of a measured system whose distribution is $P [x]$. We perform a measurement on it and obtain outcome $y$ which is also a probability variable. The error of the measurement can be characterized by a conditional probability $P[y | x]$, which describes the probability of obtaining outcome $y$ under the condition that the true value of the measured system is $x$. We note that $\sum_y P[y | x] = 1$ for all $x$, where we note that the sum should be replaced by the integral if $y$ is a continuous variable. If the measurement is error-free, $P[y | x]$ is given by the delta function or the Kronecker’s delta. We assume that $P[y | x]$ is independent of the probability distribution $P[x]$; in other words, the error is independent of the state preparation of the measured system. The joint probability of $x$ and $y$ is given by $P[x,y] = P[y|x] P[x]$, and the probability of obtaining $y$ by $P[y] = \sum_x P[x, y]$. The probability of realizing $x$ under the condition that the measurement outcome is $y$, denoted as $P[x |y]$, is given by the Bayes theorem: $$P[x|y] = \frac{P[y|x]P[x]}{P[y]}.
\label{Bayes}$$

We next discuss the information contents related to the 