# 5.1: Composite Systems


Suppose we have two systems, described by Hilbert spaces $$\mathscr{H}_{1}$$ and $$\mathscr{H}_{2}$$, respectively. We can choose orthonormal bases for each system:

$\mathscr{H}_{1}:\left\{\left|\phi_{1}\right\rangle,\left|\phi_{2}\right\rangle, \ldots,\left|\phi_{N}\right\rangle\right\} \quad \text { and } \mathscr{H}_{2}:\left\{\left|\psi_{1}\right\rangle,\left|\psi_{2}\right\rangle, \ldots,\left|\psi_{M}\right\rangle\right\}\tag{5.1}$

The respective dimensions of $$\mathscr{H}_{1}$$ and $$\mathscr{H}_{2}$$ are $$N$$ and $$M$$. We can construct $$N \times M$$ basis states for the composite system via $$\left|\phi_{j}\right\rangle$$ and $$\left|\psi_{k}\right\rangle$$. This implies that the total Hilbert space of the composite system can be spanned by the tensor product

$\left\{\left|\phi_{j}\right\rangle \otimes\left|\psi_{k}\right\rangle\right\}_{j k} \quad \text { on } \quad \mathscr{H}_{1+2}=\mathscr{H}_{1} \otimes \mathscr{H}_{2}\tag{5.2}$

An arbitrary pure state on $$\mathscr{H}_{1+2}$$ can be written as

$|\Psi\rangle=\sum_{j k} c_{j k}\left|\phi_{j}\right\rangle \otimes\left|\psi_{k}\right\rangle \equiv \sum_{j k} c_{j k}\left|\phi_{j}, \psi_{k}\right\rangle\tag{5.3}$

For example, the system of two qubits can be written on the basis $$\{|0,0\rangle,|0,1\rangle,|1,0\rangle,|1,1\rangle\}$$. If system 1 is in state $$|\phi\rangle$$ and system 2 is in state $$|\psi\rangle$$, the partial trace over system 2 yields

$\operatorname{Tr}_{2}(|\phi, \psi\rangle\langle\phi, \psi|)=\operatorname{Tr}_{2}(|\phi\rangle\langle\phi|\otimes| \psi\rangle\langle\psi|)=|\phi\rangle\langle\phi|\operatorname{Tr}(|\psi\rangle\langle\psi|)=| \phi\rangle\langle\phi|,\tag{5.4}$

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.

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