In mathematics and the foundations of quantum mechanics, the projective Hilbert space
Contents
The equivalence classes for the relation
This is the usual construction of projectivization, applied to a complex Hilbert space.
Overview
The physical significance of the projective Hilbert space is that in quantum theory, the wave functions
Rays that differ by such a
If
The same construction can be applied also to real Hilbert spaces.
In the case
so that, for example, the projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line
Complex projective Hilbert space may be given a natural metric, the Fubini–Study metric, derived from the Hilbert space's norm.
Product
The Cartesian products of projective Hilbert spaces is not a projective space. Their categorical product is equivalent to the tensor product of respective (vector) Hilbert spaces and, in quantum physics, describes states of a composite quantum system. Segre mapping is an embedding of the Cartesian product of two projective spaces into their categorical product. 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 categorical product space does not lie in its range and represents entangled states.