Partial Trace and Invariant Integration
In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W).
Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then
commutes with all operators of the form and hence is uniquely of the form . The operator R is the partial trace of T.
Read more about this topic: Partial Trace
Famous quotes containing the words partial, trace and/or integration:
“The one-eyed man will be King in the country of the blind only if he arrives there in full possession of his partial facultiesthat is, providing he is perfectly aware of the precise nature of sight and does not confuse it with second sight ... nor with madness.”
—Angela Carter (19401992)
“The land of shadows wilt thou trace
And look nor know each others face
The present mixed with reasons gone
And past and present all as one
Say maiden can thy life be led
To join the living with the dead
Then trace thy footsteps on with me
Were wed to one eternity”
—John Clare (17931864)
“The more specific idea of evolution now reached isa change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.”
—Herbert Spencer (18201903)