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The orthogonal projection onto the line spanned by a normalized vector.
For a normalized vector in a complex Hilbert space, the rank-one projector onto its span is the operator
It is an orthogonal projection, is positive semidefinite, is idempotent, and has trace one. Multiplying by a complex phase leaves unchanged, so the projector represents the associated one-dimensional complex line rather than a chosen vector on it.
A normalized state vector in a complex Hilbert space is a vector with . It lies on the unit sphere.
In quantum mechanics, and represent the same pure state, because all transition probabilities depend only on squared inner-product magnitudes. The phase-independent state is equivalently represented by the rank-one projector .
Let be a closed linear subspace of a Hilbert space . The decomposition defines the orthogonal projection by .
It is the unique bounded operator satisfying , , and . Conversely, the range of every self-adjoint idempotent bounded operator is a closed subspace.
An operator on a complex Hilbert space is positive semidefinite, written , if
for every vector . Positivity implies that is self-adjoint. In finite dimension it is equivalent to all eigenvalues of being real and nonnegative, and also equivalent to a factorization .
Positive operators are the effects in a POVM and, after trace normalization, the density operators of quantum theory.