Comment on 'Energy-time uncertainty relation for driven quantum systems'

The quantum speed limit (QSL) gives the fundamental speed limit to quantum time evolution. In time-independent quantum systems, the minimal evolution time $\tau_{{{\rm QSL}}}$ needed for the state to rotate orthogonally is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau \geqslant \max \left\{\frac{\pi \hbar}{2\Delta E} , \frac{\pi \hbar}{2 \left(E-E_0 \right)} \right\} , \nonumber \end{align} \tag{ 1 }$$

where $\Delta E$, E and E₀ are the energy variance, mean energy and ground-state energy, respectively. The first bound is called the Mandelstam–Tamm (MT) bound [1] and the second bound is called the Margolus–Levitin (ML) bound [2]. We emphasize that the MT and ML bounds are characterized by the energy variance and mean energy, respectively.

Recently, Deffner and Lutz derived the two ML bounds in time-dependent closed systems [3]. Furthermore, in [4], they derived the MT bound and the ML bound for non-Markovian dynamics. As a result, they concluded that the ML bound is tighter than the MT bound in non-Markovian systems.

In this comment, we point out the following: (i) the derivation of one ML bound for unitary dynamics in [3] is incorrect. (ii) Another ML bound for unitary dynamics in [3] appears to have no direct physical meaning. (iii) The ML bound has not yet been established in time-dependent quantum systems, except for the adiabatic case [5].

In [3], the authors used the following relation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle |\langle\psi_0|\psi_\tau\rangle|=|\langle\psi_0| U_\tau|\psi_0\rangle|=\left| \sum_n |\langle \psi_0|n \rangle|^2 \exp(-{\rm i}J_n) \right| \label{t-order} , \nonumber \end{align} \tag{ 2 }$$

where they defined that $U_\tau$ denotes the time evolution operator and $\{|n\rangle \}$ is the set of its instantaneous eigen states, with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{\hbar} \int_0^\tau {\rm d}t H_t |n\rangle \equiv J_\tau|n\rangle =J_n |n\rangle \label{t-order2}. \nonumber \end{align} \tag{ 3 }$$

Using equation (2), the authors obtained the ML bound for time-dependent closed systems

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau \geqslant \frac{\hbar}{E_\tau} \mathcal{L}(\psi_0,\psi_\tau) \label{ML-time} , \nonumber \end{align} \tag{ 4 }$$

where $E_\tau=(1/\tau)\int_0^\tau {\rm d}t \left| \langle \psi_0| H_t|\psi_0 \rangle \right|$ and $\mathcal{L}(\psi, \psi_\tau)=\arccos(|\langle \psi_0|\psi_\tau \rangle |)$.

However, equation (3) does not hold clearly. The authors identified $\newcommand{\e}{{\rm e}} \exp(-({\rm i}/\hbar) \int_0^\tau {\rm d}t H_t)$ with $U_\tau$ and ignored the time ordered product of $U_\tau$, which is never justified, except for the special case that $[H_t , H_{t'}]=0$ for any time. In order to correctly realize their idea, we must use the Magnus expansion [6]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle U_\tau=\exp\left(- \frac{\rm i}{\hbar} \Omega_\tau \right), \nonumber \end{align} \tag{ 5 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Omega_\tau = \int_0^\tau {\rm d}t_1 H_{t_1} -\frac{\rm i }{2\hbar} \int_0^\tau {\rm d}t_1 \int_0^{t_1} {\rm d}t_2 \left[ H_{t_1}, H_{t_2} \right]+\cdots , \nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Omega_\tau |n'\rangle =J_n'|n'\rangle , \nonumber \end{align} \tag{ 7 }$$

where $|n'\rangle$ is the set of instantaneous eigenstates of $\Omega_\tau$. Then, we can identify $\newcommand{\e}{{\rm e}} \exp\left(- ({\rm i}/\hbar) \Omega_\tau \right)$ with $U_\tau$ and equation (4) is modified to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau \geqslant \frac{\hbar}{\frac{1}{\tau}\left| \langle \psi_0|\Omega_\tau|\psi_0 \rangle \right|} \mathcal{L}(\psi_0,\psi_\tau) \label{ML-time2} . \nonumber \end{align} \tag{ 8 }$$

Although the derivation of equation (8) is correct, it is a formidable task to estimate the value of $\left| \langle \psi_0|\Omega_\tau|\psi_0 \rangle \right|$ via H_t in general.

In addition, the authors derived also another ML bound in appendix of [3]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau \geqslant \frac{4\hbar}{\pi^2 \bar{E}_\tau} \mathcal{L}^2(\psi_0,\psi_\tau) \label{ML-time3} , \nonumber \end{align} \tag{ 9 }$$

where $\bar{E}_\tau$ is given by $(1/\tau) \int_0^\tau {\rm d}t \left| \langle \psi_0|H_t |\psi_t\rangle \right|$. Although the derivation of equation (9) is correct, the value of $\langle \psi_0|H_t |\psi_t\rangle$ cannot be limited only from the eigenvalues of H_t. Therefore, equations (8) and (9) are mathematically correct but appear to have no direct physical meaning. The authors failed to obtain the meaningful ML bound for time-dependent closed systems in [3].

In summary, the ML bound is limited only to time-independent systems and has not yet been established in time-dependent systems except for the adiabatic case [5]. The derivation of the ML bound is based on spectrum expansion and, when we extend it straightforwardly, we obtain equation (8) which makes no sense physically. In addition, we mention that, for the classical Liouville equation, the classical ML-type bound is looser than the classical MT-type bound even in time-independent systems [7, 8]. It is also noted that the derivation of the ML bound for non-Markovian dynamics in [4] is incorrect [9]. These results might imply that the ML bound is a peculiar phenomenon to time-independent (or adiabatic) systems and not a universal property in time evolution.

The authors thank Ryo Takahashi for useful discussions. M Okuyama was supported by JSPS KAKENHI Grant No. 17J10198. M Ohzeki was supported by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan) and JSPS KAKENHI No. 16K13849, No. 16H04382, and the Inamori Foundation.