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\newcommand{\Si}{\Sigma}
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\newcommand{\PP}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$ 

One wants to estimate $n$ unknown real numbers $x_1,\dots,x_n$. Toward this end, one can measure $n$ linear combinations 
\begin{equation}
	a_{11}x_1+\dots+a_{1n}x_n,\ \dots,\ a_{n1}x_1+\dots+a_{nn}x_n
\end{equation}
of $x_1,\dots,x_n$ with coefficients $a_{ij}\in\{-1,0,1\}$. Each measurement involves a random error, which has mean zero and the same standard deviation $\si\in(0,\infty)$; all measurements are independent or, more generally, non-correlated. 
So, one knows the values of the coordinates $y_1,\dots,y_n$ of the random column vector 
\begin{equation}
	y:=Ax+\xi,
\end{equation}
where $A$ is the $n\times n$ matrix with entries $a_{ij}$, $x:=[x_1,\dots,x_n]^T$, and $\xi=[\xi_1,\dots,\xi_n]^T$ is the column vector of the errors of the $n$ measurements. 

Thus, one obtains the unbiased estimate 
\begin{equation}
	\hat x:=A^{-1}y=x+A^{-1}\xi
\end{equation}
of $x$, provided that the matrix $A$ is nonsingular; the unbiasedness means that $\E\hat x=x$. The covariance matrix of the estimation error $A^{-1}\xi$ is 
\begin{equation}
	\E A^{-1}\xi\xi^T(A^{-1})^T=\si^2(A^T A)^{-1}=:(b_{A;\,ij})_{i,j=1}^n, 
\end{equation}
so that the standard error of this estimation of $x_i$ is $\sqrt{b_{A;\,ii}}$. 

The question is this: 
>What can be said about 
\begin{equation}
	\si_n^2:=\min\big\{\max_1^n b_{A;\,ii}\,\colon A\in\{-1,0,1\}^{n\times n},\ A\text{ is nonsingular}\big\}, 
\end{equation}
the minimum of $\max\limits_1^n\mathsf{Var}\,\hat x_i$ 
over all choices of nonsingular $n\times n$ matrices $A$ with entries $a_{ij}\in\{-1,0,1\}$?

-------------------

For small enough $n$, the problem can be solved by direct calculation. In particular, for $n=2$ we have $\si_n^2=\si^2/2$ -- attained e.g. at $A=\begin{bmatrix}1&1\\1&-1\end{bmatrix}$. For $n=3$, $\si_n^2$ is $\si^2/2$ well -- attained e.g. at 
$A=\begin{bmatrix}1&1&1\\1&1&-1\\1&-1&0\end{bmatrix}$ and at $A=\begin{bmatrix}1&1&1\\1&1&-1\\1&-1&1\end{bmatrix}$; the first of these two choices of $A$ may be considered a better one, because for it $\{b_{11},b_{22},b_{33}\}=\{\frac38,\frac38,\frac12\}\si^2$, whereas for the latter choice of $A$ we have $\{b_{11},b_{22},b_{33}\}=\{\frac12,\frac12,\frac12\}\si^2$. 

This question is sparked by the somewhat similar one at [Optimal linear measurement operator][1], which in turn was inspired by the question at [Can I really double my accuracy?][2], which in turn goes back to Mosteller.  


  [1]: https://mathoverflow.net/questions/299017/optimal-linear-measurement-operator

[2]: https://mathoverflow.net/questions/297755/can-i-really-double-my-accuracy-on-variance-of-a-sum-of-random-variables