In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space
is the set of equivalence classes
given by This is the usual construction of projectivization, applied to a complex Hilbert space.
[1] In quantum mechanics, the equivalence classes
The physical significance of the projective Hilbert space is that in quantum theory, the wave functions
The Born rule demands that if the system is physical and measurable, its wave function has unit norm,
, in which case it is called a normalized wave function.
The unit norm constraint does not completely determine
with absolute value 1 (the circle group
quantum state (algebraic definition), given a C*-algebra of observables and a representation on
No measurement can recover the phase of a ray; it is not observable.
is a gauge group of the first kind.
is an irreducible representation of the algebra of observables then the rays induce pure states.
Convex linear combinations of rays naturally give rise to density matrix which (still in case of an irreducible representation) correspond to mixed states.
, the Hilbert space reduces to a finite-dimensional inner product space and the set of projective rays may be treated as a complex projective space; it is a homogeneous space for a unitary group
That is, which carries a Kähler metric, called the Fubini–Study metric, derived from the Hilbert space's norm.
[2][3] As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line
See Hopf fibration for details of the projectivization construction in this case.
The Cartesian product of projective Hilbert spaces is not a projective space.
The Segre mapping is an embedding of the Cartesian product of two projective spaces into the projective space associated to the tensor product of the two Hilbert spaces, given by
In quantum theory, it describes how to make states of the composite system from states of its constituents.
It is only an embedding, not a surjection; most of the tensor product space does not lie in its range and represents entangled states.