Hmm, I always disliked coordinate approach to (multi)linear algebra. To me, the notion of a tensor product clicked with the following definition:
Tensor product v₁⊗v₂ of vectors form V is bilinear form on the dual space V* defined by v₁⊗v₂(ω₁,ω₂) = v₁(ω₁)v₂(ω₂). Tensor product V⊗V is space of all bilinear forms on V*,
This can easily be generalized to products of forms (covariant tensors), and mixed tensor products. Getting coordinate representation in basis (E₁ ... Eₙ) is trivial: (v₁⊗v₂)ₖ,ₗ = v₁⊗v₂(εₖ,εₗ) where εₖ is dual to Eₖ...
> Tensor product V⊗V is space of all bilinear forms on V*
This is only true for finite-dimensional V. In general bilinear forms on V* are linear maps `V*⊗V* -> F` (where F is the base field) which is isomorphic to `(V*⊗V*)*`. For infinite-dimensional V this is much larger than `V⊗V`.
The general characterization is that `V⊗W` is the vector space (and associated linear map `⊗: V x W -> V⊗W`) such that for every bilinear map `h: V x W -> T` there's a unique linear g such that `h(v,w) = g(v⊗w)`. Intuitively, it adds "just enough" elements to `V x W` to encode arbitrary bilinear maps as linear ones.
I always dislike the unnecessary use of dual of dual. We should define tensor product as a quotient space of all linear combinations of v₁⊗v₂ as in https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_s...
The advantage of not using dual of dual is that it generalizes correctly to infinite dimensional vector spaces and modules.
It was a bit annoying that the first few examples give a flattened tensor product. It's explained at the end, but the dimensions are at least as important as the values and it feels wrong to skip that (isomorphism notwithstanding).
Some additional ressources on tensors I've found useful:
1) A video[0] by Michael Penn exposing this idea of "tensor product of vector spaces." It's close to what is presented in this blog post, but more rigorous/complete.
2) Two videos[1][2] by Frederic Schuller. They are each from of bigger courses (resp. ~Differential Geometry for Physics and Quantum Mechanics), but I think they are self-contained enough to be intelligible. They both present tensors in different settings, so one will have to work a little to unify all this. I like in particular how in [1] he really takes the time to first distinguish between all the different tensor product (of space, of vectors, of operators): the usual notation/terminology can be needlessly confusing for beginners.
> 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.
> 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.
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.
Tensor product is basically the mathematical way to express if condition.
Say you have 2x2 matrix A, B, C.
Any arbitrary component within the Tensor(A, B, C) is:
if A =(a1,a2) and B=(b1,b2) and C=(c1,c2): this value
The coordination is concatenation of matching dimension:
(a1b1c1, a2b2c2)
While in many computational natural science, people using tensor product to store and manipulate data. Also it is how the mathematical equations being written on paper.
But in computer science's perspective, dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system. System that their matrix is dominated by zero. Also it would be super clear if people just write if-then pseudocode instead of cryptic half-bake tensor expressions. People tend to invent their own notation while writting paper.
> Tensor product is basically the mathematical way to express if condition.
Not really, no. The way mathematicians actually express if conditions is with the word "if". The obvious pointlessly formal way to do it is with a pair of functions `ThingConditionedOn -> {0, 1}` and `{0, 1} -> Result`, but why would you?
> dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system.
Tensors are not their components, any more than locations are their coordinates. Whether you choose a sparse or dense (or symbolic) encoding does not change the object being encoded.
Very nice, but there’s a technical error in the section “in quantum physics”. A density matrix as described does not represent the state of two systems, just one system. I think the author is confusing a vector in H⊗H with a bounded operator on H.
ubavic|2 years ago
Tensor product v₁⊗v₂ of vectors form V is bilinear form on the dual space V* defined by v₁⊗v₂(ω₁,ω₂) = v₁(ω₁)v₂(ω₂). Tensor product V⊗V is space of all bilinear forms on V*,
This can easily be generalized to products of forms (covariant tensors), and mixed tensor products. Getting coordinate representation in basis (E₁ ... Eₙ) is trivial: (v₁⊗v₂)ₖ,ₗ = v₁⊗v₂(εₖ,εₗ) where εₖ is dual to Eₖ...
consilient|2 years ago
This is only true for finite-dimensional V. In general bilinear forms on V* are linear maps `V*⊗V* -> F` (where F is the base field) which is isomorphic to `(V*⊗V*)*`. For infinite-dimensional V this is much larger than `V⊗V`.
The general characterization is that `V⊗W` is the vector space (and associated linear map `⊗: V x W -> V⊗W`) such that for every bilinear map `h: V x W -> T` there's a unique linear g such that `h(v,w) = g(v⊗w)`. Intuitively, it adds "just enough" elements to `V x W` to encode arbitrary bilinear maps as linear ones.
howling|2 years ago
Y_Y|2 years ago
mbivert|2 years ago
1) A video[0] by Michael Penn exposing this idea of "tensor product of vector spaces." It's close to what is presented in this blog post, but more rigorous/complete.
2) Two videos[1][2] by Frederic Schuller. They are each from of bigger courses (resp. ~Differential Geometry for Physics and Quantum Mechanics), but I think they are self-contained enough to be intelligible. They both present tensors in different settings, so one will have to work a little to unify all this. I like in particular how in [1] he really takes the time to first distinguish between all the different tensor product (of space, of vectors, of operators): the usual notation/terminology can be needlessly confusing for beginners.
[0]: https://www.youtube.com/watch?v=K7f2pCQ3p3U
[1]: https://www.youtube.com/watch?v=OZ1WCyJmjgo
[2]: https://www.youtube.com/watch?v=4l-qzZOZt50
lisper|2 years ago
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|2 years ago
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.
xeonmc|2 years ago
sigmoid10|2 years ago
hackandthink|2 years ago
"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)
akasakahakada|2 years ago
Say you have 2x2 matrix A, B, C.
Any arbitrary component within the Tensor(A, B, C) is:
The coordination is concatenation of matching dimension: (a1b1c1, a2b2c2)While in many computational natural science, people using tensor product to store and manipulate data. Also it is how the mathematical equations being written on paper.
But in computer science's perspective, dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system. System that their matrix is dominated by zero. Also it would be super clear if people just write if-then pseudocode instead of cryptic half-bake tensor expressions. People tend to invent their own notation while writting paper.
consilient|2 years ago
Not really, no. The way mathematicians actually express if conditions is with the word "if". The obvious pointlessly formal way to do it is with a pair of functions `ThingConditionedOn -> {0, 1}` and `{0, 1} -> Result`, but why would you?
> dealing with tensor matrix is simply a waste of memory since 90% of the time people are dealing with sparse system.
Tensors are not their components, any more than locations are their coordinates. Whether you choose a sparse or dense (or symbolic) encoding does not change the object being encoded.
hackandthink|2 years ago
The author seems to be related to https://www.sandboxaq.com/.
"SandboxAQ leverages the compound effects of AI and Quantum technologies (AQ) to solve hard challenges impacting society."
This reads like a scam (don't know anything about it).
ziofill|2 years ago
saberience|2 years ago
uptownfunk|2 years ago
tempodox|2 years ago