Finite expectation-estimation task

A finite expectation-estimation task consists of a nonempty finite set $X$, a probability law $p:X\to [0,1]$ with ${\sum }_{x\in X}{p}_x=1$, and an exactly encoded real-valued function $g:X\to [a,b]$, where $a\le b$ are finite rationals. Its target is $\mu :={\mathbb{E}}_p[g]={\sum }_{x\in X}{p}_{x}g(x)\mathrm{.}$

For rational parameters $\epsilon >0$ and $0<\delta <1$, an $(\epsilon ,\delta )$ estimator is a randomized or quantum procedure that outputs a real number $\tilde{\mu }$ satisfying $\mathrm{Pr}(\mathrm{\mid }\tilde{\mu }-\mu \mathrm{\mid }\le \epsilon )\ge 1-\delta \mathrm{.}$ The task specifies the target law and success event only. A particular procedure must state how it accesses samples or coherent state preparation and must charge those resources separately.

A value that approximates an underlying real-valued observable to error $\tau$ defines a different task with target ${\mathbb{E}}_p[\tilde{g}]$; any claim about ${\mathbb{E}}_p[g]$ must add the encoding error $\tau$ to its error budget.