= Basis-copying isometry
{title2=$C|k\rangle=|k\rangle|k\rangle$}
Relative to a fixed <orthonormal basis>, the displayed linear map is an <isometry> because its <basis> images are orthonormal. Its adjoint obeys $C^*|k\rangle|\ell\rangle=\delta_{k\ell}|k\rangle$. If <vectors> are described by coefficient <functions>, this adjoint maps $|f\rangle|g\rangle$ to $|fg\rangle$, where the product is pointwise. Although $C$ copies every <basis> <quantum state>, it does not copy arbitrary <superpositions>. For example it sends $(|0\rangle+|1\rangle)/\sqrt2$ to $(|00\rangle+|11\rangle)/\sqrt2$, whereas two copies would also contain the cross terms $|01\rangle$ and $|10\rangle$.
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