"Interaction-Free" measurement involving statistical mixtures What happens in the standard "interaction free measurement" when the detector connected to a bomb is replaced with attenuation (where light is lost through a semi-transparent medium)?
Consider the following experiment:

I know already that if I put a single photon through a lossy medium with absorption A and transmission T = (1-A) - that my photon will end up in a statistical mixture of photon and vacuum:

So what happens in the example experiment with a lossy medium in a MZI (Mach-Zender interferometer)? 
I would think that since mixed states are classical probability states, I can break down this experiment into 2 separate cases: one in which the photon is absorbed by the lossy medium - and one in which the photon transmits through the lossy medium. 
But if we can break things down into these two cases, then we see that these two cases are exactly the cases considered for the "interaction free measurement" experiment:

Now here's the part that I'm particularly not sure about. In the case where the photon is not absorbed - will it experience perfect Mach-Zender interference? Or will there be a reduction in the strength of the probability amplitude associated with the photon traveling through that arm. 
I think it makes a lot of sense to use classical probability to separate the mixed state into two separate cases. So there would be a A% chance of the dud case, and a (1-A)% probability of the dud case.  
But, I've struggled a bit to fully represent the entire situation in a full-density matrix framework.
(And if anyone is interested, I can edit in my attempt at writing this experiment mathematically in this picture and highlight the parts that are not clear)
Any ideas on the proper way to treat this case?
EDIT: 
Here's an attempt at explaining my confusion in working out the math of this situation.
My state begins in a pure state, and I will expand it out as a density matrix so that I can include the part involving attenuation:
$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|1_L\rangle + |1_R \rangle) $$
$$|\Psi\rangle \langle \Psi | = \frac{1}{2}(|1_R \rangle \langle 1_R|+ |1_R \rangle \langle 1_L|+ |1_L \rangle \langle 1_R \rangle + |1_L \rangle \langle 1_L \rangle)$$
Where $|1_L\rangle$ and $|1_R\rangle$ describe the states in which the reflects and transmits from the first beam splitter. Now mode in the "right path" $|1_R\rangle$ will experience the lossy medium and turn into a statistical mixture:
$|1_R\rangle \langle 1_R| \rightarrow  A|0_R\rangle \langle 0_R|+T|1_R\rangle \langle 1_R|$
This is clear to me (single photons that become attenuated become mixed states, and I have confirmed this with a "quantum process operator" for attenuation). 
But what happens the coherence between the $|1_L\rangle$ and $|1_R\rangle$ after attenuation?
$|1_L\rangle \langle 1_R| \rightarrow ?$
I don't know how to prove it, but I can guess that it becomes: 
$|1_L\rangle \langle 1_R| \rightarrow (A|1_L\rangle \langle 0_R|+T|1_L\rangle \langle 1_R|)$
In this case if we plug this in our density matrix: 
$$\frac{1}{2}(A |0_R\rangle \langle 0_R|+T|1_R\rangle \langle 1_R|+ A|0_R \rangle \langle 1_L| + T|1_R \rangle \langle 1_L|+ A|1_L \rangle \langle 0_R \rangle + T|1_L \rangle \langle 1_R \rangle + |1_L \rangle \langle 1_L \rangle)$$
$$\frac{1}{2}( A(|0_R\rangle \langle 0_R| +|0_R \rangle \langle 1_L|+|1_L \rangle \langle 0_R \rangle) + T(|1_R\rangle \langle 1_R|+ |1_R \rangle \langle 1_L| + |1_L \rangle \langle 1_R \rangle) + |1_L \rangle \langle 1_L \rangle)$$
Now I want to consider the case only when the photon is transmitted, I'm not sure exactly how to express this as an operator, but I can project my state to dump the $|0_R\rangle$ terms (only considering the transmitted photons). The density matrix reduces to:
$$\frac{1}{2}(T(|1_R\rangle \langle 1_R|+ |1_R \rangle \langle 1_L| + |1_L \rangle \langle 1_R \rangle) + |1_L \rangle \langle 1_L \rangle)$$
Now if we incorporate the second beam splitter this density matrix becomes:
$$\frac{1}{2}\left( (\frac{1}{2}+\frac{3}{2}T)|D_1\rangle \langle D_1| + (\frac{1}{2}-\frac{1}{2}T)|D_2\rangle \langle D_2| + (\frac{1}{2} - \frac{1}{2}T)(|D_1\rangle \langle D_2|+|D_2\rangle \langle D_1|) \right)$$ 
Here I see when T = 1 I recover the normal MZI interference, but when T is not one, I see that I have imperfect MZI interference. 
I suspect that my intutition is correct, and that I not giving this system the proper mathematical treatment. 
Maybe the way that I am projecting the state into $I_L \otimes |1_R\rangle \langle 1_R |$ is incorrect? Maybe I have to renormalize somewhere in there?
 A: Mostly yes, you are correct.
There are several ways to do this calculation.
Let me start with what I find to be the simplest one, and does not require the use of density matrices at all.
Without using density matrices
$\newcommand{\ket}[1]{\lvert #1\rangle}$The main idea is to notice that a "lossy beamsplitter" is nothing but a "normal" beamsplitter in which you are ignoring one of the output arms. With this in mind, let me denote with $\ket{1},\ket2,\ket3$ the states in which the photon is in the first ("left" in the image), second ("right" in the image), or additional "hidden" spatial mode.
The state after the first beamsplitter is
$$\ket{\psi_1}=\frac{1}{\sqrt2}(\ket1+\ket2).$$
The "lossy" beamsplitter now evolves $\ket2$ into a balanced superposition of $\ket2$ and $\ket3$, leaving $\ket1$ untouched, so we get
$$\ket{\psi_1}\mapsto\ket{\psi_2}\equiv\frac{1}{\sqrt2}\ket1+\frac{1}{\sqrt2}(t\ket2+a\ket3),$$
where $\lvert a\rvert^2=A, \lvert t\rvert^2=T$.
The final beamsplitter acts by making $\ket1$ and $\ket2$ interfere but leaving $\ket3$ untouched, and therefore
$$\ket{\psi_2}\mapsto\ket{\psi_3}\equiv\frac{1}{2}\left(1+t\right)\ket1 + \frac{1}{2}\left(1-t\right)\ket2+\frac{a}{\sqrt2}\ket3.$$
The probability of seeing the photon in the second mode, that is, of $D_2$ clicking, is finally given by
$$p_2=\frac{(1-t)^2}{4}.$$
You can see that the photon does not interfere with itself as much as it would have done in a regular MZ interferometer, in which you would have had $p_2=0$.
With density matrix formalism
If we want to use density matrices for the calculation, then it is handier to use the notation $\ket{1_L},\ket{1_R}$ for when the photon is in the left/right mode, and $\ket0$ to denote the state in which no photon is in the interferometer.
Also, let me use the shortand notation $\mathbb P(\ket\psi)\equiv\ket\psi\!\langle\psi\rvert$.
After the first beamsplitter (but before the lossy one), the state is
$$\rho_1=\mathbb P\left[\frac{1}{\sqrt2}(\ket{1_L}+\ket{1_R})\right].$$
The lossy beamsplitter can now be modeled as a map of the form
$$\mathcal E(\rho)=\operatorname{Tr}_2(\mathcal U\rho\mathcal U^\dagger),$$
and $\mathcal U$ is the Hamiltonian used also in the previous section:
$$\mathcal U\ket{1_R}=t\ket{1_R}+a\ket{1_E}$$
for some extra mode $\ket{1_E}$. It follows that
$$\mathcal E(\ket{1_L}\!\langle 1_R\rvert)=a\ket{1_L}\!\langle 0_R\rvert+t\ket{1_L}\!\langle 1_R\rvert,$$
and the rest of the calculation proceed like you showed.
An easier way to perform this calculation with this formalism is as follows:
\begin{align}
\rho_2 &=\operatorname{Tr}_E\left(\mathbb P\left[ \frac{1}{\sqrt2}(\ket{1_L}+t\ket{1_R}+r\ket{1_E}) \right]\right) \\
&= \left(\frac{1+T}{2}\right)\mathbb P\left[\frac{\ket{1_L}+t \ket{1_R}}{\sqrt{1+T}}\right]
+ \frac{R}{2}\mathbb P[\ket{\text{vac}}]
\end{align}
computing the last step now is easy because we can simply work separately on the pure states whose mixture makes up $\rho_2$:
$$\rho_2\mapsto\rho_3=\left(\frac{1+T}{2}\right)\mathbb P\left[\frac{(1+t)\ket{1_L}+(1-t)\ket{1_R}}{\sqrt{2(1+T)}}\right]
+ \frac{R}{2}\mathbb P[\ket{\text{vac}}].$$
You can see in the expression for $\rho_2$ why you cannot simply consider the two different cases of "photon absorbed therefore zero interference" and "photon transmitted therefore full interference".
What you actually have is a mixture of the photon being absorbed, and the photon being transmitted and therefore only partially interfering.
This is not surprising, because you cannot simply make the photon either stay there or "magically vanish". The only way to get something equivalent to the lossy beamsplitter is to have the photon "leak" away with some probability, but such an operation is fundamentally always unitary, and therefore evolves $\ket{1_R}$ into a superposition of $\ket{1_R}$ and something else. Because of this $\ket{1_L}$ cannot fully interfere with $\ket{1_R}$ anymore, because $\ket{1_R}$ is already interfering with something else (even though you are not looking at this something else). This is an effect analogous to the so-called monogamy of entanglement, but for coherences.
