Bloch sphere - AbsoluteAstronomy.com

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system

Two-level system

In quantum mechanics, a two-state system is a system which has two possible states. More formally, the Hilbert space of a two-state system has two degrees of freedom, so a complete basis spanning the space must consist of two independent states...

(qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

), named after the physicist Felix Bloch

Felix Bloch

Felix Bloch was a Swiss physicist, working mainly in the U.S.-Life and work:Bloch was born in Zürich, Switzerland to Jewish parents Gustav and Agnes Bloch. He was educated there and at the Eidgenössische Technische Hochschule, also in Zürich. Initially studying engineering he soon changed to physics...

.

Quantum mechanics is mathematically formulated in Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

or projective Hilbert space

Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P of a complex Hilbert space H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by...

.
The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space).
In a two-dimensional Hilbert space this is simply the complex projective line

Riemann sphere

In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

, which is a geometrical sphere.

The Bloch sphere is a unit 2-sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, with each pair of antipodal points corresponding to mutually orthogonal state vectors.
The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors

and

, respectively,
which in turn might correspond e.g. to the spin

Spin (physics)

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

-up and spin

Spin (physics)

-down states of an electron.
This choice is arbitrary, however.
The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states.
The Bloch sphere may be generalized to an n-level quantum system but then the visualization is less useful.

In optics, the Bloch sphere is also known as the Poincaré sphere

Poincaré sphere

Poincaré sphere, often spelled Poincare sphere may refer to:* Poincaré sphere, in optics, is a graphical tool for visualizing different types of polarized light...

and specifically represents different types of polarizations. See the Jones Vector for a detailed list of the 6 common polarization types and how they map on to the surface of this sphere.

The natural metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

on the Bloch sphere is the Fubini-Study metric

Fubini-Study metric

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....

Definition

Given an orthonormal basis, any pure state

of a two-level quantum system can be written as a complex superposition of the basis vectors

and

.
Since global phase factors do not have any physical meaning, we can take the coefficient of

to be real and non-negative.
Thus

has the normalized representation

with

and

.
Except in the case where

is one of the ket vectors

the representation is unique.
The parameters

and

, re-interpreted as spherical coordinates, specify a point

on the unit sphere in

.
For mixed states, any two-dimensional density operator

can be expanded using the identity and the Hermitian, traceless

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

,
where

is called the Bloch vector of the system.
The eigenvalues of

are given by

.
As density operators must be positive-semidefinite, we have

.

For pure states we must have

,
in accordance with the previous result.
Hence the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

A generalization for pure states

Consider an n-level quantum mechanical system. This system is described by an n-dimensional Hilbert space

Hilbert space

H_n. The pure state space is by definition the set of 1-dimensional rays of H_n.

Theorem. Let U(n) be the Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

of unitary matrices of size n. Then the pure state space of H_n can be identified with the compact coset space

To prove this fact, note that there is a natural

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

of U(n) on the set of states of H_n. This action is continuous and transitive on the pure states. For any state

, the isotropy group of

, (defined as the set of elements

of U(n) such that

) is isomorphic to the product group

In linear algebra terms, this can be justified as follows. Any

of U(n) that leaves

invariant must have

as an eigenvector. 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

, which is isomorphic to U(n - 1). From this the assertion of the theorem follows from basic facts about transitive group actions of compact groups.

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

Now the (real) dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

of U(n) is n². This is easy to see since the exponential map

is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of self-adjoint complex matrices has real dimension n².

Corollary. The real dimension of the pure state space of H_n is
2n − 2.

In fact,

Let us apply this to consider the real dimension of an m qubit quantum register. The corresponding Hilbert space has dimension 2^m.

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

The geometry of density operators

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

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

. However, while the Bloch sphere parametrizes not only pure states but mixed states for 2-level systems, for states of higher dimensions there is difficulty in extending this to mixed states. The topological description is complicated by the fact that the unitary group does not act transitively on density operators. The orbits moreover are extremely diverse as follows from the following observation:

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

We note here that, in the literature, one can find non-Bloch type parametrizations of (mixed) states that do generalize to dimensions higher than 2.

Definition

A generalization for pure states

The geometry of density operators

See also