How can one imagine entanglement in a non-mathematical way?

What is entanglement?

Briefly, entanglement is a type of dependence between subsystems of a composite system.

Quantum mechanics can be viewed as a variant or extension of probability theory. In this view, state vectors are analogous to probability distributions, superpositions correspond to distributions that are non-deterministic (i.e. that do not assign $1$ to a single outcome), composite states correspond to joint distributions, the partial trace corresponds to marginalization and entanglement corresponds to correlation and dependence. States that lack entanglement, i.e. the product states correspond to independent distributions.

The similarity extends to some of the metrics used to quantify the amount of entanglement. For example, classical information theory measures the amount of dependence between random variables using quantities such as entropy and mutual information. Quantum entanglement is measured using analogous quantities of von Neumann entropy and quantum mutual information.

Note that the above is not meant to imply that entanglement is classical correlation. However, probabilistic dependence is a familiar, classical concept that is most closely analogous to entanglement. The differences between classical probabilistic dependence and quantum entanglement are studied by quantum information theory.

How systems become entangled?

Entanglement arises and disappears as a result of interaction. Therefore, it can be thought of as quantum correlations that exist between two systems due to their past interactions.

How to imagine entanglement?

Part of the difficulty of imagining entanglement is that its closest classical analog - dependence between probability distributions - is also nontrivial to imagine. However, there are many effective ways to think about dependence of classical random variables. For example, one can represent it using tree diagrams. Such tools can be repurposed to represent quantum states by replacing probabilities with amplitudes. Visualization of amplitudes is admittedly harder than of probabilities, but it can be done using techniques such as domain coloring from complex analysis.

Any technique used to imagine or represent a superposition can be used to imagine or represent entanglement. The key is to use composite states in superposition. This is related to the fact that there is no mention of or provision for entanglement in the postulates of quantum mechanics. Instead, entanglement arises as a byproduct of superpositions of composite states.

What happens when the spins of two electrons are entangled?

The spins interact and as a result the joint state picks up the type of quantum correlations described above. For example, an interaction might make the spins parallel without making them point in any particular direction.

In a sense, the interaction determines the composite state without fixing the states of the subsystems. This possibility has a more precise formulation in terms of von Neumann entropy which in this case is zero for the composite state and non-zero for the subsystems. One way this is often expressed is that in quantum mechanics it is possible to know everything there is to know about a composite system without knowing anything about the parts.

Consequently, when one of the spins is measured, a measurement result associated with the other can be inferred.

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A little bit of math beyond this point

The analogy above becomes a very close visual resemblance if we allow a little bit of math back into the picture. Specifically, when we consider probability density functions (PDFs) and how they capture lack or presence of dependence we see an analogy to the way expressions for state vectors capture lack or presence of entanglement. In classical probability, a joint PDF of two independent variables is a product of individual PDFs

$$ p(x, y) = p(x) p(y) $$

and every PDF which cannot be written this way incorporates some dependence between the variables. Similarly, a state vector of unentangled systems is a product of individual state vectors

$$ |psi_{x,y}rangle = |psi_{x}rangle|psi_{y}rangle $$

and every state which cannot be written this way incorporates some entanglement between the subsystems.

Using shadow puppets.

Imagine making shadow puppets. However in this setup, instead of one you have two screens and two torches (US: flashlights), pointing 90 degrees apart so that the image formed by torch 1 is projected onto screen 1 and the image formed by torch 2 is simultaneously projected onto screen 2.

 screen 1   screen 2
   /           
  /             
 /               
        mm              <-  hand

      /    
torch 2    torch 1

(Apologies for the terrible ASCII art.)

Now any movement of your hand changes both images in a correlated way. In a sense, the images are entangled - if you observe image 1 to have a certain configuration, then only a small subset of possibilities in the total configuration space of image 2 are valid, and vice versa.

The analogy is imperfect, because you cannot for example poke image 1 with a stick and have correlated changes occur in image 2. But at least no maths.

The entanglement principle implies the use of some form of elementary building blocks, and of those blocks exist several versions.

Most of them are just some forms of combinations from other blocks, but some of them are absolutely unique. Unique in such a sense, that there exists exactly only one of each of the unique blocks. Those are the prime blocks. They are very special blocks, since all other blocks are formed from combinations of those prime blocks. This is because of the entanglement rule. This rule exactly states, that the different prime blocks can be entangled to create a combined block and in addition. It is possible to entangle two prime blocks, of which you had only the exact information about one of the prime blocks. But this information alone is enough to deduce all information about the second prime block by only having access to the combined block.

https://de.wikipedia.org/wiki/No-Cloning-Theorem

This is a good question, but one that probably doesn't have a satisfying answer, as of yet.

One can say what entanglement is not; how you should not imagine it. In particular, it is not a causal link between observations/systems (as per no-communication theorem etc.). It is, instead, a form of correlation between observations made on different systems.

You can imagine classical correlation with an example such as the following: you have a bunch of coloured red and blue marbles. You distribute these into two boxes, close the boxes, and give them to Alice and Bob, which then go on their separate ways. Now the content of the boxes are correlated: whenever Alice opens her box to observe the number of marbles in her box, assuming she knows the total amount of marbles distributed between her and Bob, she can "instantaneously" infer which marbles are in Bob's box.

That's the same way entanglement works: measuring a state on one part of an entangled system you immediately know something about the state of the other part. You are not transferring information, you are just learning something about the other side of the system due to prior knowledge about their underlying correlations. Except in the quantum case it is not quite that simple. You cannot think of the situation as there being some shared classical correlation which allows you to learn properties of B's state by observing something about A's state. Quantum correlations are (can be) stronger than that. This is what allows to perform shenanigans such as teleportation or superdense coding. A system is entangled precisely when the above picture of "classical correlations" stops being sufficient to describe observations.

In particular, the intrinsically destructive nature of quantum measurements oughts to be taken into account. You see this from the fact that entanglement cannot be observed without comparing correlations observed in different measurement bases. This is I think something that makes understanding entanglement tricky: it's not about correlations between measurement results; it's about correlations between the states themselves, that is, between measurement results in any choice of measurement basis.

I don't think one can say much more than this without entering into the realm of speculation and less-than-mainstream physical interpretations. I personally never found/stumbled upon a satisfying (and accurate) way to visualise entanglement without relying on the underlying math (but of course, whether one finds a given visualisation "satisfying" is subjective).

A great (but slightly lengthy) explanation that's very understandable and with minimal math written by John Bell: https://hal.archives-ouvertes.fr/jpa-00220688/document