relation between tensor product and matrices

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relation between tensor product and matrices

Mathematics
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i think im just confused on the definition of the tensor product
I am assuming that is the right matrix

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ok i think this is useful the book defines what these basis vectors look like
we can make tensor product between two Hilbert spaces
it said that in earlier examples \[|00> = (1,0,0,0)^T, |01> = (0,1,0,0)^T , |10> = (0,0,1,0)^T , |00> = (0,0,0,1)^T \] there are column vectors like that
so is assuming \[|0> = (1,0)^T and |1> =(0,1)^T \]right
the basis of the tensor product \[{H_1} \otimes {H_2}\] is the set: \[{e_i} \otimes {f_j}\] where \(e_i\) is the basis for \(H_1\), and \(f_j\) is the basis for \(H_2\)
of course \(e_i\) are many vectors, namely \(i=1,...,n\) if the dimension of \(H_1\) is \(n\)
are we varying i and j for ei and fj for all the basis in H1 and H2 and adding them together to get some particular H1 and H2
okay i see what u mean
yes! a generic vector of \[{H_1} \otimes {H_2}\] can be written as follows: \[\psi = \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{e_i} \otimes {f_j}} \right)} } \]
an observable \(A_1\) of the space \(H_1\) can be written as follows: \[{A_1} \otimes 1\] where \(1\) is the identity of \(H_2\)
okay so thsi is is like writing A1 in our new dimension space of H1 * H2
correct! Similarly for anobservable \(B_2\) of \(H_2\): \[1 \otimes {B_2}\] using Dirac notation, the tensor product between states of a particle with spin 1/2 and another particle with spin +1, can be written as follows: \[\left| { + \frac{1}{2}, + 1} \right\rangle \]
namely: \[\left| { + \frac{1}{2}} \right\rangle \oplus \left| { + 1} \right\rangle \doteq \left| { + \frac{1}{2}, + 1} \right\rangle \]
as you wrote before
oops.. \[\left| { + \frac{1}{2}} \right\rangle \otimes \left| { + 1} \right\rangle \doteq \left| { + \frac{1}{2}, + 1} \right\rangle \]
so |00> being a state is really just |0,0>
yes!
here is the action of the observable: \[{A_1} \otimes {A_2}\]: \[\left( {{A_1} \otimes {A_2}} \right)\psi \doteq \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{A_1}{e_i}} \right) \otimes \left( {{A_2}{f_j}} \right)} } \]
I think tat your matrix is the representative of a quantum observable, for example, which belongs to \(H1\) or to \(H2\).
for example it can be one of the Pauli matrices
when we are working with the product tensor of two Hilbert spaces, we refer to the observables of each spaces involved in such tensor product

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