The partial trace is another generalization of the trace that is operator-valued.

The state of is the partial trace of over the basis of system :

The if part of the implication follows immediately from the fact that the partial trace is a positive map.

In linear algebra and functional analysis, the partial trace is a generalization of the trace.

The partial trace generalizes to operators on infinite dimensional Hilbert spaces.

The operator R is the partial trace of T.

The partial trace can be viewed as a quantum operation.

We see that this is satisfied if is as defined above via the partial trace.

The advantage of taking the partial trace is that this procedure is indifferent to the environmental basis chosen.

To see this, a partial trace is performed over the bath to give a description of the system alone: