Why can’t I use quantum teleportation to transmit data FTL 1/4 of the time?

Question

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ Assume there is an entangled pair $(q_1, q_2)$ owned by Alice and Bob, respectively, and some qubit $q_0$ in state $\ket{\psi}$ that Alice wants to teleport. Let Alice perform all the necessary operations to teleport $q_0$, namely, $\text{CNOT}(q_0, q_1)$, $H(q_0)$ (I'm not sure if this is sufficient, or if Alice has to measure her two qubits to collapse their superposition and complete the teleportation, but this isn't relevant to the question. Assume she does measure them if it is necessary). Now the state of $q_2$ should equal $\ket{\psi}$, or be closely related to it through one of the bell states. Assume that Alice and Bob coordinated on what time Alice would complete the teleportation, so that Bob is aware the teleportation has occurred.

What is keeping Bob from assuming that $q_2$ is in some particular bell state, and measuring $q_2$? It would seem that would allow faster than light communication 25% of the time. In fact, Bob could even produce imperfect clones of $q_2$, and my understanding is that he could somehow account for the imperfection of these clones. These imperfect clones would then allow him to extract more information from the single teleportation, and, assuming he knows the sort of thing he’s looking for, could provide an even higher chance that he receives meaningful information out of this communication - even if no classical information is sent from Alice.

What prevents this from working?

Edit

According to Holevo's Theorem, one can only retrieve up to $n$ classical bits given $n$ qubits. However, as I understand it, this does not prevent one from storing $n$ classical bits into a single qubit, imperfectly cloning it $n - 1$ times, and thus retrieving $n$ classical bits out. Given this, we can send a single qubit through teleportation and the receiver gets an accurate message approximately 25% of the time (less than this of course, due to the error introduced by the imperfect cloning).

In regards to the user not knowing whether the information is correct and thus it being no use, consider the classical case of $n$ one-way radios. Only 25% of the radios send the correct message, on channel $x$, the rest send random noise. Say the message is a recorded English sentence of some substantial length (say 20 words). An observer of this message, flipping through the channels, would be able to tell with high certainty which of these radios is transmitting the correct message. How does this differ in the quantum case, such that we cannot apply the same logic?

Answer 1

Suppose Alice wants to send Bob a 1000 bit message. To receive the message, Bob flips 1000 coins and writes down the results as 0s and 1s. About 50% of the random bits in the message that Bob generated are the same as in the intended message. Clearly this is an even better faster-than-light communication method than teleportation, because it succeeds 50% of the time instead of 25% of the time! /sarc

When you are transmitting information, getting a bit right 50% of the time (or 25% of the time in the case of superdense coded quantum bits) is not enough. That's as bad as random noise. You have to do better than the noise floor to transmit information.

Answer 2

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$If you could encode an arbitrary amount of bits into a single qubit, and then retrieve those bits, then yes, quantum teleportation would allow you to send a fully-accurate message 25% of the time, which is better than random chance, and would count as faster than light communication.

However, although you can encode an arbitrary amount of information into the state of a single qubit, due to Holevo's theorem, you can only ever get a single bit of classical information out.

Even imperfect cloning does not allow you to get around this, as commenters have mentioned, as the imperfect clones are entangled and thus measurement of one collapses them all, limiting the amount of useful information one can retrieve. This is stated in the paper "Quantum copying: Beyond the no-cloning theorem". In fact, even Quantum Computation and Quantum Information makes the following strong and damning statements (emphasis added) "only if infinitely many identically prepared qubits were measured would one be able to determine $\alpha$ and $\beta$." and "the laws of quantum mechanics prevent [one] from determining the state when [one] only has a single copy of $\ket{\psi}$."

Therefore, Holevo's theorem does prevent your single-qubit-with-arbitrary-encoded-information scheme from allowing faster than light communication. And, since due to Holevo's theorem you can only get one classical bit out of one qubit, that means that in order to send an $n$ bit message, you must send $n$ qubits. Since these qubits each have a 25% chance to be in a particular bell state, and they do not necessarily agree on the bell state, that means that only 25% of your bits will be correct, and you don't know which ones. As other answers have pointed out, this is worse than random chance and thus can't be considered communication.