Bloch sphere

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

[1] Mathematically each quantum mechanical system is associated with a separable complex Hilbert space

) represent the same state, the level of the quantum system corresponds to the dimension of the Hilbert space and pure states can be represented as equivalence classes, or, rays in a projective Hilbert space

The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors.

The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors

This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one: Given this constraint, we can write

Any two-dimensional density operator ρ can be expanded using the identity I and the Hermitian, traceless Pauli matrices

Specifically, as a basic feature of the Pauli vector, the eigenvalues of ρ are

Let U(n) be the Lie group of unitary matrices of size n. Then the pure state space of Hn can be identified with the compact coset space To prove this fact, note that there is a natural group action of U(n) on the set of states of Hn.

) is isomorphic to the product group In linear algebra terms, this can be justified as follows.

Since the corresponding eigenvalue must be a complex number of modulus 1, this gives the U(1) factor of the isotropy group.

The other part of the isotropy group is parametrized by the unitary matrices on the orthogonal complement of

The important fact to note above is that the unitary group acts transitively on pure states.

This is easy to see since the exponential map is a local homeomorphism from the space of self-adjoint complex matrices to U(n).

In fact, Let us apply this to consider the real dimension of an m qubit quantum register.

The real dimension of the pure state space of an m-qubit quantum register is 2m+1 − 2.

form a basis and have diametrically opposite representations on the Bloch sphere, then let be their ratio.

The vector with tail at the origin and tip at P is the direction in 3-D space corresponding to the spinor

The coordinates of P are Formulations of quantum mechanics in terms of pure states are adequate for isolated systems; in general quantum mechanical systems need to be described in terms of density operators.

The density operator describing the mixed-state of a 2-level quantum system (qubit) corresponds to a point inside the Bloch sphere with the following coordinates: where

The topological description is complicated by the fact that the unitary group does not act transitively on density operators.

Suppose A is a density operator on an n level quantum mechanical system whose distinct eigenvalues are μ1, ..., μk with multiplicities n1, ..., nk.

Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to In particular the orbit of A is isomorphic to It is possible to generalize the construction of the Bloch ball to dimensions larger than 2, but the geometry of such a "Bloch body" is more complicated than that of a ball.

The most concise explanation for why this is the case is that the Lie algebra for the group of unitary and hermitian matrices

is a real unit vector in three dimensions, the rotation of the Bloch sphere about this axis is given by: An interesting thing to note is that this expression is identical under relabelling to the extended Euler formula for quaternions.

Ballentine[12] presents an intuitive derivation for the infinitesimal unitary transformation.

This is important for understanding why the rotations of Bloch spheres are exponentials of linear combinations of Pauli matrices.

We define the infinitesimal unitary as the Taylor expansion truncated at second order.

are unitary Hermitian matrices and have eigenvectors corresponding to the Bloch basis,

Bloch sphere
Bloch sphere centered at the origin of . A pair of points on it, and have been chosen as a basis. Mathematically they are orthogonal even though graphically the angle between them is π. In those points have coordinates (0,0,1) and (0,0,−1). An arbitrary spinor on the Bloch sphere is representable as a unique linear combination of the two basis spinors, with coefficients being a pair of complex numbers; call them α and β . Let their ratio be , which is also a complex number . Consider the plane z = 0, the equatorial plane of the sphere, as it were, to be a complex plane and that the point u is plotted on it as . Project point u stereographically onto the Bloch sphere away from the South Pole — as it were — (0,0,−1). The projection is onto a point marked on the sphere as .