How Do Ket Vectors Change When Working With Bra Tensor Products?

[QITStudent]

Hey guys, I need some help understanding some very basic quantum manipulation techniques.

If I am working out the bra version of a tensor product on some ket vectors, do the two ket vectors change from |v1>|v2> to <v1|<v2|, or to <v2|<v1|.

Or, to put in context, if I measure an operator with an eigenvector AxB on state |v1>|v2> , is the probability that I get the eigenvalue associated with this eigevector

<v1|<v2|AxB|v1>|v2> or <v2|<v1|AxB|v1>|v2>?​

Many thanks for looking through this with me

 

[physicus]

If |v_1\rangle|v_2\rangle is a state is the tensor product \mathcal{H}_1 \otimes \mathcal{H}_2 of two Hilbert spaces, then the corresponding bra is in the tensor product of the dual spaces of the Hilbert spaces, namely \mathcal{H}_1^* \otimes \mathcal{H}_2^*. Thus, it is \langle v_1|\langle v_2|.
Btw, a more precise notation is |v_1\rangle\otimes|v_2\rangle\equiv|v_1\rangle|v_2\rangle and \langle v_1|\otimes\langle v_2|\equiv\langle v_1|\langle v_2|