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The quantum no-cloning theorem states that one cannot "build" a perfect cloning device for arbitrary quantum systems.

There also exists a famous thought experiment where Alice transmits information to Bob super-luminously using a quantum cloning device, which is resolved by the no-clone theorem. Essentially, there is an electron-positron pair in the singlet state. The positron travels to Alice, the electron to Bob. If Alice measures the positron in the spin down direction, Bob makes a lot of copies of the electron using a cloning device, and then measures them. If he gets all spin up, he knows Alice made the measurement. If he gets a 50-50 mix, he knows Alice did not make the measurement. If he does this fast enough, and Alice is far enough away, one might consider that information has travelled faster than the speed of light.

However, the cloning device he makes is restricted to be able to perfectly clone only plus and minus states, and fails to clone arbitrary linear combinations, i.e his device could not reproduce the electron-half of the singlet state if Alice did not make a measurement.

Here is my question. What if Bob just says, if I get all the same answer, she made the measurement, which is possible because his cloning device can clone states in either the up or down state, just not a linear combination. If he gets ANY mixture at all, he says she did not make the measurement. Why does the no-clone theorem prevent this, and why is this not a violation of prohibited super-luminal information travel?

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    $\begingroup$ Bob makes a lot of copies of the electron using a cloning device, and then measures them. Sorry, there is no cloning device. $\endgroup$
    – Sofia
    Commented Nov 29, 2014 at 16:01
  • $\begingroup$ there is no arbitrary state cloning device, but it is possible to create a device that will clone, per say only spin up or spin down states, just not linear combinations $\endgroup$
    – Nick Murphy
    Commented Nov 29, 2014 at 16:35
  • $\begingroup$ Nick, you say,"i.e his device could not reproduce the singlet state". His device is not supposed to reproduce the singlet state, which is a two-particle state, but the opposite of the single-particle state that Alice obtained. Next, how can you clone a state without destroying it, i.e. collapsing it on something? After that, one more clone, and one more clone will give you the same result. $\endgroup$
    – Sofia
    Commented Nov 29, 2014 at 18:30
  • $\begingroup$ Sofia, you are right in that he is not trying to clone the singlet state. I messed up there. However, there are ways to clone certain states without destroying the system, check out any proof of the no clone theorem. The issue is that if one chooses a device to clone certain basis states, it will never be able to properly clone linear combinations $\endgroup$
    – Nick Murphy
    Commented Nov 30, 2014 at 9:35
  • $\begingroup$ Nick, I looked at the no-cloning theorem. I am with you about cloning a state with a definite polarization, along z. But, I don't see what happens when the particle has no defined polarization, (Alice DIDN'T measure). How can Bob get some clone without collapsing his particle's state? Can you be more clear? Does Bob bring some particle into interaction with his particle and then measure the presumed clone? That causes collapse of both. Does he bring many particles to interact with his particle, and then measure? That collapses all the particles at once. $\endgroup$
    – Sofia
    Commented Nov 30, 2014 at 12:50

3 Answers 3

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Something is wrong with your scheme - it just can't work at all (we don't even need to invoke no-cloning):

The positron travels to Alice, the electron to Bob. If Alice measures the positron in the spin down direction, Bob makes a lot of copies of the electron using a cloning device, and then measures them. If he gets all spin up, he knows Alice made the measurement.

Alice cannot measure "in the spin down-direction". She can only measure along an axis and the result will say either up or down. If the state was not prepared such that Alice will always measure spin down (in which case Alice measurement doesn't tell her anything, because the state was spin-down before she measured), she will always measure a mixture of spin up and down. In order to have "all spin-downs", she'd have to postselect - but this result needs to be transmitted to Bob. In any case, Bob measures a mixture of spin ups and downs regardless of what Alice measures, or the state was preprared such that Bob always measures spin down (regardless of any measurement of Alice).

EDIT: Before you expand, let me comment on a version of this instantaneous communication protocol from here: scientific American

The crucial element is that there is no "spin-down measurement". As I said, such thing is impossible. Instead, the scheme works as follows:

Let's assume that Alice and Bob share a maximally entangled electron-positron pair and we assume that they are totally anti-correlated (i.e. choosing a basis, if the outcome is up for Alice, it will be down for Bob and so on).

Alice has two measurements: spin up/down or left/right. Note that these are two different directions which she measures, each of which has two outcomes! Now Alice chooses one of the bases and measures the electron and if we assume entanglement, the positron will then have the opposite spin. If Bob could clone the particle, he could measure in the up/down direction. If he gets a mixture, Alice measured left/right, if he gets either up or down all the time, Alice measured up/down. Since a cloning machine cannot exist, this is a contradiction.

Your idea was to take a cloning machine for the spin up/down direction. This is of course possible, but the outcome will always be the same: If Alice measured in the up/down direction, his positron will be up or down. Since Alice will measure up 50% of the time, Bob will measure 'down' 50% of the time. His state looks completely mixed. If Alice measures in the left/right direction, Bob's state will be either 'left' or 'right' - in any case, if he measures up/down, he'll get 'down' 50% of the time. His outcome will be the same every time.

You may think: But what if I apply up/down cloning? The problem is that this cloning machine can only work the following way: You measure the particle and reprepare it according to the result you got and this is a perfect cloning machine since, if you have a pure state in the up/down basis, then it is either 'up' or 'down', hence a measurement will reveal which of the two it is. So applying this machine will produce a string of states that will produce the same outcome all the time.

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  • $\begingroup$ Martin: "a version of this instantaneous communication protocol [...] Alice and Bob share a maximally entangled [...] pair and we assume that they are totally anti-correlated (i.e. choosing a basis, if the outcome is up for Alice, it will be down for Bob [...]) Now Alice chooses one of the bases and measures the electron [...] If Bob could clone the particle [then ...]" -- I wonder whether Bob is allowed to use operator $\hat B$ such that $$ \hat B|\text{blank},~\psi_p\rangle=\hat B|\text{blank},(p_u~\text{up}+p_d~\text{down})\rangle :=|(|p_u|~\text{up}+|p_d|~\text{down}),\psi_p\rangle$$ ... $\endgroup$
    – user12262
    Commented Dec 1, 2014 at 20:05
  • $\begingroup$ I don't quite understand what your operator is supposed to do. The state $|blank,\psi_p\rangle$ means "any state on Alice system" and $\psi_p$ on Bob's side? $\endgroup$
    – Martin
    Commented Dec 1, 2014 at 20:22
  • $\begingroup$ Martin: "what your operator is supposed to do" -- Bob would use $\hat B$ (which is not "cloning") to estimate "his (positron) state $\psi_p$". Note: $$\hat B |\text{blank},\text{exp[} i\alpha\pi\text{]}\text{up}\rangle := |\text{up},\text{exp[}i\alpha\pi\text{]}\text{up}\rangle $$,$$\hat B |\text{blank},\text{exp[} i\beta\pi\text{]}\text{down}\rangle := |\text{down},\text{exp[}i\beta\pi\text{]}\text{down}\rangle $$,$$ \hat B |\text{blank},\text{exp[}i\gamma\pi\text{]}(\text{up}\pm\text{down})\rangle :=|(\text{up}+\text{down}),\text{exp[} i\gamma\pi\text{]}(\text{up}\pm\text{down})\rangle $$. $\endgroup$
    – user12262
    Commented Dec 1, 2014 at 20:45
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    $\begingroup$ I'm afraid your map is just not linear, since you have to work with tensor products. Since we can't implement any nonlinear map, this can't exist. This is the same argument as the no-cloning theorem. $\endgroup$
    – Martin
    Commented Dec 1, 2014 at 22:07
  • $\begingroup$ Martin: "I'm afraid your map is just not linear" -- That's right. (The equations of my previous comment should be helpful to recognize that.) I still need to convince myself that the suggested operator/map is (therefore) not unitary either; referring in particular to this Wikipedia derivation. But anyways, that's a wrinkle I had not considered. So: thanks for bringing this to my attention. $\endgroup$
    – user12262
    Commented Dec 2, 2014 at 6:14
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Suppose the particle is in a state of $\frac{1}{\sqrt{2}}$up + $\frac{1}{\sqrt{2}}$down. There is a 50% chance of measuring it up and a 50% chance of measuring it down. If you run this through the imperfect cloner multiple times, it won't have half the clones be up and half be down. Suppose the first clone is up. Since it measured the particle spin up, it is now spin up, and every additional clone will be spin up.

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  • $\begingroup$ I think this is correct - regardless of what Alice does, Bob will measure the same spin state over and over again (up-up-.. with p=1/2. or down-down-... with p=1/2) on his cloned state. Right? $\endgroup$
    – innisfree
    Commented May 18, 2015 at 11:50
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Don't really care about the scheme or the statistics. People will talk a lot about these things, but the idea behind math and "theorems" (and, ironically, syntax) is to get around reasoning at a low level that the mind is poor at tracking. This means that "thought experiments" which resort to details in order to challenge an hypothesis are prone to be sort of like magic tricks: they seem to survive only as long as understanding is lacking.

For example, there once was a time when the smartest people on the planet were doing thought experiments about algebra, the properties of polygons, and transmuting lead into gold. They were typically either proven wrong or trivialized eventually, and the thought experiments vanished. So the mere existence of thought experiments--or even the drive to come up with them--is usually a discrete, infallible sign that the "theory" doesn't hang together, or is clouded with noise that makes it needlessly complex (Quantum Computing is likely the latter--and that is what we are talking about: quantum signal processing).

So I'm not addressing the scheme, but I'd like to highlight a few things in terms that are a bit more relevant than a physical science "theorem" in an unresolved domain: unclassified quantum computing/photonics.

A) Quantum superposition $=$ Quantum entanglement $=$ Quantum Coherence

B) It has recently been acknowledged that Quantum Coherence $=$ Optical Coherence (i.e., photon coherence is more or less the same thing as quantum coherence)

And, with holography and holographic techniques being regularly used and vanishing from scientific discourse under the auspices of national security for the last 60 years, it is perhaps safe to say that Photon Coherence is recorded and replayed all the time, and coherent (entangled) elections to boot! (every time a technique relying on electron holography is used, in fact)

So I don't know about this "no-cloning theorem." It sounds kind of like "no-fish on Fridays": one of those things people just say and do, but which no one can really find any justification for. Any how, I am sure there are many papers out there. But I am not sure why holography works, if coherence cannot be cloned. The universe--and anything that relies on a beam splitter--simply does not make sense otherwise.

And I am sure that our only source for the no cloning theorem are texts and symbols that only the most entrenched can decipher...that and inconclusive experiments with newly invented measurement techniques.

Holography, however, has been around since 1941, and is something you can do in your backyard as long as you spend a few bucks.

And I'll leave it with this: the arguments I have heard about the no cloning theorem (other than symbolic gibberish that purports to overcome issues in experimental error) relate to causality. And, where they relate to causality, they fall back on the old philosophical trope: how can a tree fall in the woods without someone there to observe it without violating causality?

And all of superposition, communication, light limited communication etc. follows from this philosophical kernel that, I think we can acknowledge, is meaningless when viewed under the light of a normal math proof.

Any iff mathematical concept must be proven in both directions: if you take schrodinger's cat and examine the superposition of the rest of the world relative to the cat when the cat's master decides to open the chest and take the cat out based on the decay rate of another isotope outside the chest, then infinite cats live and die, infinite scientists open the chest, and the universe explodes.

It is, however, possible for there to be only one universe if some aspect of the cat, the scientist, the isotope in the chest, and the isotope next to the scientist are zero distance from each-other, and in constant communication through some global information singularity. But that would introduce something else entirely. Which is the realm of sci-fi, and of course, infinite cats, scientists, Amys and Bobs living, dying and randomly sending FTL versions of Shakespeare with infinite quantum monkeys are much more reasonable.

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