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.

"cat in box"="dead" + "alive".

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:

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.
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?
What decides which option we should pick?

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

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