Partial Trace For Operators On Hilbert Spaces
The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and let
be an orthonormal basis for W. Now there is an isometric isomorphism
Under this decomposition, any operator can be regarded as an infinite matrix of operators on V
where .
First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum
converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined.
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