Defining Krauss operators with normal distribution

[Danny Boy]

I am interested in defining Krauss operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of basis states ##\{ |x \rangle\}_x## and a set of quantum measurement operators of the form $$A_C = \sum_x \sqrt{Pr(x|C)} |x \rangle \langle x|.$$ I want to prove that ##A_C## defines valid Krauss operators. If I consider the Normal Distribution $$Pr(x|C) = \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-C)^2}{2 \sigma^2}}~~~~\text{where } x \in \{-M,-M+1,...,M \}$$ where ##C## is the mean, ##x## the random variable and ##\sigma^2## the variance, then $$\int_{-\infty}^\infty Pr(X=x|C)\,dC = \frac{1}{\sqrt{2 \pi \sigma^2}}\int_{-\infty}^{\infty}e^{-\frac{(x-C)^2}{2 \sigma^2}}dC = 1.$$ thus it follows that

$$\int_{C}A_C^{\dagger}A_CdC = \int_{-\infty}^{\infty} \sum_{x}Pr(x|C) |x \rangle \langle x | dC = \sum_x \bigg[\int_{-\infty}^{\infty}Pr(x|C)dC\bigg]|x \rangle \langle x | = \sum_x |x \rangle \langle x | = 1 $$

hence we have shown $$\int_{C}A_{C}^{\dagger}A_C dC = 1$$ hence ##A_C## satisfies the condition necessary for ##\{A_C \}_C## to be Krauss operators.

Please advise if my working and conclusion that ##\{A_c \}_C## are valid Kraus operators is correct?

Thanks.

 

[eljose79]

As a fellow scientist, I would like to provide some feedback on your post. Your working and conclusion seem to be correct. You have correctly shown that the integral of the measurement operators over all possible outcomes is equal to 1, which is a necessary condition for the operators to be valid Krauss operators. Additionally, your use of the Normal Distribution as a probability distribution is a valid choice for defining quantum measurements peaked at some basis state.

However, I would like to suggest some improvements to your post. Firstly, it would be helpful to define the notation you are using, such as specifying the meaning of the subscript "C" in ##A_C## and the use of summation over "x". This will make it easier for others to understand your post.

Secondly, it would be beneficial to provide some context for your work. Why are you interested in defining these Krauss operators? Are you working on a specific problem or experiment? This will help others understand the motivation behind your work and provide more meaningful feedback.

Overall, your post is well-written and your conclusion is correct. However, providing more context and defining your notation would make it easier for others to understand and provide feedback on your work. Keep up the good work!