms a mixture of a part of the state with Bloch vector and a part of the state with Bloch vector , then the mixture is described by the Bloch vector . So one can write all states as convex combinations of pure states, and the Bloch sphere also shows that the state space is a convex set whose extreme points are the pure states. ${\displaystyle p}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle 1-p}$ ${\displaystyle {\vec {w}}}$ ${\displaystyle p\cdot {\vec {v}}+(1-p)\cdot {\vec {w}}}$

Geometric interpretation

If and are spin states for the spin quantum number  1/2, for example parallel position and antiparallel position of an electron in the magnetic field , then in the superposition state the expected value of the (vectorial) spin operator points in the direction indicated by the assigned point on the Bloch sphere. ${\displaystyle \left|\uparrow \right\rangle }$ ${\displaystyle \left|\downarrow \right\rangle }$

Web links

Commons : Bloch spheres  - collection of images, videos and audio files

• Eric W. Weisstein : Bloch Sphere . In: MathWorld (English).