1
$\begingroup$

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ 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?

$\endgroup$
  • 1
    $\begingroup$ The issue is, effectively, knowing which bits of communication have succeeded. If I have a long string and I know a random (approximate) quarter of the bits are correct, how does that help me? You need to wait for a classical message to tell you which ones the correct ones are. $\endgroup$ – DaftWullie May 14 at 7:12
  • 1
    $\begingroup$ Even if you could do that (you can't get the information back out again. there are other stack exchange questions about this), the fundamental issue is still the same: the receiver doesn't know whether the data they have received is correct or not, so it's no use to them. $\endgroup$ – DaftWullie May 14 at 12:49
  • 1
    $\begingroup$ “ However, as I understand it, this does not prevent one from storing 𝑛 classical bits into a single qubit, imperfectly cloning it 𝑛−1times, and thus retrieving 𝑛 classical bits out”. That is exactly what it prevents. When you clone, roughly speaking, it’s the same part of the state that gets copied many times, not different parts. $\endgroup$ – DaftWullie May 14 at 16:47
  • 1
    $\begingroup$ To put another way: the clones are highly entangled. You measure one, and they all collapse. $\endgroup$ – DaftWullie May 14 at 17:39
  • 1
    $\begingroup$ Random guessing works even 50% of the time! $\endgroup$ – Norbert Schuch May 16 at 10:28
5
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • $\begingroup$ It does not succeed 50% of the time, it succeeds far less than that since the probability that all of the bits are equal to the message Alice has is (1/2)^1000 (1/2 correct * 1/2 correct * ...). Yes, 50% of the bits will be the same as the intended message, however I'm not certain that's relevant here. I could encode each of those 1000 bits in a single qubit, teleport it, and, assuming I had a way to get all of them out of that single qubit, I could have a 25% chance of getting the intended message without transmitting classical information. That's far better than random chance. $\endgroup$ – Techmaster21 May 14 at 12:36
  • $\begingroup$ @Techmaster21 Could you expand on how to encode 1000 bits in a single qubits and then retreive the information? (If you can do that, not only you broke multiple theorems in information theory, but you're going to collect a lot of prizes!) I guess you could encode however many bits you want in the amplitudes of the states, but you can't measure those with a single shot measurement. Maybe the problem is your proposed protocol for that. $\endgroup$ – user2723984 May 14 at 15:18
  • $\begingroup$ @user2723984 See the last paragraph of this answer physics.stackexchange.com/a/383044. I didn't intend to propose using a single-shot measurement, but rather a series of measurements on copies of the teleported qubit. $\endgroup$ – Techmaster21 May 14 at 15:28
  • 1
    $\begingroup$ I think you should post "why can't I store $n$ qubits in 1 qubit and use an approximate cloning procedure to retreive the information" as a different question, as I think that's what your problem boils down to, note that I'm pretty sure such a procedure would violate Holevo's theorem, as you start from a single qubit and succesfully extract $n$ bits of information from it, even if to do it you used some other (initially "blank") qubits $\endgroup$ – user2723984 May 14 at 15:35
  • 1
    $\begingroup$ @Techmaster21 the last paragraph of that post assumes you already have a large number of copies of the qubit, in which case you can reconstruct the amplitudes, but if you have a single copy you have to produce all of these clones, doing so exactly is impossible by the no-cloning theorem, and I guess (though it would be interesting to see it directly without referencing higher theorems) that any approximate cloning procedure wouldn't be enough (i.e, the error in the cloning would grow faster than the uncertainty in the amplitudes shrink) $\endgroup$ – user2723984 May 14 at 15:43
0
$\begingroup$

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$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.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.