What are classical analogies for the notions of superposition, entanglement, and interference?

I'm working on a conceptual explanation of quantum computing properties and have used analogies to make the ideas more accessible. I'd appreciate feedback on the validity of these analogies from those more experienced in this field.

Superposition: Imagine a coin flipped in the air, representing all its possible outcomes simultaneously—both heads and tails. This is akin to a quantum bit, or qubit, which can exist in multiple states simultaneously until it is measured, at which point it collapses to a single outcome.

Entanglement: Consider two coins tossed such that the result of one coin instantaneously informs us about the result of the other, regardless of the distance between them. This analogy aims to illustrate quantum entanglement, where the state of one qubit is intrinsically linked to another, showing correlation without direct causal influence.

Interference: Picture tossing two coins and somehow manipulating the toss such that certain outcomes are more likely than others when they land (i.e., (Heads,Heads) is more likely than all other outcomes). This mirrors the quantum concept of interference, where the probabilities of outcomes can be adjusted to increase the likelihood of a desired result.

Do these analogies represent the concepts of superposition, entanglement, and interference in quantum computing? Any insights or suggestions for improvement would be greatly appreciated.

• Commented Apr 25 at 19:00
• you can't really find a good classical analogy.. that's kinda the point of quantum mechanics. And I wouldn't call these ones accurate, no. The one about superposition is very common and you'll find it used in most popular treatments of quantum mechanics. You can find some discussion about it in quantumcomputing.stackexchange.com/q/2443/55. The other two are just wrong: entanglement is a type of correlation, but it can be stronger than what classical correlations allow; and interference is not just about having probabilistic outcomes
– glS
Commented Apr 25 at 19:41

Unfortunately, classical analogies generally can't capture the interesting properties of quantum systems and these analogies are no exception. One challenge is that quantum states can look like they obey classical randomness if you don't follow the right steps. This means that classical analogies for quantum mechanics sometimes right, but never in interesting ways.

To be more specific, here's where things go wrong (you will need familiarity with density operators):

Superposition Classical randomness: A coin flipped in the air can be modelled by an unknown value (a "random variable") $$X$$, which can be treated as either heads or tails until it is observed. Saying that $$X$$ can be both heads and tails "at the same time" is not very well defined, but we can reason about random variables in a way that accounts for all possible values they might take. And a convenient way for doing so is to use density operator notation from quantum mechanics, $$\rho_X := \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix}. \tag{1}$$ This object can be "measured" in exactly the same way as a quantum state is, and until it is measured it exists as a mixture of $$|0\rangle$$ and $$|1\rangle$$. But this object obeys all of the laws of QM and can be treated as a quantum system(!), just not a very interesting one. Compare this to the density matrix for the state $$|+\rangle = (|0\rangle +|1\rangle)/\sqrt{2}$$, $$|+\rangle \langle +| = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}. \tag{2}$$ This state exists as a superposition of $$|0\rangle$$ and $$|1\rangle$$, which is closely tied to those off-diagonal elements. The problem is, if all you do is measure $$\{|0\rangle, |1\rangle\}$$ (i.e. look at the result of a coinflip), you won't see a difference between (1) and (2), so the analogy has managed to trick us in this case.

Entanglement Classical correlation: Just like we modelled a coinflip with a random variable $$X$$, we can model classical correlations between two coins that have the property that you mentioned. Simply flip an unbiased coin $$X_1$$, give it to someone else (so that you don't see the result) to prepare a coin with the same outcome, $$X_2 = X_1$$. Then send $$X_2$$ as far away as you like. Looking at either coin immediately reveals the result of the other. As before, this scenario has a density matrix representation, $$\rho_{X_1 X_2} = \begin{pmatrix} 1/2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/2 \end{pmatrix}. \tag{3}$$ So this system exists as a classically correlated mixture of $$|00\rangle$$ and $$|11\rangle$$ until a measurement is performed. We can compare this to a genuinely entangled state $$|\Phi\rangle = (|00\rangle +|11\rangle)/\sqrt{2}$$ represented as $$|\Phi\rangle \langle \Phi| = \begin{pmatrix} 1/2 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1/2 & 0 & 0 & 1/2 \end{pmatrix}. \tag{4}$$ This state is entangled, which is closely tied to the off-diagonal elements again. But if all you ever do is check for heads or tails (i.e. perform a measurement in a single basis), these states (3) and (4) behave in the same way, and so the analogy has again tricked us. But there isn't a widely accepted classical analogy that fully captures the interesting properties of entanglement, which require e.g. using different bases for measurement.

Constructing analogies like this might be more pleasant than dealing with the unfamiliarity and un-intuitiveness of quantum mechanics, but classical analogies cannot really convey non-classical concepts.

• Thank you very much! I'm fine with the analogies being inaccurate to a certain extent (they are just analogies after all), but I don't want people to say that those analogies are wrong. Thus, my only question now is: are my analogies fundamentally wrong and cannot be used? I admit I still want to use them even thought they do not fully capture the properties :-) But if they are plain wrong, then of course, I will not use them. Commented Apr 25 at 18:07
• In my opinion, the analogies are fundamentally wrong. I think an analogy should clarify a complicated concept by providing an example of a simple system that follows the same rules in aspects of the system that matter. e.g. A wall clock provides an analogy for modular arithmetic. But your examples obey different rules than quantum mechanics, they just look the same. Its kind of like saying a light bulb is an analogy for how the sun works... Commented Apr 25 at 18:47