How can one impliment Bennett's partial measurement onto a binomial subspace for state distillation?

Question

I'm reading the seminal paper on entanglement distillation by Bennett et. al.

The idea is that Alice and Bob have $n$ identical copies of an imperfect (but pure) Bell state. The initial state is therefore:

$$ \Psi(A,B) = \prod_{i=1}^n(\cos \theta |0_A0_B \rangle + \sin \theta |1_A1_B \rangle) $$

Naturally, when you expand this out, you'll get $2^n$ terms with $n+1$ binomial coefficients. The authors state that Alice (or Bob) can perform "an incomplete Von Neuman measurement projecting the state into one of the $n+1$ orthogonal subspaces."

For the two qubit case, it's clear that Alice could perform a parity measurement on her qubits to project onto $\{|0_A0_B\rangle,|1_A1_B\rangle\}$ or $\{|0_A1_A\rangle, |1_A0_B\rangle\}$. My question is, how would you implement such a projection in the $n$ qubit case?

Detailed responses are much appreciated.

Answer 1

Alice has to project onto the subspace with some number $n$ of $1$'s in their state $\lvert x_1,x_2,\dots,x_N\rangle$. This can be done by first running a circuit which adds the value of all $x_i$s and stores it in an ancilla $\lvert a\rangle$. (I.e., this is a classical circuit doing $a\to a+1$ controlled by the values of $x_i$.) Then, run a classical circuit which checks whether $a=n$ and outputs the result on another ancilla $\lvert b\rangle$. Then, Alice measures $\lvert b\rangle$ in the computational basis. If she gets $1$, she has projected onto that subspace.