= Teleportation as an identity channel on a reference
{title2=$\mathcal T_A\otimes\operatorname{id}_R=\operatorname{id}_{A\to B}\otimes\operatorname{id}_R$}
Ideal corrected <quantum teleportation> transfers an input system to the receiver while preserving its entire joint <density operator> with an arbitrary reference. Expand a joint <pure state> as $\sum_j|j\rangle|r_j\rangle$. Every teleportation branch acts with the same known correction on each system basis vector and does nothing to the reference vectors; undoing that correction restores the complete sum. Linearity extends the result to mixed states. The sender's measured carrier no longer retains the original reference correlations.
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