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:
“And meanwhile we have gone on living,
Living and partly living,
Picking together the pieces,
Gathering faggots at nightfall,
Building a partial shelter,
For sleeping and eating and drinking and laughter.”
—T.S. (Thomas Stearns)
“Muhammad is the Messenger of God,
and those who are with him are hard
against the unbelievers, merciful
one to another. Thou seest them
bowing, prostrating, seeking bounty
from God and good pleasure. Their
mark is on their faces, the trace of
prostration....
God has promised
those of them who believe and do deeds
of righteousness forgiveness and
a mighty wage.”
—Qur’An. Victory 48:35, ed. Arthur J. Arberry (1955)
“Look back, to slavery, to suffrage, to integration and one thing is clear. Fashions in bigotry come and go. The right thing lasts.”
—Anna Quindlen (b. 1952)