Partial trace

Summary

The partial trace takes an operator on a tensor product $\mathcal{H}_A \otimes \mathcal{H}_B$ and traces out the factor $\mathcal{H}_B$, giving an operator on $\mathcal{H}_A$. It is the unique linear map that sends $X \otimes Y$ to $\tr(Y)\, X$; in infinite dimensions one also asks that it be continuous in the trace norm. In quantum mechanics it produces the reduced state of a subsystem from the state of a composite system. The reduced state gives exactly the statistics of all measurements made on the subsystem alone.