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In response to some remarks made in the thread "How do particles become entangled?", as well as a number of private messages I have received, I feel there is some need to post some information on the notion of a "tensor product".

Below, a rather intuitive look at the idea of the "tensor product" is taken. For simplicity, the vector spaces involved are assumed to be finite-dimensional. The infinite-dimensional case can be accommodated with only some minor amendments to the presentation.

(Note: The usual symbol for thetensor productis an "x" with a "circle" around it, but below I will use the symbol "x".)

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Let U and V be finite-dimensional vector spaces overCwith bases {u_{i}} and {v_{j}}, respectively. For each u_{i}and v_{j}, define an object "u_{i}x v_{j}", and construe the full collection of these objects to be abasisfor a new vector space W. That is,

W ≡ {∑_{ij}α_{ij}(u_{i}x v_{j}) | α_{ij}ЄC} ,

where, bydefinition,

if∑_{ij}α_{ij}(u_{i}x v_{j}) = 0 ,thenα_{ij}=0 for all i,j .

The above then makes W a vector space overCsuch that

Dim(W) = Dim(U)∙Dim(V) .

However ... had we chosen adifferentset of basis vectors for U or V, then the vector space W thereby obtained would be formallydistinctfrom the one obtained above. There would be no way to 'link' the bases for each of the two W's.

Let us now introduce some additional 'structure' on the operation "x", such that all W's obtained by the above construction will be formallyidenticalno matterwhichbases are chosen for U and V. Specifically, weextendthe definition of "x" to be, thus allowingbilinearanyvector of U to be placed in the left "slot", andanyvector of V to be placed in the right "slot". We do this as follows:

For any u,u'ЄU , v,v'ЄV , and αЄC, let

(u + u') x v = (u x v) + (u' x v) ,

u x (v + v') = (u x v) + (u x v') ,

α(u x v) = (αu) x v = u x (αv) .

Now all W's areoneandthe same.

The next thing we need is an inner product <∙|∙> on W. Let <∙|∙>_{1}and <∙|∙>_{2}be the inner products on U and V, respectively. Then, for any u x v and u' x v' Є W , define

<u x v|u' x v'> ≡ <u|u'>_{1}∙<v|v'>_{2}.

Finally, extend <∙|∙> to thewholeof W by "antilinearity" in thefirstslot and "linearity" in thesecondslot.

It now follows that <∙|∙> is an inner product on W.

Moreover, if {u_{i}} and {v_{j}} areorthonormalbases of U and V respectively, then {u_{i}x v_{j}} is anorthonormalbasis of W.

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# On the Tensor Product

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