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

TL;DR

This paper explores the time-energy uncertainty relation through hypothesis testing, optimizing measurement strategies to determine when state discrimination becomes fundamentally limited, revealing the role of optimal measurements in quantum uncertainty.

Contribution

It introduces a hypothesis testing framework for the time-energy uncertainty relation, identifying optimal measurements and conditions under which state discrimination is inherently difficult.

Findings

Optimal measurement is a specific projective measurement involving operators Q_0 and 1-Q_0.

The power of the hypothesis test decreases significantly when the time interval is too short, satisfying a specific inequality.

The study links the time-energy uncertainty to the limits of quantum state discrimination using optimized hypothesis testing.

Abstract

It is difficult to discriminate between the state of initial time $t_0$ and that of time $t_0+\Delta t$. The task of discriminating between the two states can 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 the process of optimization of this hypotheses testing and the condition that discrimination between the two states become difficult even when optimized test was executed. In addition, We consider time-enegy uncertainty relation from the viewpoint of optimized hypotheses testing. Optimization consists of two steps. The first step is to optimize the decision rule based on the outcome of the given measurement by the Neyman-Pearson theorem. The second step is to select optimum measurement in order to maximize the power of test. The optimum measurement 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 be impossible to discriminate between the two states when the power of this test declines. 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, the time-energy uncertainty relation presents the time interval in which even the optimum measurement hardly detect modification of the initial state .