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dan815
 one year ago
relation between tensor product and matrices
dan815
 one year ago
relation between tensor product and matrices

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dan815
 one year ago
Best ResponseYou've already chosen the best response.0i think im just confused on the definition of the tensor product

dan815
 one year ago
Best ResponseYou've already chosen the best response.0I am assuming that is the right matrix

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ok i think this is useful the book defines what these basis vectors look like

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2we can make tensor product between two Hilbert spaces

dan815
 one year ago
Best ResponseYou've already chosen the best response.0it 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

dan815
 one year ago
Best ResponseYou've already chosen the best response.0so is assuming \[0> = (1,0)^T and 1> =(0,1)^T \]right

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2the 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 \(H_2\)

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2of course \(e_i\) are many vectors, namely \(i=1,...,n\) if the dimension of \(H_1\) is \(n\)

dan815
 one year ago
Best ResponseYou've already chosen the best response.0are 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

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2yes! a generic vector of \[{H_1} \otimes {H_2}\] can be written as follows: \[\psi = \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{e_i} \otimes {f_j}} \right)} } \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2an observable \(A_1\) of the space \(H_1\) can be written as follows: \[{A_1} \otimes 1\] where \(1\) is the identity of \(H_2\)

dan815
 one year ago
Best ResponseYou've already chosen the best response.0okay so thsi is is like writing A1 in our new dimension space of H1 * H2

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2correct! 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: \[\left { + \frac{1}{2}, + 1} \right\rangle \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2namely: \[\left { + \frac{1}{2}} \right\rangle \oplus \left { + 1} \right\rangle \doteq \left { + \frac{1}{2}, + 1} \right\rangle \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2as you wrote before

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2oops.. \[\left { + \frac{1}{2}} \right\rangle \otimes \left { + 1} \right\rangle \doteq \left { + \frac{1}{2}, + 1} \right\rangle \]

dan815
 one year ago
Best ResponseYou've already chosen the best response.0so 00> being a state is really just 0,0>

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2here is the action of the observable: \[{A_1} \otimes {A_2}\]: \[\left( {{A_1} \otimes {A_2}} \right)\psi \doteq \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{A_1}{e_i}} \right) \otimes \left( {{A_2}{f_j}} \right)} } \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2I think tat your matrix is the representative of a quantum observable, for example, which belongs to \(H1\) or to \(H2\).

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2for example it can be one of the Pauli matrices

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.2when 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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