What is the difference between having a single-qubit state and knowing a result of a measurement you want to perform on it?

Question

In the quantum teleportation protocol Alice can send Bob an unknown quantum state $|\psi\rangle$. If the only thing Bob does with $|\psi\rangle$ is to measure it in some basis, I guess it would be simpler for Alice to just do the measurement herself and send the result to Bob. That would not require a shared entangled pair but Alice would need to know Bob's favorite basis. Is there something Bob can do with an actual state $|\psi\rangle$ which can not be done with information that Alice sends to Bob?

More generally, due to Holevo bound only a single bit of information can be encoded in a qubit. What is then the difference between teleporting an actual quantum state $|\psi\rangle$ and sending the result of some measurement of this state?

Answer 1

There are lots of different situations one can talk about from a cryptography perspective. But here's one that has a huge practical relevance:

There are physical realisations of quantum computers which, individually, are limited in the number of qubits they can use. For example, ion traps. For the sake of argument, assume you can hold 10 qubits in each ion trap. How am I to do a quantum computation between more than 10 qubits? Take two traps. Every time I need to perform a two-qubit gate between qubits in different traps, teleport one qubit from one trap to the other. Then keep going with your protocol. That way, you can accomplish protocols that require entangled states of large numbers of qubits that Alice, with a single ion trap, would simply be unable to create or replicate.