[lisper]

> But why the tensor product? Why is it that this construction—out of all things—describes the interactions within a quantum system so well, so naturally? I don’t know the answer

That's an odd thing for the author to say, because s/he gives the answer later in the very same passage:

> but perhaps the appropriateness of tensor products shouldn't be too surprising. The tensor product itself captures all ways that basic things can "interact" with each other!

That is the answer. It's the tensor product because there are logically no other possibilities. The tensor product says everything you can possibly say about the interactions of two systems whose states are described by a (possibly infinite) set of numbers and whose interactions correspond to some basic constraints, like being time-reversible. It just so happens that nature behaves according to those constraints, and that is why the tensor product describes the behavior of nature.

[codethief]

> It's the tensor product because there are logically no other possibilities.

There are many other possibilities, unless you can provide satisfactory answers (from first principles) to the following questions: Why would we expect superpositions of quantum states to be encoded as a vector sum of the individual state vectors? Why is time evolution in quantum mechanics a linear operation on those state vectors?

If those things weren't true, tensor products would be utterly useless to describe product states.

[consilient]

> Why would we expect superpositions of quantum states to be encoded as a vector sum of the individual state vectors?

Because that's what the word superposition means. If you don't have linear dynamics you don't have superpositions that aren't sums of other states, you just don't have superpositions.

> Why is time evolution in quantum mechanics a linear operation on those state vectors?

This, on the other hand, is an open physical question. An answer to "Why QM and not some completely different theory" is probably not in the cards, but as long as we're only considering "nearby" theories, nonlinearity gets you either superluminimal communication (bad) or basis-dependent observables (worse) depending on which bullets you bite.

[codethief]

> > Why would we expect superpositions of quantum states to be encoded as a vector sum of the individual state vectors?

> Because that's what the word superposition means.

No, that's not what superposition means a priori. I can think of many other ways to implement (mathematically) the idea of a system being in "two states at the same time", apart from vector addition. Yes, if you do Stern-Gerlach often enough, you might convince yourself that the vector space structure is a sensible choice but I take issue with OP's statement that

> there are logically no other possibilities

as if things had been obvious right from the get-go.

[consilient]

> No, that's not what superposition means a priori.

The word was originally used to describe the decomposition of waveforms into sums of sinusoids, which is as canonical an example of a linear system as you can get.

> the idea of a system being in "two states at the same time", apart from vector addition.

But that's not what's going on. A system is only ever in one state at a time: the ability to treat it as a sum (modulo the norm) of other states is linearity of all operators. This has immediate observable consequences: nonlinear operators can distinguish between different ensembles realizing the same mixed state.

[codethief]

> > No, that's not what superposition means a priori.

> The word was originally used to describe the decomposition of waveforms into sums of sinusoids, which is as canonical an example of a linear system as you can get.

You are right, I was being a bit too sloppy with my usage of the term "superposition". I guess once people realized that a QM system being in "two states at the same time" is just a linear sum like for waves, they started calling it a superposition. Anyway, my point (in my original comment) was a completely different one: You still have to assume all that linear structure to start talking about how canonical the tensor product is.

> But that's not what's going on. A system is only ever in one state at a time: the ability to treat it as a sum (modulo the norm) of other states is linearity of all operators

Again, you are right, that's why I put it in quotes. Nevertheless, if we start just from the observation that a system can occupy two states "simultaneously" (in the sense that sometimes we measure one, sometimes the other state) we might think of other ways to encode that beyond vector sums, e.g. Cartesian products without any linear structure.

Anyway, I don't think we disagree fundamentally, we're merely arguing about terminology.

[lisper]

A useful illustration: polarized light can be considered to be in a superposition if you use a basis rotated 45 degrees to the polarization axis. So whether or not polarized light is in a superposition depends entirely on how you choose to look at it. It's not a reflection of the underlying physical reality.

[lisper]

We don't expect these things, we merely observe them. Indeed, the fact that QM is linear and hence time-reversible is violently at odds with everyday experience, so it is emphatically not the case that we "expect" these things. This just turns out to be how they are. The tensor product is merely the most compact description of these observations, kind of like how untyped lambda calculus turns out to be a compact description of universal computation (which is also not a thing that one would a priori expect).

[codethief]

> We don't expect these things, we merely observe them.

Sure, but your original claim was that

> It's the tensor product because there are logically no other possibilities. The tensor product says everything you can possibly say about the interactions of two systems whose states are described by a (possibly infinite) set of numbers and whose interactions correspond to some basic constraints, like being time-reversible.

I merely wanted to point out that your claim sounded quite broad and you need to assume many things about the mathematical structure of QM here (based on established observations of course). So, unless you simply take those for granted, you would have to come up with an explanation for them in order for there to be

> logically no other possibilities

In this case, though (if you take all those observations for granted), your claim becomes almost tautological IMO.

[lisper]

It is a tautology. The heavy lifting is being done by the phrase "whose interactions correspond to some basic constraints". Maybe the word "basic" implies more simplicity than is warranted, though if you actually write down what the constraints are, it's a short list and they are not very complicated. That those constraints lead to the tensor product is tautological. I'm not saying this is a Deep Insight, only that it is the answer to "Why the tensor product?"

BTW, just because something is a tautology doesn't mean it can't lead to deep insights. Darwinian evolution is a tautology too: if you have a variety of self-reproducing systems, then the ones that are better at reproducing will make more copies than those that are worse. Well, duh! That's what "better at reproducing" means! The thing that makes it a Deep Insight is that this tautological observation can actually explain some very complex data. Likewise, that the tensor product is the mathematical construct that describes linear interactions between systems that obey conservation laws is a tautology. What makes that a Deep Insight is not that, but the fact that the resulting relatively simple math makes surprising predictions (entanglement in particular) that turn out to be confirmed by experiment.

[xeonmc]

So in short, tensor product is just saying “Everything may contribute to everything. Linearly.“

[lisper]

Yep, pretty much.