What experimental proof of quantum superposition do we have?

Question

My question is both naive and subtle. Naive because I don't know much more than the layman about physics and in particular quantum physics. Subtle because physics is an attempt to model the world, and as a computer scientist, with a strong interest in machine learning but also formal logic and models, a model is just that, a model. Not necessarily reality. It is not because a model fits reality that the model is the truth about reality.

From my understanding, we know that:

• the quantum physics model has not been contradicted on the notion of superposition that it introduces

• there are experiments that can be explained using the quantum model where the classical model fails

Am I correct to say that this only proves that the classical model, is just that, a model, and therefore incomplete? And that for those cases quantum mechanics has a better predictability power.

Now the question: Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model)? Or is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't?

I would like to know if objects of our world can be in two states at the same time or that it is just more practical for predictability purposes to model things this way.

Answer 1

"Being in superposition" is not an objective property of a quantum mechanical state. Quantum mechanical states live in a Hilbert space, where, since it is a vector space, every state can be expressed as the sum of other states. That is what we mean by "superposition": The sum of two states is a again a state.

But as long as you don't choose a basis of this vector space as your reference for what an "unsuperposed" state is, asking whether a state is "in superposition" doesn't make any sense. A state of definite position is not a superposition of other states of definite position, but it is an infinite superposition of states of definite momentum. Every state is a superposition of states that belong to a basis where it itself isn't a basis vector.

So "quantum superposition" is not some sort isolated postulate of quantum mechanics, it is built right into the basic mathematical structure of the space of states. You cannot remove "superposition" from this formulation of the theory any more than you can remove real numbers from classical mechanics. So there is no experimental test like "Quantum mechanics with superposition" vs. "Quantum mechanics without superposition" where you could compare the predictions of two well-defined theories.

Also, note that this superposition really is about a technical property in the mathematical formalism: Our ability to form sums of states. The formalism itself makes no direct claim about how you should think about this, and indeed different quantum interpretations may disagree whether "the object is in both states at once" is really the correct natural language interpretation of the mathematical fact in the formalism. But since (most) quantum interpretations do not change the experimental predictions of quantum mechanics, none of these different ontologies of quantum superposition can be experimentally tested.

Therefore, every experimental test of quantum mechanics is an experimental test of "quantum superposition", if you so wish. The notion of superposition cannot be separated from the rest of quantum mechanics, it is too fundamental for that. Whether that "really means" an object "is in two states at the same time" is not a question physics can answer.

Answer 2

I guess you might know that if you have a linear equation $\mathcal{L}$ and two solutions of it, then a superposition of these solutions is also a solution of this linear equation. $$ \mathcal{L}(f(x)) = 0, \quad \mathcal{L}(g(x)) = 0 \quad \implies \quad\mathcal{L}(a \cdot f(x) + b \cdot g(x)) = 0,\quad a,b \in \mathbb{C} $$

Hence you can think of each of your basis solutions, $f,g$ as basis vectors and any possible solution as a vector space being spanned by these two basis vectors.

In quantum mechanics, the time evolution of the state of a system is given by a linear equation called Schrödinger's equation, and we hence can express solutions (evolution) of states as a superposition of the evolution of "basis" states. But obviously, you can also just choose your basis such that the state from which you want to describe the evolution is in your basis to make it no more be a superposition, basically doing a change of basis.

Let's say for example that you describe the states $s_1$ and $s_2$ as a superposition of the sates $s_3$ and $s_4$:

$$ \left\{\begin{array}{ll} s_1 = \frac{1}{\sqrt{2}}(s_3 + s_4) \\ s_2 = \frac{1}{\sqrt{2}}(s_3 - s_4) \end{array}\right. \qquad \iff \qquad \left\{\begin{array}{ll} s_3 = \frac{1}{\sqrt{2}}(s_1 + s_2) \\ s_4 = \frac{1}{\sqrt{2}}(s_1 - s_2) \end{array}\right. $$

Is for example $s_1$ a superposition? Or is it a basis state? You see that superposition is not an absolute term. It only makes sense to talk about superposition of something.

Knowing this, you understand that there is no "quantum weirdness" arising only from superposition, most of the simple physics equations that you might have seen are linear and their solutions can be expressed as superposition of other solutions.

There are a lot of phenomenons which can only be explained with superposition, like most the behaviour of sound waves or electromagnetic waves. If you want a concrete example, take a look at this nice explanation of the Young's double slit experiment for a classical wave.

We also find similar patterns if we use electrons or whatever elementary particle in the double slit experiment, which shows that indeed the description of the evolution of these particle is described by a superposition of the evolution of the particle going through one slit and the evolution of the particle going through the other slit, both solutions interfering together. So this is experimental proof that we need superposition in quantum mechanics and obviously there are thousands of other (quite more complicated) examples.

The weirdness of quantum mechanics comes from what a quantum state describes:

• In classical mechanics, when you have a sound wave which is a superposition of two different pure sounds waves with two frequencies, $f_1,f_2$, and you want to know which frequencies your sound is made of, you can simply measure your sound and determine easily which frequencies is made of and even what are the amplitude and phase of these two sounds waves in your final sound wave.

• In quantum mechanics, if you are in the state $s_1 = \frac{1}{\sqrt{2}}(s_3 + s_4)$ and you want to determine how much of $s_3$ or $s_4$ you have in your state, you can't just measure $s_3$ and then $s_4$. This is because of the influence that a measurement has on your system. The two coefficients squared in front of $s_3$ and $s_4$ actually describe the probability of measuring one or the other. The problem is that if for example you measure (with a 50% chance) $s_3$ then your state "collapses" into the sate $s_3$. This collapse is really where the quantum weirdness lies, and no one understands completely what happens during such a collapse, even the best physicists. It makes is impossible to reconstruct completely your state by just measuring $s_3$ a lot of time to see what is the probability of measuring it. This gives rise to a lot of weird phenomenons which could be shown experimentally, the most intriguing one being in my opinion the entanglement of particles.

Now to answer you other questions:

What experimental proof of quantum superposition do we have?

There are plenty of experimental proof of experiments which require superposition, like the Young double slit experiment

is classical model just a model, and therefore incomplete ?

Yes, quantum mechanics is a better description of nature, than classical physics. Classical physics is therefore incomplete. However, all physics theories are basically models, and you can never be sure that they are complete. But the actual description of quantum mechanics also has some problems, for example we can't solve the equations when a lot of particles are involved, and there are also contradiction with general relativity. It is therefor also just a model, but a very good one.

Answer 3

The answers to your questions are immediate consequences of the definition of a quantum state.

1. Every quantum state is a superposition for the same reason that every integer is a sum of other integers. One could try to test this empirically by checking that 8=3+5 and then that 9=2+7 and then that 10=(-3)+13, but to do that would be entirely to miss the point. The result is an immediate consequence of the definitions, and it would be pointless to "test" it in this way.

2. "I would like to know if objects of our world can be in two states at the same time." No they can't, for exactly the same reason that a bachelor cannot be married. Once again, this is a direct consequence of the definition and it would be pointless to test. (And this, incidentally, has pretty much nothing to do with quantum mechanics in particular; it has only to do with the broad general meaning of the word "state".)

If you are trying to ask whether quantum mechanics accurately describes the world, there's of course a lot of empirical evidence that bears on this question. There is also a lot of empirical evidence that bears on the question of whether arithmetic accurately describes the world. That evidence is interesting. But nobody has found it necessary to test whether arithmetic --- or quantum mechanics --- conforms to its own definition.

Answer 4

Wave Nature

As many have pointed out, the double-slit experiment is perhaps the canonical demonstration of superposition and a macroscopically visible demonstration of quantum mechanics. But the fundamental point of the double-slit experiment is to demonstrate that matter is wave-like. And to say that it is wave-like is to say that it is extended in space, periodic in some sense, and has the ability to interfere with itself. These concepts simply do not exist in classical theories which assume that matter is point-like, which is basically the diametrically opposite view.

If you set up a screen with two vertical slits, and shoot paintballs through the slits, as long as the slits are wide enough to accommodate the paintballs, you will see two vertical lines of paint on the target behind the slits. And this comports with the classical prediction using classical point particles.

If you create a seawall and make two small holes in it, and then start generating waves on one side of it, the waves will hit the wall, pass through the holes, and interfere with itself on the other side, creating a pattern of peaks and troughs that is far more complicated than the original wave. So classical theory can deal with waves just fine. It just doesn't admit that point particles are waves.

The difference between classical and quantum mechanics is that QM says that everything is a wave, and thus, has a wavelength. And thus, everything can, in principle, be made to interfere with itself. What this means in practice can be a bit subtle, but suffice it to say that this prediction is noticeably absent from classical theories.

Experiment

Now, when we pass electrons through the double-slit experiment, an easy objection is to say that the interference happens because multiple electrons are passing through at the same time and interfering with each other. Thus, individual electrons are not really waves...the experimental artifact is due to collections of electrons acting as a wave together (just like ocean water hitting the seawall). However, this objection is defeated by passing electrons through the experiment one at a time. Classical theory cannot make a wave out of a single particle.

But what does it mean for a particle to pass through two slits as a wave? The way we detect the electron on the target is by, say, a glowing phospor. Since an electron can only hit one phosphor, what does it mean to say that it's wave-like? At this point, it's looking very particle-like, and it's easy to imagine that it only took one well-defined path through the experiment. The problem is which path did it take. The interference pattern does not show up with a single electron, because you can't make a pattern from a single data point. You need many electrons to build up a pattern. And what you see is that instead of just hitting the target in two well-defined lines, the electrons are spread out horizontally, but in clumps. How do they know to do that? If the electrons are taking a simple direct path through one slit or the other, why would they strike the back target in one of a dozen or more clusters? Where did those clusters come from? The QM answer is that the electron takes both paths simultaneously, thus causing it to interfere with itself, with the final point of impact determined by a probability distribution that varies in intensity across the target. And so far, nobody has come up with a more economical explanation than that.

Music

Classical theory says that a charge like an electron orbiting another charge like a proton is accelerating and thus should be constantly emitting radiation and spiraling into each other. Classical theory tells us that our precisely localized electron point particles behaving according to Newtonian mechanics should cause all matter to collapse on itself in a blinding flash of light faster than you can say: "By Grabthar's Hammer!" The fact that it doesn't tells us that classical mechanics is missing something important. Quantum mechanics says that electrons don't spiral into atomic nuclei because they are not point particles, but waves. And bound states are much like waves in a musical instrument: the set of frequencies allowed is highly constrained by the size and geometry of the instrument (e.g., where it is fretted). And this conveniently tells us why electrons not only don't spiral into the nucleus, but also why they absorb and emit radiation at particular frequencies.

So when an electron is bound to a nucleus, where is it? Well, you could detect it at any number of locations, just like you could detect it in many places on the target screen in the double-slit experiment. But where you do detect it is not predicted by classical theory (which says that the place you detect it is crashing into the nucleus). Saying that the electron is definitely in one place or another before you've measured it is not a well-defined operation, because the electron is in a superposition of states. So you could say that pretty much all atomic theory since the description of the photoelectric effect is more or less a "proof" of quantum mechanics in general, and superposition in particular.

Answer 5

is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't?

Here are two examples from my side.

(1) Quantum dots produce entangled photons in their polarisation. That this is the case has been statistically tested. There is no other way, as we are forced to make do with measuring equipment that irretrievably destroys the superposition. In some cases we get the expected result, in others not. Nevertheless, the quantum dot and the entangled photon pairs produced are established findings. But now things are getting interesting.

The assertion that the state is only formed when we measure is a statement of faith in quantum mechanics or an interpretation of the mathematical apparatus. What breaks down is our ignorance of the state. The state only manifests itself during the measurements (but only statistically), but is already present when it is created.

(2) The photons in the double-slit experiment are always photons, no matter where we measure them. The photons are diffracted even if they only hit one edge and even if this does not happen in a bunch but individually. The intensity distribution on the observation instrument then of course only appears statistically.

Interaction with the edge is not mentioned anywhere, which is strange. Planck based his blackbody theory precisely on the fact that there are standing waves in the resonant cavity where the electric field of the EM wave at the interface must be zero. And these Plank corpuscles, known as electro-magnetic quanta aka photons, do not interact with the surface electrons of the edge? And the research on phonon excitations in solids does not give us the idea that the deflection of the photons could have anything to do with resonance processes at the edge?

Answer 6

Now the question: Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model)?

Science doesn't prove statements. Scientists notice problems with existing explanations, guess solutions to those problems and criticise the solutions until only one is left and it has no known criticisms. Criticism includes looking for failure to explain experimental results but isn't limited to such problems.

Your second question:

Or is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't?

can be answered.

As David Deutsch explained in Chapter 2 of The Fabric of Reality, the only existing explanation of single particle interference involves multiple versions of the interfering particle. The only existing explanation of entanglement experiments also involves multiple versions of the entangled systems and the macroscopic systems that measure them:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

Answer 7

Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model) ?

No, although there is a lot of experimental evidence that is very difficult to explain without assuming a superposition of states (but maybe that is just because we have not been clever or imaginative enough). On the other hand, neither has it been "somehow proven" that any physical entity does not exist in a superposition of states. Indeed, some interpretations of quantum mechanics require that all objects are in a superposition of states all of the time. So, even after we open the box, Schrodinger's cat is still both alive and dead at the same time, and it is a limitation of our perception (due to the entanglement between our state and the cat's) that makes it appear that the cat is in one state or the other but not both.

Generally speaking, physicists are happy if they have a model which successfully predicts some aspects of the behaviour of the physical world (and the wider the scope of the model, the happier they are), but they do not look too deeply into philosophical questions about underlying reality. At the moment we have two very successful models in physics - quantum mechanics and general relativity - but we know that they must be incomplete because they are not consistent with one another.

Answer 8

The famous double slit experiment is the best experiment demonstrating quantum interference. The standard interpretation of quantum mechanics involves "wavefunctions" which allow for states to be in "superpositions." This model explains how these wavefunctions can interfere in a way such that you get results consistent with the double slit experiment.

When you don't measure the system, the different paths of the superposition state interfere in a wave-like way. When you measure it, you force the system to become one of the superpositions, and the interference between superpositions consequently disappears.

If the double-slit is a bit too difficult mathematically: A quantum mach-zhender interferometer is a similiar experiment which has much simpler math, but all the same concepts.

Answer 9

Schrödinger never believed that the quantum field potential didn't have a specific value even if it could not be specifically measured. The cat thought experiment was made to ridicule the Copenhagen interpretation.

Because Schrödinger's equation is linear, it can model multiple possible states at the same time. Though we might not know which state we are in until it affects things at a macro level. i.e., we make a measurement.

Superposition may be hard to prove, given we can interpret QM through quantum Bayesianism. Which doesn't require superposition to be anything in reality.

So if such an experiment could be made, one would probably also have to reject Schrödinger's equation.