I would like to solve the following problem. I have a $N\times 1 $ complex vector, for example with $N=6$:
$$v=(a \; b \; c\; d \; e \;f)^T$$
and its complex conjugate
$$\bar{v}=(\bar{a} \; \bar{b} \; \bar{c}\; \bar{d}\; \bar{e} \;\bar{f})^T$$
I can access any value of the product of a subset of $v$ with the complementary subset of $\bar{v}$. For example:
$$(a+b+c)(\bar{d} + \bar{e} + \bar{f})=\alpha_1\\ (a+b+d)(\bar{c} + \bar{e}+\bar{f})=\alpha_3\\ (a+c+d)(\bar{b} + \bar{e}+\bar{f})=\alpha_2\\...$$
In each case, the subsets are of size $N/2$, and each element appears only once. If we consider the elements of $\bar{v}$ as unknowns as well, we have $2N$ unknowns, and if I'm correct there are $N$ choose $N/2$ (or $N!/(N/2!)^2$) such products. Therefore we can produce enough equations to close the system starting from $N=6$.
Is there a general way to find $v$ form the $\alpha_i$?
I would like to find a way to automatise this in code for any $N$, where the list of $\alpha_i$ is given, with for each $\alpha_i$ some information encoding the corresponding subset (for example a vector of 1 and 0 indicating which elements are taken from $v$ and which from $\bar{v}$, respectively). I tried to express the problem in a compact algebraic way but I couldn't put it in a form that would lend itself to common functions (I work on Matlab).