Here we discuss the meaning of a partial trace and show how to perform this operation. We provide explicit examples and a python implementation that is intended to be instructive, but is not (at all) optimized for efficiency.

We first consider the canonical example of two coupled two-level systems, sites a and b.

The basis states of this system are , , , , for and the density matrix is:

The reduced density matrix for one of these sites can be computed by the partial trace. For site a the reduced density matrix is:

$

\rho_{a} = \begin{bmatrix}

\rho_{00, 00} + \rho_{01,01} & \rho{10,10} + \rho{11

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