# Projective Hilbert space

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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 $v \ne 0$, for the relation $\sim$ given by

$v \sim w$ when $v = \lambda w$ for some non-zero complex number $\lambda$.

The equivalence classes for the relation $\sim$ are also called rays or projective rays.

This is the usual construction of a projective space, applied to a complex Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions $\psi$ and $\lambda \psi$ represent the same physical state, for any $\lambda \ne 0$. It is conventional to choose a $\psi$ from the ray so that it has unit norm, $\langle\psi|\psi\rangle = 1$, in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine $\psi$ within the ray, since $\psi$ could be multiplied by any $\lambda$ with absolute value 1 and retain its normalization. Such a $\lambda$ can be written as $\lambda = e^{i\phi}$ with $\phi$ called the global phase. This freedom means that projective representations of quantum states are important in quantum theory.

The same construction can be applied also to real Hilbert spaces.

In the case $H$ is finite-dimensional, that is, $H=H_n$, 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

$P(H_{n})=\mathbb{C}P^{n-1}$

so that, for example, the two-$\mathbb{R}$-dimensional projective Hilbert space (the space describing one qubit) is the complex projective line $\mathbb{C}P^{1}$. 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.