# Can $2^n$ bits be sent with $n$ instances of quantum teleportation?

So, right now these are two pieces of information I've been told are correct:

1. Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits
2. $$n$$ qubits can store $$2^n$$ classical bits of information

If both of these are true, using, for example, 16 classical bits and 8 preshared qubits, could Alice send Bob 256 classical bits of information via teleportation?

My understanding is that both teleportation (which I understand fairly well) and storing exponentially many bits with multiple qubits (which I don't know much about) involve entanglement. Would this mean that these two properties are sort of mutually exclusive, and you can only have one at once (preventing more than 2 classical bits/qubit from being sent through teleportation)? Or is it indeed possible to send exponentially more data through a classical communication channel by presharing qubits for teleportation?

1. Quantum teleportation can send a single qubit from Alice to Bob, with two classical bits

Correct, on the condition that Alice and Bob also have an entangled qubit pair shared between them. This entanglement resource is typically called 1 ebit.

1. $$n$$ qubits can store $$2^n$$ classical bits of information.

This part isn't quite right. If you consider $$n$$ qubits, then we can have a superposition of up to $$2^n$$ (computational basis) states. If we say each of those components can have either $$0$$ or $$1$$ coefficients, ignoring normalization, then we can encode a bit of information on each of those $$2^n$$ states. In principle, you can have $$2^n$$ complex coefficients with arbitrarily high precision, which could let you encode arbitrarily large quantities of classical information.

Now to answer your question. Unfortunately, no, you can't send exponentially more classical data with quantum teleporation. While you can encode arbitrary amounts of information into a single qubit, there's a thing called Holevo's theorem which says that with $$n$$ qubits, you can at best extract $$n$$ classical bits of information due to the invasive nature of measurement in quantum mechanics. Since you only get a single shot at gaining information from the quantum state before you irreversibly change it, the amount of classical information you can extract is quite small compared to the amount you could theoretically encode.

However, all is not totally lost. A procedure related to quantum teleportation is called (quantum) superdense coding which gives a procedure for sending 1 qubit and using an entangled pair (1 ebit) to communicate 2 classical bits of information, so you can use quantum resources to more efficiently communicate quantum information, it's just twice as efficient instead of exponentially so.

• Concerning the superdense coding, you can send two classical bits with one qubit but you also need one qubit of Bell pair. This means that you still need two qubits to send two classical bits. As you pointed out, information content of qubit is actually one bit. Commented Jan 13, 2023 at 7:26