I am confused about the order of operations with kets, as well as when one can permute the order of the kets. Question 1: When I have a term as follows: a^2|+>|d+><+|<d+| ...why exactly am I able to switch the order of the bras and kets as follows? a^2|+><+|d+><d+| I get confused about when we are able to change the order of the bras/kets and when we have to preserve their order. Question 2: And why don't we look at the previous state and say that <+|d+> = 0? In that case the entire term would be zero, which is obviously not the case. I get that the states can't be called orthogonal because they refer to different objects, and hence to different Hilbert spaces...is that the correct answer? What, then, is <+|d+> equal to? Those are my questions. For the setup to this question, read on...This example is taken from Zurek 1991 (Decoherence and the Transition from Quantum to Classical). A particle in state |K> = a|+> + b|-> interacts with a quantum detector/system, and becomes |K>= a|+>|d+> + b|->|d-> where |d+> and |d-> stand for the up and down states of the quantum detector. (we assume the detector is not macroscopic, so we avoid any confusion about the existence of Macroscopic Quantum States). When I create the density matrix for this state, I get |K><K| = a^2|+>|d+><+|<d+| + b^2|->|d-><-|<d-| +....plus a couple other off-diagonal terms.