Assume Alice has a set of $N$ non-zero complex numbers $\mathcal{A}=\{A_1, A_2, \ldots A_N\}$ . Bob does not have any magnitude/phase information on these numbers.
Bob is allowed to associate another set of unit magnitude complex numbers $\mathcal{B}=\{B_1, B_2, \ldots B_N\}$ in a one-to-one manner with numbers in $\mathcal{A}$. Alice randomly pick $k$ numbers indexed by $j_1,j_2,\ldots, j_k$ from $\mathcal{A}$ . She does the following operation and give it to Bob:
$\hspace{9cm}Y=\sum_{i=1}^{k}B_{j_i}A_{j_i}.$
Bob's aim is to gather information on the value of $k$, i.e., the number of complex numbers from set $\mathcal{A}$ being used in the summation $Y$.
Bob can operate in stages. i.e., in each stage he can specify a different $\mathcal{B}$ and Alice uses that to compute $Y$. Thus, after each stage Bob will have a different weighted summation result $Y$. Note that $j_1,j_2,\ldots, j_k$ remains same across stages.
My question: What is the maximum information about $k$ Bob can achieve in 2 stages?
My attempt: I can distinguish $k=0$, $k=1$, and $k>1$ cases using 2 stages if I choose $\mathcal{B}=\{1,1,....1\}$ in stage 1 and $\mathcal{B}=\{e^{mj2\pi/N}\}_{m=1}^{N}$ in stage 2. This is because, if $k=1$, the $Y$ values will have the same magnitude in the 2 stages. This property does not hold for $k>1$.
Can someone tell me if I can do better (gather more info about $k$) in 2 stages? Or achieve the same ($k=0$, $k=1$, and $k>1$)-classification in just 1 stage?