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 projectivization, 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 (the U(1) action) 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. For example, a density matrix of a pure quantum state is represented by a projective ray; it is impossible to recover the phase from a density matrix. The same is true for its generalisation, pure states in a representation[clarification needed] of a C*-algebra.

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 \mathrm{U}(n) or orthogonal group \mathrm{O}(n), 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 projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line \mathbb{C}P^{1}. This is known as the Bloch sphere. See Hopf fibration for details of the projectivization construction in this case.

Complex projective Hilbert space may be given a natural metric, the Fubini–Study metric, derived from the Hilbert space's norm.

Product[edit]

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.

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