Counterfactual Quantum trajectories: Given that my photo detector clicked, what would have happened with a different type of a detector?

Abstract

Quantum trajectory theory, also known as quantum state filtering, enables us to estimate the state of a quantum system conditioned on the measurement we perform. In cases where we measure the fluorescence from a driven two-level atom with an inefficient photo detector, the conditioned state of the atom is generally not pure, except immediately after a photon detection since then we know that the atom is in the ground state. For the detection schemes such as homodyne measurement the state is never pure since it gives rise to quantum state diffusion and not quantum state jumps. In these scenarios questions can be asked as: 
Given that I did use a photo detector and did see a particular time sequence of detections, how would the atom have behaved if instead I had chosen to measure the fluorescence using a homodyne detection scheme?
These questions are called counterfactual questions. Counterfactuality has played significant roles and has a long history in philosophy of trying to make sense of such questions. There are various approaches in how to evaluate such counterfactual questions. One such influential and attractive approach is that of David Lewis where he has a generalized analysis for counterfactuals. 

Analysis 2. A counterfactual " If it were that A, then it would be that C" is (non-vacuously) true if and only if some (accessible) world where both A and C are true is more similar to our actual world, overall, than is any world where A is true, but C is false. 

To evaluate our atom counterfactual problem we use his approach under the two main considerations:

1) To avoid any big, widespread, diverse violations of law. 

The antecedent of our counterfactual ( the thing that we propose to change) is our choice of measurement and that is within the laws of Quantum theory. 

2) Maximize the spatiotemporal region throughout which perfect match of particular fact prevails. 

Thus, in evaluating the counterfactual problem, any information not collected by the primary detector can be modeled as photon absorptions and should be held fixed, under the above consideration. 

Denoting these other 'clicks' , described by some list of times M, and using the actual observed record of photon-counts denoted by the list of times, N, we can calculate a conditional probability of M given N. Following this we evaluate a second conditional probability with which we are most likely to observe a homodyne record over time ,Y , if we were ( counterfactually) making a homodyne measurement given M ( since M remains fixed). Conditioning the actual state (the state conditioned on all the measurements in the counterfactual case) of the atom on these probabilities and performing an ensemble average over all possible M and Y would give us the best ( relative to trace-mean-square-deviation cost function) estimate of the counterfactual state which answers the question.