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.
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