Quantum teleportation requires sending classical information. So you can't do it faster than the speed of light.
Quantum Teleportation (simplified)
Alice and Bob have a pair of qubits, $A$ and $B$. $A$ and $B$ are entangled so that:
- If you measure $A$ along the X axis and also measure $B$ along the X axis, the answer will be the same.
- If you measure $A$ along the Z axis and also measure $B$ along the Z axis, the answer will be the same.
In other words, their X-parity is SAME and their Z-parity is also SAME.
Alice wants to give Bob a qubit $Q$. To do this she will compare $Q$ to $A$ along the X and Z axes, then tell Bob how the comparison went. Because $A$ and $B$ were SAME along those axes, she ends up telling Bob how to change $B$ to get $Q$.
Alice starts by measuring the X-parity of $Q$ and $A$. She finds out if they are SAME-X or DIFFERENT-X. Because $A$ and $B$ agreed along the X axis, she's actually finding out if $B$ and $Q$ agree along the X axis. If they're different-X, she yells out "HEY BOB! THE X-PARITY IS WRONG. FLIP $B$ OVER TO FIX THAT!". If $A$ and $Q$ agree along the X axis, she instead yells "X AXIS OKAY!".
Then Alice does the same thing with the Z-parity. She compares $Q$ and $A$ along the Z axis, which is actually telling her whether $B$ agrees or disagrees with $Q$ along Z. If they differ, she yells out "BOB! THE Z-PARITY IS WRONG. FLIP $B$ OVER THE OTHER WAY TO FIX THAT!". Otherwise she yells out "Z AXIS OKAY!".
After Bob hears both of Alice's yells, and has fixed any wrong parities, $B$'s state has been overwritten with $Q$'s original state. It's literally the quantum equivalent of a one-time pad cipher. (Well, except that $Q$ and $A$ get totally trashed by the measurements that Alice did.)
Notice that the process required yelling. Bob had to be told which corrections to apply. That's why quantum teleportation can't be done faster than light speed. Yells don't move faster than light.