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

1. dan815

|dw:1443895296128:dw|

2. dan815

i think im just confused on the definition of the tensor product

3. dan815

I am assuming that is the right matrix

4. dan815

ok i think this is useful the book defines what these basis vectors look like

5. Michele_Laino

we can make tensor product between two Hilbert spaces

6. dan815

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

7. dan815

so is assuming $|0> = (1,0)^T and |1> =(0,1)^T$right

8. Michele_Laino

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 $$H_2$$

9. Michele_Laino

of course $$e_i$$ are many vectors, namely $$i=1,...,n$$ if the dimension of $$H_1$$ is $$n$$

10. dan815

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

11. dan815

okay i see what u mean

12. Michele_Laino

yes! a generic vector of ${H_1} \otimes {H_2}$ can be written as follows: $\psi = \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{e_i} \otimes {f_j}} \right)} }$

13. Michele_Laino

an observable $$A_1$$ of the space $$H_1$$ can be written as follows: ${A_1} \otimes 1$ where $$1$$ is the identity of $$H_2$$

14. dan815

okay so thsi is is like writing A1 in our new dimension space of H1 * H2

15. Michele_Laino

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: $\left| { + \frac{1}{2}, + 1} \right\rangle$

16. Michele_Laino

namely: $\left| { + \frac{1}{2}} \right\rangle \oplus \left| { + 1} \right\rangle \doteq \left| { + \frac{1}{2}, + 1} \right\rangle$

17. Michele_Laino

as you wrote before

18. Michele_Laino

oops.. $\left| { + \frac{1}{2}} \right\rangle \otimes \left| { + 1} \right\rangle \doteq \left| { + \frac{1}{2}, + 1} \right\rangle$

19. dan815

so |00> being a state is really just |0,0>

20. Michele_Laino

yes!

21. Michele_Laino

here is the action of the observable: ${A_1} \otimes {A_2}$: $\left( {{A_1} \otimes {A_2}} \right)\psi \doteq \sum\limits_i {\sum\limits_j {{\psi _{ij}}\left( {{A_1}{e_i}} \right) \otimes \left( {{A_2}{f_j}} \right)} }$

22. Michele_Laino

I think tat your matrix is the representative of a quantum observable, for example, which belongs to $$H1$$ or to $$H2$$.

23. Michele_Laino

for example it can be one of the Pauli matrices

24. Michele_Laino

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