Thank you both for your response. I managed to clear the logic after some time. It's really annoying to get stuck like that. I started thinking in terms of operators and then, progress.
I had a really hard time interpreting what this really means in terms of vector spaces...
So I get
Tr(ρABlog(ρA\otimesρB)) = Tr(ρAB(logρA\otimesIB + IA\otimeslogρB))
This is obviously wrong if you exploit the trace of a tensor product and the trace of an identity matrix which is just the number of dimensionality of each vector space. Where am I loosing it?
Thank you for your response, it was very helpful. Back to basics of tensor product relations. My next problem is with the interpretation of ρAB. I do not want to make the assumption that ρAB = ρA\otimesρB. However, this assumption is made in the literature without comment at times, while one is...
I'm having problems understanding the trace of tensor products when the density matrix is expressed in its reduced density operators. The proof of subadditivity is quite simple.
S(ρAB||ρA\otimesρB) = Tr(ρABlogρAB) - Tr(ρABlogρA\otimesρB) = TrAB(ρABlogρAB) - Tr(ρAlogρA) - Tr(ρBlogρB)
This...