Teleportation as an identity channel on a reference (${\mathcal{T}}_A\otimes {\mathrm{id}}_R={\mathrm{id}}_{A\to B}\otimes {\mathrm{id}}_R$)

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 ${\sum }_j|j⟩|r_j⟩$. 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.