Projective Hilbert space
Encyclopedia
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 vectors v in H, with v ≠ 0, for the relation given by
The equivalence classes for ~ are also called projective rays.
This is the usual construction of projective space
, applied to a Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory
, the wave functions ψ and λψ represent the same physical state, for any λ ≠ 0. There is not a unique normalized wavefunction in a given ray, since we can multiply by λ with absolute value
1. This freedom means that projective representation
s enter quantum theory.
The same construction can be applied also to real Hilbert spaces.
In the case H is finite-dimensional, that is,
, the set of projective rays may be treated just as any other projective space; it is a homogeneous space
for a unitary group
or orthogonal group
, in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes
so that, for example, the two-
-dimensional projective Hilbert space (the space describing one qubit
) is the complex projective line
. This is known as the Bloch sphere
.
Complex projective Hilbert space may be given a natural metric, the Fubini-Study metric
. The product of two projective Hilbert spaces is given by the Segre mapping.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and the foundations of quantum mechanics
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 projective Hilbert space P(H) of a complex 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...
H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by
- v ~ w when v = λw for some non-zero complex number λ.
The equivalence classes for ~ are also called projective rays.
This is the usual construction of projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, applied to a Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory
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 wave functions ψ and λψ represent the same physical state, for any λ ≠ 0. There is not a unique normalized wavefunction in a given ray, since we can multiply by λ with absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1. This freedom means that projective representation
Projective representation
In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
s enter quantum theory.
The same construction can be applied also to real Hilbert spaces.
In the case H is finite-dimensional, that is,
, the set of projective rays may be treated just as any other projective space; it is a homogeneous spaceHomogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
for a unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
or orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes
so that, for example, the two-
-dimensional projective Hilbert space (the space describing one qubitQubit
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....
) is the complex projective line
. This is known as the Bloch sphereBloch sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....
.
Complex projective Hilbert space may be given a natural metric, 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....
. The product of two projective Hilbert spaces is given by the Segre mapping.