Quantum weak measurement parameter estimation vs Classical Estimation

[tworitdash]

I am not an expert in quantum theory. I want to carry out some parameter estimation on a set of data I have. I have a model for the data with the parameter(s) of interest as variable(s).

The data available is sporadic, meaning non-statistical or techniques involving no prior knowledge on the parameter of interest may result in very bad estimate of the parameter. For example, a Doppler frequency estimator with DFT with very sparse data in time. So, we can call the bad estimates for the parameters as bad(weak) estimates. I am currently looking at Bayesian principles with Markov Chain Monte Carlo techniques to estimate a probability distribution of parameters.

However, I came across parameter estimation with quantum weak measurement recently. I am curious how can I translate what people use in the parameter estimation for the weak measurements in a classical setting.

I found this paper for the same.

Hofmann, H. F. (2011). On the estimation of interaction parameters in weak measurements. AIP Conference Proceedings, 1363(October 2011), 125–128. https://doi.org/10.1063/1.3630162Here, they say that the estimation problem can be defined by this following equation.

$$ \hat{E_m} = \sqrt{w_m}(\hat{1} + \epsilon k_m \hat A) $$

Where $\hat{E_m}$ is the quantum statistics of the measurements, $w_m$ are the output probabilities, $\hat{A}$ is the observable, $k_m$ is the correlation between the outcome and the effects, $\epsilon$ is an interaction parameter. How can I relate these quantities with a classical estimation problem. For example, my data are represented as $z$, the model I have is $s$ and the parameter I want to estimate is $\Theta$. So,

$$ x = f(\Theta) $$
$$ z = f(\Theta) + noise $$

A bad estimate of $\Theta$ with a blind estimator (without any prior knowledge) can be named as $\Gamma$.

So, the likelihood of $z$ given $\Theta$ is represented as $p(z|\Theta)$, the prior probability of $\Theta$ is represented as #p(\Theta)# and the joint probability of data and the parameter is

$$ p(z, \Theta) = p(\Theta) p(z|\Theta) $$

There is also a joint probability mentioned in that paper, but I am confused.

Is $m$ all the values permitted for the parameter? Or $m$ are the weak estimates for the parameters? If someone could look into the paper and give me some insight with some relationship with what I defined as a classical estimation problem, that would be great. Thanks!

 

[eljose79]

Dear forum post author,

Thank you for your interest in using quantum weak measurement for parameter estimation. As a scientist with knowledge in quantum theory, I can provide some insights into how you can relate the concepts in the paper to a classical estimation problem.

Firstly, the parameter $m$ in the paper refers to the different measurement settings or outcomes that are used to estimate the parameter of interest. In your classical estimation problem, this would correspond to the different values of the parameter $\Theta$ that you are trying to estimate.

Next, the weak estimates for the parameters in the paper refer to the quantum statistics $\hat{E_m}$, which are obtained through weak measurements. In your classical estimation problem, these weak estimates would correspond to the observed data $z$.

The interaction parameter $\epsilon$ in the paper represents the strength of the interaction between the system and the measurement apparatus. In your classical problem, this could correspond to the noise level or the precision of your measurement instrument.

In the paper, the authors use a Bayesian approach to estimate the parameters. This means that the prior probability distribution $p(\Theta)$ is used to incorporate any prior knowledge or assumptions about the parameter. In your classical estimation problem, this would correspond to any prior knowledge or assumptions you have about the parameter $\Theta$.

Finally, the joint probability mentioned in the paper is the joint probability distribution of the data and the parameter, which is given by $p(z, \Theta) = p(\Theta) p(z|\Theta)$. In your classical estimation problem, this would correspond to the likelihood of the data given the parameter and the prior probability of the parameter.

I hope this helps to clarify the concepts in the paper and how they relate to a classical estimation problem. If you have any further questions, please do not hesitate to ask. Good luck with your research!