![Bloch_sphere](https://upload.wikimedia.org/wikipedia/commons/6/6b/Bloch_sphere.svg)
In quantum
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
and
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, the Bloch sphere is a geometrical representation of the
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
space of a
two-level quantum mechanical system
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a sys ...
(
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
), named after the physicist
Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of ne ...
.
Quantum mechanics is mathematically formulated in
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
or
projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by
:w \sim v if and only if v = \ ...
. The pure states of a quantum system correspond to the one-dimensional subspaces of the corresponding Hilbert space (and the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, the space of all such states is the
complex projective line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
This is the Bloch sphere, which can be mapped to the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
.
The Bloch sphere is a unit
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
, with
antipodal points
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
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
and
, respectively, which in turn might correspond e.g. to the
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
-up and
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s 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.
For historical reasons, in optics the Bloch sphere is also known as the
Poincaré sphere Poincaré sphere may refer to:
* Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light
** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system
* Poincaré homology sph ...
and specifically represents different types of
polarizations. Six common polarization types exist and are called
Jones vectors. Indeed
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
was the first to suggest the use of this kind of geometrical representation at the end of 19th century,
[
] as a three-dimensional representation of
Stokes parameters
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoher ...
.
The natural
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
on the Bloch sphere is the
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Edua ...
. The mapping from the unit 3-sphere in the two-dimensional state space
to the Bloch sphere is the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
, with each
ray of
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s mapping to one point on the Bloch sphere.
Definition
Given an orthonormal basis, any
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
of a two-level quantum system can be written as a superposition of the basis vectors
and
, where the coefficient of (or contribution from) each of the two basis vectors is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. This means that the state is described by four real numbers. However only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directly
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
), so that there is redundancy in this description. We can take the coefficient of
to be real and non-negative. 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:
:
, or equivalently
.
Given this constraint, we can write
using the following representation:
:
, where
and
.
The representation is always unique, because, even though the value of
is not unique when
is one of the states (see
Bra-ket notation)
or
, the point represented by
and
is unique.
The parameters
and
, re-interpreted in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
as respectively the
colatitude
In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non- ...
with respect to the ''z''-axis and the
longitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...
with respect to the ''x''-axis, specify a point
:
on the unit sphere in
.
For
mixed states, one considers the
density operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
. Any two-dimensional density operator can be expanded using the identity and the
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
,
traceless
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace o ...
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
,
:
,
where
is called the Bloch vector.
It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of the
Pauli vector, the eigenvalues of are
. Density operators must be positive-semidefinite, so it follows that
.
For pure states, one then has
:
in comportance with the above.
As a consequence, 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.
''u'', ''v'', ''w'' representation
The Bloch vector
can be represented in the following basis, with reference to the density operator
:
:
:
:
where
:
This basis is often used in
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
theory, where
is known as the
population inversion
In science, specifically statistical mechanics, a population inversion occurs while a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energ ...
. In this basis, the numbers
are the expectations of the three
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, allowing one to identify the three coordinates with x y and z axes.
Pure states
Consider an ''n''-level quantum mechanical system. This system is described by an ''n''-dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. 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
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of U(''n'') is ''n''
2. 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 ''n''
2.
Corollary. The real dimension of the pure state space of ''H''
''n'' is 2''n'' − 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
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
quantum register In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.
Definition ...
is 2
''m''+1 − 2.
Plotting pure two-spinor states through stereographic projection
![Riemann Spin2States](https://upload.wikimedia.org/wikipedia/commons/6/6e/Riemann_Spin2States.jpg)
Given a pure state
:
where
and
are complex numbers which are normalized so that
:
and such that
and
,
i.e., such that
and
form a basis and have diametrically opposite representations on the Bloch sphere, then let
:
be their ratio.
If the Bloch sphere is thought of as being embedded in
with its center at the origin and with radius one, then the plane ''z'' = 0 (which intersects the Bloch sphere at a great circle; the sphere's equator, as it were) can be thought of as an
Argand diagram
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. Plot point ''u'' in this plane — so that in
it has coordinates
.
Draw a straight line through ''u'' and through the point on the sphere that represents
. (Let (0,0,1) represent
and (0,0,−1) represent
.) This line intersects the sphere at another point besides
. (The only exception is when
, i.e., when
and
.) Call this point ''P''. Point ''u'' on the plane ''z'' = 0 is the
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of point ''P'' on the Bloch sphere. 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
:
:
:
.
Mathematically the Bloch sphere for a two-spinor state can be mapped to a
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
or a complex 2-dimensional
projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by
:w \sim v if and only if v = \ ...
, denotable as
. The complex 2-dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(of which
is a projection) is a representation space of
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
.
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. The Bloch sphere parametrizes not only pure states but mixed states for 2-level systems. 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
is the probability of the individual states within the ensemble and
are the coordinates of the individual states (on the ''surface'' of Bloch sphere). The set of all points on and inside the Bloch sphere is known as the ''Bloch ball.''
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 μ
1, ..., μ
''k'' with multiplicities ''n''
1, ..., ''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
:
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.
Rotations
A useful advantage of the Bloch sphere representation is that the evolution of the qubit state is describable by rotations of the Bloch sphere. The most concise explanation for why this is the case is that the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
for the group of unitary and hermitian matrices
is isomorphic to the Lie algebra of the group of three dimensional rotations
.
Rotation operators about the Bloch basis
The rotations of the Bloch sphere about the Cartesian axes in the Bloch basis are given by
:
Rotations about a general axis
If
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
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
.
:
Derivation of the Bloch rotation generator
Ballentine
[Ballentine 2014, "Quantum Mechanics - A Modern Development", Chapter 3] 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
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. Hence a brief treatment on this is given here. A more complete description in a quantum mechanical context can be found
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
.
Consider a family of unitary operators
representing a rotation about some axis. Since the rotation has one degree of freedom, the operator acts on a field of scalars
such that:
:
:
Where
We define the infinitesimal unitary as the Taylor expansion truncated at second order.
:
By the unitary condition:
:
Hence
:
For this equality to hold true (assuming
is negligible) we require
:
.
This results in a solution of the form:
:
Where
is any Hermitian transformation, and is called the generator of the unitary family.
Hence:
:
Since the Pauli matrices
are unitary Hermitian matrices and have eigenvectors corresponding to the Bloch basis,
, we can naturally see how a rotation of the Bloch sphere about an arbitrary axis
is described by
:
With the rotation generator given by
See also
*
Atomic electron transition
Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a few na ...
*
Gyrovector space
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
*
Poincaré sphere (optics)
Polarization ( also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of th ...
*
Versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s
* Specific implementations of the Bloch sphere are enumerated under the
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
article.
References
{{DEFAULTSORT:Bloch Sphere
Quantum mechanics
Projective geometry