Time-energy measure for quantum processes

Quantum mechanics sets limits on how fast quantum processes can run given some system energy through time-energy uncertainty relations, and they imply that time and energy are tradeoffs against each other. Thus, we propose to measure the time energy as a single unit for quantum channels. We consider a time-energy measure for quantum channels and compute lower and upper bounds of it using the channel Kraus operators. For a special class of channels (which includes the depolarizing channel), we can obtain the exact value of the time-energy measure. One consequence of our result is that erasing quantum information requires $\sqrt{(n+1)/n}$ times more time-energy resource than erasing classical information, where $n$ is the system dimension.

Figure 1

The point $\langle a|b\rangle$ is initially obtained as a linear combination of four points at eigenangles ${\theta }_j,j=1,\cdots ,4$ of $U$ based on Eq. (19). A new eigenangle ${\theta }_2^{\prime }$ can be found such that $\langle a|b\rangle$ is a linear combination of ${\theta }_2^{\prime }$, ${\theta }_3$, and ${\theta }_4$.

Figure 2

The optimal $U$ for the max time energy consists of two nontrivial eigenangles ${\theta }_1$ and ${\theta }_2$, which form a vertical line passing through the point $\langle a|b\rangle$. Two other rotated lines about the point are shown. The dashed line (green) and the dotted line (blue) have max time energy ${\theta }_1^{\prime }$ and $|{\theta }_2^{\prime }|$, respectively, which are both greater than the optimal value ${\theta }_1$.