Thinking Like a Theorist Number 4: Quantum mechanics and your grandfather

We recently had a physics problem set on the Hyperloop of Elon Musk. In Musk's article describing the Hyperloop he comments: "Short of figuring out real teleportation, which would of course be awesome (someone please do this), the only option for super fast travel is to build a tube over or under the ground that contains a special environment." This got me thinking about quantum teleportation and some of the marvelous properties quantum systems have for transmitting information. Hence this TLAT brings up a common question about information transfer in quantum mechanics. I hope that the community will clear up anyone's misconceptions and enlighten those of us who haven't thought about measurement and information deeply yet.

The introduction

In beginning classical physics every object we deal with has well defined intrinsic values, at every instant of time, for all the quantities one could possibly measure. For example, imagine a ball tossed through the air. Whether or not we look at or weigh the ball at every instant, we still model the ball as having a definite position, shape, mass, color, size, etc. at every moment in time. We call the quantities that can be measured the "observables" assigned to the object. And, we call a configuration of the ball at a moment in time the "state" of the ball. So for example a ball at a time $t_0$ might have a state $S_0$ given by a list of the values of the observables. Let's limit the observables to position, velocity, and color of the ball for the time being. Hence a state for the ball would be defined by the observable list

$S_0=(x_0,v_0,red)$.

At a later time $t_1$ the state might be

$S_1=(x_1,v_1,blue)$.

(For the mathematically inclined, note that the list of quantities really looks like a strange type of vector. It is, states of objects are mathematically described by elements of vector spaces.) For this discussion, whatever the state is doesn't matter - what matters is that in classical physics at every moment of time there is a definite value for each of the observables that go into describing the state. Note also that once we've given the list of values for every possible observable associated with the object, we've actually completely defined the object from the viewpoint of physics. Any other number we assigned to the object would not be an observable and hence can have no bearing or relevance for physics.

One of the most puzzling aspects of quantum mechanics for the beginning student (and for the advanced sometimes) is that objects at certain times do not necessarily have definite values for their observables. Sometimes objects can exist in states that are sums of the states with definite values, i.e. an object can be in a state

$S_2=a \times S_0 + b \times S_1$

where $a$ and $b$ are complex numbers. We call such a state a "superposition", which is a fancy word for summing certain types of objects. You may be familiar with superposition after having dealt with it in electromagnetism to find the electric field of multiple charges. It's the same basic concept - a state is the sum of two other states just as an electric field of two charges is the sum of the electric field of each charge.

We can talk about superpositions meaningfully because quantum mechanics only predicts the probabilities for the values of various observables when a measurement is made. This in turn is possible because if you think about it, a physical theory that describes our world doesn't actually mathematically require that observables are well defined at all times. Certainly when we actually measure the ball only one color must be returned in both quantum and classical mechanics. However, if we measure the ball at $t_1$ and later at $t_2$ physics merely must give us a rule for calculating (perhaps non-deterministically) the values of the observables at $t_2$ given their values at $t_1$, which we then match to experiment. In other words, only when we measure at $t_2$ can we possibly care about the ball - between that time and $t_1$ the ball could turn rainbow and zip off to Mars and back, split into two and recombine, etc. I don't care, it's not being measured. Hence we don't need to always assign a definite value to each observable to do physics. In quantum mechanics the probability of getting a particular value of an observable at $t_2$ depends on $a$ and $b$, in classical mechanics the ball can only ever be in $S_0$ or $S_1$ and never in a superposition. In either case, we get a definite rule for calculating what we should observe.

The most famous example of this behavior is Schrodinger's Cat, which is an unfortunate feline that has been placed in an impenetrable box with a radioactive element. If the element decays a Geiger counter records the event and a machine kills the cat. If there is no decay the cat stays alive. The decay of the element (and everything else, but the element matches with people's intuition) is controlled by quantum mechanics, if we don't measure whether the decay happened or not the entire cat exists in a superposition of the "dead" and "alive" states, i.e.

Hence the cat is neither dead nor alive according to quantum mechanics.

This strikes most people as strange because it contradicts their classical intuition. It gets even stranger to most people in the following thought experiment. Imagine two boxes, box 1 and box 2 each with one of our colored balls in it. Both balls are originally green and the boxes are next to each other and can communicate solely with each other (we cut a hole for example). We again place a radioactive element in the combined box with the balls. If it decays, a machine in the box spray paints both balls red. We wait a little while and then separate the boxes by a large distance. Each ball is still in the superposition state

$a \times "red" + b \times "green"$

since we haven't measured anything about the balls yet. We call the two balls "entangled" as their observables are entangled together - if I measure one ball to be red the other must also be red, etc. due to our spray paint machine.

After we separate the boxes we open one up and look at the ball, and see that it is green. We therefore know that the other ball must also be green. However, the instant before the measurement the other ball was in a superposition, so the state of the ball far away changed "instantaneously" from a superposition to green even though nothing was done to it locally. This seems like it would cause a problem as we would be "teleporting", transmitting information using quantum mechanics faster than the speed of light. Transferring information faster than light is a no-no as it means one can create causal paradoxes like preventing one's own birth. This brings us to three questions for the Brilliant community:

1. Did we just transfer information infinitely fast? Why or why not? What classical physics assumptions do we need to give up to prevent information from traveling infinitely fast? (In other words, what fallacies if any were present in the way I presented the argument above?) Note: you can answer this question without any deep knowledge of quantum mechanics.

2. Is there an adjustment one can make to our definition of causality that doesn't break the world, can mimic quantum mechanics, yet keep the balls behaving classically?

3. What decides which option we should pick?

Note by David Mattingly
6 years, 7 months ago

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No, we didn't transfer information instantaneously or infinitely fast. As you said that would cause causal paradoxes [That sounded weird!].

Alt text

Quantum teleportation is a misnomer. It doesn't involve things suddenly disappearing and then reappearing somewhere else.

I see the thought experiment this way:

There is nothing FTL happening here because nothing is actually moving. No information is transferred. Let's say you separated box 1 and box 2 really far away. You have box 1 with you and you assigned one of your friends with box 2 and told him not to open it. After you open box 1, you instantaneously know about the quantum state of the ball in box 2. But remember that your friend doesn't know what color the ball in box 2 is. Now if you want to transfer that information, you'll have to phone, email or shout out to your friend. In other words, you'll have to use slower-than-light channels to transfer that information (the quantum state of ball 2).

Is there something wrong with my reasoning? I'll really appreciate it if anyone offers any kind of feedback.

- 6 years, 7 months ago

Well, you're right. No information can actually be practically sent through this method.

- 5 years, 12 months ago

I'd like to clear up what I think is a misconception about quantum mechanics. The original post does not (properly) address it, and I spotted it in a reply already.

The misconception is that in QM, you do not (completely) know the state of a system. I feel that this is wrong. A wavefunction, or another superpositional form like $\alpha |dead\rangle + \beta |alive\rangle$, is an exact description of the state of a system. It represents everything there is to know about the system. The uncertainty in the outcome of an observation is an artifact of the measurement process, not of the quantum state.

Going back to Schrödinger's cat: classically, one would say the cat is either dead or alive, we just don't know which it is. Then, when we have a look, and find that the cat is alive, we would argue that the cat must have been alive all along.

In the QM description however, we do know the exact state of the cat. The superposition description is not a reflection of ignorance. We don't know the outcome of a measurement, but that's impossible. When we do the measurement, then the outcome does not provide any new information about the state of the cat prior to the measurement, unlike in the classical case.

Disclaimer: there is no consensus on how QM should be interpreted, so what I said is not necessarily undisputably true. It does however conform to the interpretation I was taught, which I believe is the most commonly adopted one.

- 6 years, 7 months ago

1. "We don't know the outcome of a measurement, but that's impossible". Could you elaborate?

2. Btw, even I, from whatever I have read about the quantum world, believe that the whole problem of state changes occurs because of measurements. Tell me if I am wrong, but I think in Hisenberg's principle thingy, the uncertainty is because of measurement. Every electron does have an exact momentum and position, but the very act of measurement (bombarding by photons) changes the measured value of momentum and position and only that's why the momentum and position can't be measured accurately. Am I right?

3. I think saying that we do know the exact state, I believe it'd be better to say that the system has an exact state (but we don't fully know it). I know this contradicts the very purpose of your writing. But if we did know the exact state of the system, then it'd mean we'd have measured it (else how do you know it?). But since we have measured some state, the state has changed and we don't know the new state after the measurement.

- 6 years, 7 months ago

Before I proceed to answer your questions, I should repeat that there are many interpretations. My answers will be in accordance with the popular Copenhagen interpretation.

1. It is simply impossible to predict the outcome of a (single) measurement. If you keep resetting the system in the same state, doing everything in exactly the same way, then the outcomes of the measurements will differ. The collection of all outcomes will however statistically conform to the probability distribution as given by the wavefunction (or in the case of the cat, by the coefficients $\alpha$ and $\beta$ of the superposition $\alpha|dead\rangle + \beta|alive\rangle$).

2. (According to the Copenhagen interpretation), that's wrong. Classically, you would attribute an exact position and momentum to an electron, and that gives you the state of the electron. But according to QM, there is no such thing as an electron with an exact position and an exact momentum. You simply have to let go of that concept. The "exact" equivalent of the classical attributes is the wave function; it represents the state of the electron.

3. The idea that you can somehow still work with definite but unknown properties, the so-called deterministic or realistic view, is shaken by Bell's theorem. It involves an experiment for which the realistic view gives you a prediction that conflicts with QM. I remember a specific version where the realistic idea of definite states gives you $2$ as answer, whereas pure QM predicts that the answer can be as high as $2\sqrt2$. Experiments seem to rule in favor of QM (the accuracy is still an issue). In order to save the realistic view, you'd have to give up on something else, like locality (the idea that you can't have instantaneous action over a distance, which nicely refers back to the original idea of the topic).

About knowing/measuring a state: you can know a state by having full knowledge and understanding of how you prepared it. For instance, with the cat, if you know the properties of the radioactive element, then you can work out the coefficients $\alpha$ and $\beta$. Alternatively, you can in a way measure a state, by preparing the state multiple times and performing the measurements. The outcomes then give you a statistical picture of the wavefunction.

- 6 years, 7 months ago

You're right!

There is no such thing as an electron with exact position and momentum.

The thing is: 'realism' is very intuitive while QM is not.

It turns out the universe has no problem giving up 'realism' because it could care less about what we find intuitive.

Intuition is built upon experience. If we experienced quantum mechanical stuffs from our childhood, we would find QM very intuitive.

- 6 years, 7 months ago

Ok. Understood. This quantum stuff is just way too high level stuff for me, as of now. I'll keep poking my nose in it, again and again however! Thanks for the detailed point-wise answer :D

- 6 years, 7 months ago

Great article, but I have a question regarding the 2-balls example that you gave. First, we didn't transfer information infinitely fast because that would be faster than the speed of light and that would be unacceptable.

Second, Isn't the spray paint machine kind of a "Local Hidden Variable" shared between the two balls? and Isn't Bell's Theorem refute this theory as explanation for entanglement? Please correct me if I am getting something wrong.

Thanks,

- 6 years, 7 months ago

First thing: I am a noob in quantum mechanics. Second thing: From what I have read about quantum mechanics, it seems to me (and correct me if I am wrong) that the only difference between classical mechanics and quantum mechanics is that in classical, we say every object has always got a definite state. In quantum, however, unless we measure the state, the state of the object can be anything. In the Schrodinger cat experiment, we say the cat is neither alive nor dead till we actually measure its state - we just can't say anything about the cat. However, in reality, it can't be neither alive nor dead (this is where I must be wrong, since obviously Schrodinger and all others since ~1920 aren't fools to work on quantum mech!).

I think I understand quantum mech ( quite wrongly, probably - I didn't measure my state :D ) - "we can't say anything unless we measure", but then in the Cat experiment, if it is claimed that the cat is actually neither dead nor alive, then I must say that this I don't understand.

- 6 years, 7 months ago

- 6 years, 7 months ago

I didn't know anything about quantum mechanics until my summer at University of Pennsylvania, but I believe you are right. They taught us there about how electrons move quickly in wave motions, and since we can't measure perfectly, we can only predict the probability of am electron being in a certain position (which gives you those orbitals from chem class) and also that quantum states change, but that a particle is only in one state at a time. I'd like to think the professors weren't dumbing it down for me...

- 6 years, 7 months ago

Is this just the EPR paradox reworded? Either way, if we let information travel faster then light, then we violate causality, - this above all other things shouldn't be done. However i don't know how else to resolve the problem - a quick internet search talks about wavefunction collapse and 'violating locality' - can someone please explain to me what this means? It also talks about conjugate properties like position and momentum, is the way you set up the argument flawed because youre only considering the colour? Is the answer simply that you can allow the state to change from "undecided" to "green" for the other ball instantly, since nothing physical has actually happened, its only the underlying probability thingy that has made its mind up...

"How can one account for something that was at one point indefinite with regard to its spin (or whatever is in this case the subject of investigation) suddenly becoming definite in that regard even though no physical interaction with the second object occurred, and, if the two objects are sufficiently far separated, could not even have had the time needed for such an interaction to proceed from the first to the second object?" Gah, quantum mechanics confuses me, but which ever way I look at it, quantum entanglement happens, and that seems to somehow involve some kind of information transfer happening instantly weather I like it or not..

Please correct me if any or all of this random speculation is wrong, Id love for someone to tell me whats actually happening :')

- 6 years, 7 months ago

I don't think information is transferred, since the color of the ball is a random event, therefore we can't do anything to influence the color of the ball, which is the "information" transmitted. If we have no control over the "information" transmitted, then I suppose we can't use it to communicate at all.

- 6 years, 7 months ago

That's what I thought as well, but I once heard that it is not impossible to decide the colour what colour one ball is, with which you immediately decide on the colour of the other ball as well. Maybe someone else can confirm this.

- 6 years, 7 months ago

I am certainly not any type of physicist or anything, but I have talked to one of my teachers a little and parallel universes have come up. I was wondering if it could be possible that parallel universes are created, some in which the balls remain green and and equal number in which the balls become red. When we open one box and see that the ball is green, we realize we are in one of the universes in which the balls remained green. This would mean that the unopened box doesn't have to receive any information; it is green because it is in the green ball universe. So I suppose when we look at the ball, we don't determine what color the ball is, but which universe we are in. Does any of this make sense or is it at all possible?

- 6 years, 7 months ago

This is what the Everett Many Worlds interpretation of QM suggests (in essence), but here you are using it in the wrong context. Many Worlds says the new Universes only come into existence when you observe it, so I have no idea whether that counts as faster than light or not.

- 5 years, 12 months ago

I think, in practice, the mathematics implies faster than light so :( to classical and Special Relativity in particular. But I'm not a physicist so I'm not too sure.

- 5 years, 12 months ago

This is a slightly different topic but mostly related to sub-atomic chemistry. According to the QM model, electrons are orbiting a nucleus. All matter are made of atoms. Wood is hard. But if we touch wood, we feel rigid. So if we touch wood, we are actually in contact with the matter(atoms,molecules,electrons or whatever) if it is in an orbiting state, it should definitely collide with the nucleus if we touch wood. please help me if I am wrong.

- 6 years, 7 months ago

The electromagnetic potential doesn't even allow you to touch wood (or whatever it is you're attempting to touch), remember, like charges repel.

- 5 years, 12 months ago

True - nothing ever comes into contact with one another due to Pauli's Exclusion Principle.

- 5 years, 12 months ago

Yes, but the principle force is the Electromagnetic one in the case he described.

- 5 years, 12 months ago

Yes, you're right!

- 5 years, 12 months ago

By the way, this is not the method in which the electron does not lose energy and gets electromagnetically attracted to the nucleus (it's what's being acted against in this case). For this physical paradox, please refer to the De Broglie Hypothesis.

- 5 years, 12 months ago

Disclaimer: I am not a physicist.

- 5 years, 12 months ago

Actually, you can prove that any causal (if there exists such a word) theory of entanglement will get predictions wrong when you look at the entanglement of spin (i.e. Bell's Theorem). I don't have time (nor skills) to prove this right here, right now, but if you refer to these sets of videos:

Again, I refer you to this set of videos as well as this channel itself if you want a rudimentary but fairly comprehensive introduction to Quantum Mechanics (and some of the mathematics required). (Warning to the extremely talented: these videos contain little to no mathematics so it won't answer everything.)

- 5 years, 12 months ago

Actually there are answers to your questions if you just make a step outside of quantum mechanical world to join the quantum-relativistical mechanics.

1) No, information can't be transfered between systems FTL

2) Yes, there is

I'll try to make it simple. What brought you to all these contradictions is the fact that you're considering a relativistic behavior (things moving at the speed of light) while using quantum mechanics. If you watch at Scrhoedinger's equation: (assuming c=h slash=1) $i\frac{d \Psi }{dt}=H \Psi$ You are watching at hamiltonian H, which is the energy of the system. To include relativity in quantum mechanics (for a free particle) H is not just kynetic energy, but it also includes mass. $H=\sqrt{p^2+m^2}$ By doing so you face a lot of technical issues, which can be solved and lead to Klein-Gordon's equation and Dirac's equation which describe behavior of bosons and fermions.

To answer your question: after having included this stuff in the new hamiltonian and found the eigenfunctions what you can see is that if you have two events which are quantum mechanically related, but they are at a space-time distance in which they can be seen happening at the same time (space-type events) the corresponding observables cannot be related, because they always commute

I hope I answered your questions. I'm not 100% comfortable yet in explaining QRM in english.

- 4 years, 5 months ago

Long story short:

all these questions are answered in relativistic quantum mechanics, where it can be proved that no information can be transfered FTL. Paradoxes come from the fact that you try to apply 'classical' quantum mechanics to light, which is intrinsically relativistic. Unfortunately it's pretty hard to show this without maths.

- 4 years, 5 months ago