The expectation value of M after the subsystem A is prepared in and that of when the composite system is prepared in ρ should be the same, i.e. the following equality should hold: We see that this is satisfied if is as defined above via the partial trace. Furthermore, such operation is unique. Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map is completely positive and trace-preserving. The density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1. It has a spectral decomposition: Its easy to see that the partial trace also satisfies these conditions. For example, for any pure state in , we have Note that the term represents the probability of finding the state when the composite system is in the state . This proves the positive semi-definiteness of . The partial trace map as given above induces a dual map between the C*-algebras of bounded operators on and given by maps observables to observables and is the Heisenberg picture representation of . Comparison with classical case Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz-Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation. Linear algebra Functional analysis