Projective measurement using two mode squeezed state? Let me define two mode squeezed states as $ \left | \xi \right>_n=\exp\left(\xi \hat{a}^\dagger \hat{b}^\dagger-\hat{a} \hat{b} \xi^\star\right)\left | n,0 \right>$ where $\left|n,0\right>$ is the number state with $n$ photons in mode 1 and none in mode 2. Now, let me define a projective
measurement $\{\left | \xi \right>_n\left<\xi\right|,\mathbb{I}-\left | \xi \right>_n\left<\xi\right|\}$. Is it possible to implement this projective measurement in the lab? have you seen any paper which has been discussed such a measurement?
 A: If you are happy with destructive measurements, this can be done in principle in the lab. Let $\rho$ be the input state, and write $|\xi\rangle = S|n,0\rangle$.


*

*Undo the squeezing transformation $S$, i.e., transform $\rho'=S^\dagger\rho S$. Note that $S^\dagger$ is just another squeezing transformation which can in principle be implemented in the lab.   Correspondingly, this means that now you need to perform the measurement $\{|n,0\rangle\langle 0,n|,I-|n,0\rangle\langle 0,n|\}$. 

*Use photon number resolving detectors to measure both modes.  Again, this is something which can be done in principle (to a certain accuracy).  Count the outcome $(n,0)$ towards the first outcome, and all others towards the second.
Formally speaking, what you are looking for is the probability $p_\xi$ for the first outcome, which we can rewrite as
$$
p_\xi = \langle \xi|\rho|\xi\rangle
=\langle n,0|S^\dagger \rho S|n,0\rangle
 = \langle n,0| \rho'|n,0\rangle\ ,
$$
where the last term is exactly what the proposed measurement determines.
