Partial Trace As A Quantum Operation
The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product of Hilbert spaces. A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product The partial trace of ρ with respect to the system B, denoted by, is called the reduced state of ρ on system A. In symbols,
To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is . However one chooses to define a reduced state, there should be consistency of measurement statistics. 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 it is the unique such operation.
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 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 .
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