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If Alice and Bob share an entangled state like the EPR state, by sending only two classical bits, Alice can transfer one qubit to Bob by means of quantum teleportation (see also Section 1.3.7 of http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf).

Are two classical bits optimal for this protocol?

If Alice can do this by sending only one classical bit, does this violate Holevo's bound or any other rules?

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Alice sent 1 bit to Bob.

But that bit contained 1 qubit (via too good teleportation).

But that qubit contained 2 bits (via superdense coding).

But those 2 bits contained 2 qubits (too good teleportation).

But those 2 qubits contained 4 bits.

But those 4 bits contained 4 qubits.

But those 4 qubits contained 8 bits.

But...

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    $\begingroup$ I think this does show that if you can teleport 1 qubit using 1 bit, then you can teleport n qubits using 1 bit for any n. But how do you prove that you can't do that? $\endgroup$ Commented Jul 4, 2022 at 5:27
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    $\begingroup$ @benrg it violates no signalling. Have the receiver just guess the seed bit instead of waiting to receive it. Because the message can be long it's possible to tell if the guess was right and transmission succeeded. Have multiple independent seeds to boost chance of success arbitrarily close to 100%. $\endgroup$ Commented Jul 4, 2022 at 17:38

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