Suppose I have a quantum system $S$ ("system") with Hamiltonian $H_S$ and initial density matrix $\rho_S(0)$. I allow $S$ to interact with another system $P$ ("probe"), which has Hamiltonian $H_P$ and initial state $\rho_P(0)$, via an interaction Hamiltonian $H_I$. Then I measure $P$ in the basis of operator $\hat{Q}$.
Suppose my classical readout device is imperfect: if $P$ is in state $\lvert q \rangle$ which is an eigenstate of $\hat{Q}$ with eigenvalue $q$, then my readout device spits out numbers $q_\text{readout}$ according to a statistical distribution which depends on $q$. For example, we might have a case where the readout value is Gaussian distributed about $q$, i.e. $$P(q_\text{readout} | q) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp \left[ - \frac{(q - q)^2}{2 \sigma^2} \right] \, .$$$$P(q_\text{readout} | q) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp \left[ - \frac{(q_\text{readout} - q)^2}{2 \sigma^2} \right] \, .$$
Given the Hamiltonians $H_S$, $H_P$, $H_I$, the initial states $\rho_S$ and $\rho_P$, the function $P(q_\text{readout}|q)$, and a realized measured value $q_\text{readout}$, what concepts/approach does one use to find out the state of the combined system $S \otimes P$ after the measurement? How does the result change if the measured value $q_\text{readout}$ is ignored?
An allowed simplification would be to take the state of the combined system $\rho_{S P}$ after the interaction step as a known quantity. In other words, we're not so much interested in computing the evolution of $S \otimes P$ under the interaction $H_I$. However, but I think that whether or not $H_I$ commutes with $H_S$ winds up being important.
Notes
- While the example probability distribution (i.e. the Gaussian) is continuous, the spectrum of $\hat{Q}$, and/or the distribution $P(q_\text{readout}|q)$ may be discrete. I suppose it's even possible to have one continuous and the other one discrete!
Resources
- A Straightforward Introduction to Continuous Quantum Measurement by Jacobs and Steck.