5.1: Composite Systems
( \newcommand{\kernel}{\mathrm{null}\,}\)
Suppose we have two systems, described by Hilbert spaces H1 and H2, respectively. We can choose orthonormal bases for each system:
H1:{|ϕ1⟩,|ϕ2⟩,…,|ϕN⟩} and H2:{|ψ1⟩,|ψ2⟩,…,|ψM⟩}
The respective dimensions of H1 and H2 are N and M. We can construct N×M basis states for the composite system via |ϕj⟩ and |ψk⟩. This implies that the total Hilbert space of the composite system can be spanned by the tensor product
{|ϕj⟩⊗|ψk⟩}jk on H1+2=H1⊗H2
An arbitrary pure state on H1+2 can be written as
|Ψ⟩=∑jkcjk|ϕj⟩⊗|ψk⟩≡∑jkcjk|ϕj,ψk⟩
For example, the system of two qubits can be written on the basis {|0,0⟩,|0,1⟩,|1,0⟩,|1,1⟩}. If system 1 is in state |ϕ⟩ and system 2 is in state |ψ⟩, the partial trace over system 2 yields
Tr2(|ϕ,ψ⟩⟨ϕ,ψ|)=Tr2(|ϕ⟩⟨ϕ|⊗|ψ⟩⟨ψ|)=|ϕ⟩⟨ϕ|Tr(|ψ⟩⟨ψ|)=|ϕ⟩⟨ϕ|,
since the trace over any density operator is 1. We have now lost system 2 from our description! Therefore, taking the partial trace without inserting any other operators is the mathematical version of forgetting about it. This is a very useful feature: you often do not want to deal with every possible system you are interested in. For example, if system 1 is a qubit, and system two is a very large environment the partial trace allows you to “trace out the environment”.
However, tracing out the environment will not always leave you with a pure state as in Eq. (5.4). If the system has interacted with the environment, taking the partial trace generally leaves you with a mixed state. This is due to entanglement between the system and its environment.