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It is difficult to discriminate between the state of initial time $t_0$ andthat of time $t_0+\Delta t$. The task of discriminating between the two statescan be interpreted as the hypothesis testing problem as to a state $\rho$deciding whether $\rho$ is $\rho_{t_0}$ or$\rho_{t_0+\Delta t}$ We show theprocess of optimization of this hypotheses testing and the condition thatdiscrimination between the two states become difficult even when optimized testwas executed. In addition, We consider time-enegy uncertainty relation from theviewpoint of optimized hypotheses testing. Optimization consists of two steps.The first step is to optimize the decision rule based on the outcome of thegiven measurement by the Neyman-Pearson theorem. The second step is to selectoptimum measurement in order to maximize the power of test. The optimummeasurement is proved to be $\{ |0\rangle \langle 0|,1-|0\rangle \langle 0|\}$and the power of test in the optimized test holds$\gamma_{Max}=1-\exp(-\frac{2n}{\hbar^2}\Delta t^2\Delta H^2)+o(\Delta t^2)$when $\Delta t \ll 1$ satisfies,where n is the number of the data. It will beimpossible to discriminate between the two states when the power of this testdeclines. This condition becomes $\frac{2n}{\hbar^2}\Delta t^2\Delta H^2\ll 1$in the optimized hypotheses testing.  It is remarkable that the optimum measurement is composed of the operators$Q_0$ and $1-Q_0$ in the proof of time-energy uncertainty relation. Namely, thetime-energy uncertainty relation presents the time interval in which even theoptimum measurement hardly detect modification of the initial state .

Considerations in the Time-Energy Uncertainty Relation from the Viewpointt of Hypothesis Testing