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The tensor product, demystified (2018)

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Re: The tensor product, demystified (2018)

#21

Post by lisper »

xeonmc wrote: post #20
lisper wrote: post #16

> 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…

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

Yep, pretty much.

Re: The tensor product, demystified (2018)

#22

Post by sigmoid10 »

hackandthink wrote: post #13
sigmoid10 wrote: post #3

A nice, high school level writeup of how to calculate this product with ordinary vectors, but it totally drops the ball on the necessity and its use in physics. It would probably be best to ignore the entire last paragraph and instead read up on pure vs. mixed quantum states if you actually care about that.

It's in the text: "The tensor product itself captures all ways that basic things can 'interact' with each other!" Tensor Product is also the way to go when combining classical probabilistic systems. And you need the tensor product already for pure states in QM. (mixed states need density matrices)

>The tensor product itself captures all ways that basic things can 'interact' with each other!

In quantum physics, "interacting" usually has a different meaning. So one should use these terms more carefully.

>And you need the tensor product already for pure states in QM.

No. Pure states are just vectors (or more precise: rays) in Hilbert space. The usual inner product is sufficient to work with them. An outer (=tensor) product of these states will just give you a density matrix with tr(ρ^2)=1.

Re: The tensor product, demystified (2018)

#23

Post by consilient »

codethief wrote: post #18
lisper wrote: post #16

> 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…

> 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…

> 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.

Re: The tensor product, demystified (2018)

#24

Post by codethief »

consilient wrote: post #23
codethief wrote: post #18

Earlier quoted context omitted.

> 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…

> 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…

> > 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.

Re: The tensor product, demystified (2018)

#25

Post by codethief »

lisper wrote: post #19
codethief wrote: post #18

Earlier quoted context omitted.

> 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…

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 desc…

> 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.

Re: The tensor product, demystified (2018)

#26

Post by consilient »

codethief wrote: post #24
consilient wrote: post #23

Earlier quoted context omitted.

> 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…

> > 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…

> 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.

Re: The tensor product, demystified (2018)

#27

Post by lisper »

codethief wrote: post #25
lisper wrote: post #19

Earlier quoted context omitted.

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 desc…

> 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-reversibl…

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.

Re: The tensor product, demystified (2018)

#28

Post by hackandthink »

sigmoid10 wrote: post #22
hackandthink wrote: post #13

Earlier quoted context omitted.

It's in the text: "The tensor product itself captures all ways that basic things can 'interact' with each other!" Tensor Product is also the way to go when combining classical probabilistic systems. And you need the tensor product already for pure states in QM. (mixed states need density matrices)

>The tensor product itself captures all ways that basic things can 'interact' with each other! In quantum physics, "interacting" usually has a different meaning. So one should use these terms more carefully. >And you need the tensor product already for pure states in QM. No. Pure states are just vectors (or more precise: rays ) in Hilbert space. The usual inner product is sufficient to work with them. An outer (=tens…

"Product states are multipartite quantum states that can be written as a tensor product"

"In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state."

https://en.wikipedia.org/wiki/Separable_state

Re: The tensor product, demystified (2018)

#29

Post by sigmoid10 »

hackandthink wrote: post #28
sigmoid10 wrote: post #22

Earlier quoted context omitted.

>The tensor product itself captures all ways that basic things can 'interact' with each other! In quantum physics, "interacting" usually has a different meaning. So one should use these terms more carefully. >And you need the tensor product already for pure states in QM. No. Pure states are just vectors (or more precise: rays ) in Hilbert space. The usual inner product is sufficient to work with them. An outer (=tens…

"Product states are multipartite quantum states that can be written as a tensor product " "In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state ." https://en.wikipedia.org/wiki/Separable_state

Keyword is can. There's a reason why most university level physics curricula defer quantum density matrices (and by extension tensor algebra) to more advanced classes. There's a lot of mathematical legwork required before you can actually make use of that.

Re: The tensor product, demystified (2018)

#30

Post by codethief »

consilient wrote: post #26
codethief wrote: post #24

Earlier quoted context omitted.

> > 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…

> 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 (m…

> > 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.

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