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Physics LibreTexts

5.1: Composite Systems

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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=H1H2

An arbitrary pure state on H1+2 can be written as

|Ψ=jkcjk|ϕj|ψkjkcjk|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,ψϕ,ψ|)=Tr2(|ϕϕ||ψψ|)=|ϕϕ|Tr(|ψψ|)=|ϕϕ|,

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.


This page titled 5.1: Composite Systems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform.

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