Consider the(adsbygoogle = window.adsbygoogle || []).push({}); Gaussian position measurement operators$$\hat{A}_y = \int_{-\infty}^{\infty}ae^{\frac{-(x-y)^2}{2c^2}}|x \rangle \langle x|dx$$ where ##|x \rangle## are position eigenstates. I can show that this satisfies the required property of measurement operators: $$\int_{-\infty}^{\infty}A_{y}^{\dagger}A_{y}dy = 1$$ where ##1## is the identity operator. I am interested in defining an analogue of this continuous Gaussian measurement operator for some discrete eigenstates ##|m\rangle##. These are discrete eigenstates rather than continuous as above, hence I'm not sure how to define the Gaussian part of ##\hat{A}_y##. I've considered the Binomial distribution $$Pr(X = k) :=

\begin{pmatrix}

n \\

k

\end{pmatrix}p^{k}(1-p)^{n-k}$$ for ##p=0.5##, since this is a discrete probability density function which resembles the normal distribution as ##n \to \infty##, but I'm not sure how to implement this since this distribution is only centered at positive integers. Does anyone have an idea of what would be a reasonable way to define theGaussian part of this measurement operatorfor the discrete case?

Thanks or any assistance, let me know if any clarity is needed.

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# A Discrete measurement operator definition

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