Commutator, where have I gone wrong?

[raisin_raisin]

This is for the Pauli Matrics 0 and 1 are different Hilbert Spaces
$$\left[(I-Z)_{0}\otimes(I-Z)_{1} , Y_{0}\otimes Z_{1}\right]$$
$$=\left((I-Z)_{0}\otimes(I-Z)_{1}\right)\left(Y_{0}\otimes Z_{1}\right)-\left(Y_{0}\otimes Z_{1}\right)\left((I-Z)_{0}\otimes(I-Z)_{1}\right)$$
$$=\left((I-Z)_{0}Y_{0}\otimes(I-Z)_{1} Z_{1} - Y_{0}(I-Z)_{0} \otimes Z_{1}(I-Z)_{1}$$
$$= (Y_{0} -Z_{0}Y_{0}) \otimes (Z-I)_{1} - ((Y_{0} + Y_{0}Z_{0})\otimes (Z-I)_{1}$$
$$= (Y_{0} -Z_{0}Y_{0}) \otimes (Z-I)_{1} - ((Y_{0} - Z_{0}Y_{0})\otimes (Z-I)_{1} =0$$

I think this is wrong because I have done it long hand (i.e multiplying the matrices) and also I really, really don't want zero for an answer :).

Thanks

 

[dioib]

Fourth line, second term, I think the plus should be turned to minus..

 

[raisin_raisin]

Fourth line, second term, I think the plus should be turned to minus..

Thanks for your reply, sorry I still can't see it though, could you explain why?
Thanks again.

 

[gabbagabbahey]

Thanks for your reply, sorry I still can't see it though, could you explain why?
Thanks again.

You really don't see why

$$-Y_0(I-Z)_0\neq-(Y_0+Y_0Z_0)$$

?

 

[raisin_raisin]

You really don't see why

$$-Y_0(I-Z)_0\neq-(Y_0+Y_0Z_0)$$

?

Oops, thanks!