Q3: States in tensor product space that are not simple product states If you stare at the state (0) in Q2 (b) for sometime after putting on your thinking hal, you will notice something strange! This state cannot be factorized into a tensor product of linear combination of states from the two spaces. There is a name of such states, they are called entangled. To see this in concrete terms consider: NA) = aula) fi) + a12le1) If2) + a21lea) 1/1) + azalea) Ifa). Here bases {e} and {f} belong to two spaces V and W. A state is not entangled if you can write it as follows: === |A) anlei) fi) + a12le1) |f2) + a21le2) |fi) + a2zle2) |f2) = (a1lei) + a2e2)) (bilfi) +b2lff2)) (1) Here (a1, a2), (b1, b2) are coefficients of expansions in the V and W spaces respec- tively and (a11, a12-021 and a22) are coefficient of expansions in VW. We would like to find a condition on the aj coefficients to determine if a state is entangled or not. For this, we make the following matrix out of the a, coefficients: A = 011 012 021 022 (a) Now using Eq. (1) find out what value of determinant of A corresponds to product states or states that are not entangled. (b) Using the condition determined in (a), find out if the following state is entangled or not: (*-|-|-+|(+)-(0) (c) Using the condition found for the determinant in (a), write the general form of the product state or the unentangled state.