= Classical communication lower bound for exact qubit teleportation
{title2=$c\geq2\ \text{classical bits}$}
Exact <quantum teleportation> that preserves an arbitrary reference requires at least two classical <bits> from sender to receiver, even with preshared <quantum entanglement>. For the essential reduction, encode two classical <bits> in the four <Bell states> by <Pauli gates> on the sender's half of a <Bell pair>. <Teleportation as an identity channel on a reference> transfers this half while preserving the encoding. The receiver then has both halves and recovers both <bits> by a <Bell-basis measurement>, as in <superdense coding>.
The classical <bit> bound itself has a short <no-communication theorem> proof. For $N$ possible messages and $K$ possible classical transmissions, let $\sigma_{m,c}$ be the receiver's subnormalized conditional <density matrix> before decoding. The <no-communication theorem> gives $\sum_c\sigma_{m,c}=\tau$, independent of $m$, so $0\leq\sigma_{m,c}\leq\tau$. For each transmission $c$, let $\{E_{m|c}\}_m$ be the decoding <POVM>. Averaging over equiprobable messages,
$$
p_{\mathrm{success}}=\frac1N\sum_{m,c}\operatorname{Tr}(E_{m|c}\sigma_{m,c})
\leq\frac1N\sum_c\operatorname{Tr}\left(\sum_m E_{m|c}\tau\right)=\frac KN.
$$
Perfect decoding of four messages requires $K\geq4$, hence at least two classical <bits>. This applies to the preshared resources of the proposed <quantum teleportation> protocol as well as the <Bell pair> used for the encoding.
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