Quantum estimation of a parameter affected by the measurement

Question

Usual textbook quantum mechanics tells us that observable quantities are associated with Hermitian operators. However, there are many instances where we seek to measure parameters encoded in the quantum state that do not have an associated Hermitian observable. This is where quantum parameter estimation (QPE) theory enters. To my knowledge, there are essentially two types: frequentist and Bayesian quantum parameter estimation. The celebrated quantum Cramér-Rao bound is perhaps the most famous result in frequentist, asymptotic QPE.

In the context of a single copy of the system whose state contains some parameter I want to learn, clearly the iid (independent, identically distributed) paradigm drops. Further, it is clear that the process of: i) coupling the system to my probe, ii) letting them interact, iii) collect my probe, iv) measure my probe will inevitably "affect" the system, and therefore not only the parameter's assumed pre-value, but even its functional dependency on the state.

All this makes me wonder: is there a "quantum parameter estimation" framework that allows for these type of situations? My initial line of thought was: Assuming someone has prepared the state $\rho_\lambda$ in such a way that the parameter $\lambda$ has a true, fixed, real value (within some acceptable error, naturally (real numbers are not physical from an operational point of view), and even assuming that I know the functional encoding of the parameter, could I view the fact that upon the first probing I will inevitably change the state to something else as making the initially local estimation fuzzier? That is, is there a theory of quantum parameter estimation where limited resources (i.e. single copy and perhaps some restriction on the type of available POVM) are formalised as a "noisy" transition from local estimation to global estimation?

Answer 1

When doing a quantum measurement on a single quantum system, you will inevitably project the state onto one of the basis of the vector basis. This has nothing to do with QPE. Then if you want to know the state in details, you would need to do several measurements from systems "created with the same process", regardless the operator is Hermitian or no.

Consider the Stern-Gerlach experiment where you project the spin over z-direction. For one atom you can only observe it going up or down. To know the state of the input you would need to observe many of them. For the spin it is relatively easy but if you want to measure non-Hermitian states this is more challenging. (see this post). A good example is $\hat{N}=\hat{a}^\dagger \hat{a}$. You need several measurements to determine the distribution, and for instance a Bayesian approach would allow you to guess the distribution with more and more precision as you measure states of new molecules. But for this you would need to have an intuition of the distribution, its mathematical form as a function of some physical parameters you measure (is that what you mean by "functional encoding of the parameter" ?). An example of determination of a quantum operator using QPE and POVM can be found here. This make sense when you consider the fidelity of quantum gates, you want to determine the result with the highest certainty possible with a limited amount of measurements.

Of course you would also need to consider any lmitations such as decoherence, quantum noise ... Check the references of the last paper.