I Phase of |0> when it appears in a product

[Swamp Thing]

In a beam splitter with one vacuum input, the output looks like $\frac{1}{\sqrt 2}(|1_A\rangle +i|0_B \rangle)\frac{1}{\sqrt 2}(-i|1_A\rangle +|0_B \rangle)$.
If there is some further processing, the vacuum $i|0_B\rangle$, along with the i , could end up multiplied with some non-vacuum term.

Do we need to keep track of that "i" that originates from such a vacuum term? If so, what is the physical interpretation?

 

[kith]

Please make it a habit to cite your sources and explain your notation.

The state you have given doesn't make sense to me. Are you really familiar with what a products of ket vectors signifies?

 

[Swamp Thing]

Sorry about that, perhaps a bit sloppy with the subscripts there. I was referring to something like eqn. 7 https://www.google.co.in/url?q=http...YbSzkg&usg=AFQjCNF6nL9nD39y2OmysV_AMNPtnsjDrQ

 

[Swamp Thing]

Actually, they do keep track of the vac term in that reference so I guess that answers my question.