Quantum-imaginarity-based quantum speed limit

TL;DR

This work introduces imaginarity-based quantum speed limits (ISLs) that quantify how quickly a quantum system can generate or erase the imaginary part of its state. By grounding the bounds in three imaginarity measures—relative entropy imaginarity $\mathscr{M}_r(\rho)=S(\operatorname{Re}\rho)-S(\rho)$, trace-distance imaginarity $\mathscr{M}_{tr}(\rho)=\tfrac{1}{2}\|\rho-\rho^T\|_1$, and geometric imaginarity $\mathscr{M}_g(\rho)=1-\max_{\sigma\in\mathscr{R}}F(\rho,\sigma)$—the authors derive three corresponding ISLs (Theorems 1–3) that bound the minimum evolution time in terms of changes in imaginarity and dynamical speeds (e.g., $\Lambda_T$, $\Lambda_{\mathrm{tr}}$, $\Lambda_g$). They show attainability and tightness in key scenarios, such as dephasing and dissipative dynamics, and extend the framework to Liouville-space and stochastic-approximate transformations, illustrating the practical role of imaginarity as a resource in quantum dynamics. The results complement geometric and phase-based QSLs, offering operational bounds tied to the manipulation of the imaginary components of quantum states. Overall, the work connects resource theories of imaginarity with fundamental speed limits, with potential impacts on quantum control, computation, and sensing.

Abstract

The quantum speed limit sets a fundamental restriction on the evolution time of quantum systems. We explore the relationship between quantum imaginarity and the quantum speed limit by utilizing measures such as relative entropy, trace distance, and geometric imaginarity. These speed limits define the fundamental constraints on the minimum time necessary for quantum systems to evolve under various dynamical processes. As applications the dephasing dynamics and dissipative dynamics are analyzed in detail. The quantum speed limit in stochastic-approximate transformations is also investigated. Our quantum speed limits provide lower bounds on how fast a physical system evolves to attain or lose certain imaginarity, with potential applications in efficient quantum computation designs, quantum control and quantum sensing.

Figures (5)

• Figure 1: The illustration of geometric quantum speed limits is as follows: The black curve represents the path $\gamma$ in the quantum state space, describing a general evolution from the initial state $\rho_0$ to the final state $\rho_T$, with time $t \in [0, T]$ as the parameter. Using a metric on the quantum-state space, the length of this path is denoted by $\ell_\gamma(\rho_0, \rho_T)$. The red curve $\zeta$ corresponds to the geodesic connecting $\rho_0$ and $\rho_T$, and its length is given by $\mathcal{L}(\rho_0, \rho_T)$.
• Figure 2: The evolution time $T$ under the dephasing channel is compared to the speed limit time $T_{\mathrm{ISL}}$ [as defined in \ref{['isl']}], and the dynamics comes from the master equation specified by \ref{['dephasing']}, with the parameters set as $\gamma_t = 2$ and $\omega_0 = 0$. The black solid, blue dot and pink dash lines represent the cases of $\theta=\frac{\pi}{2}$, $\frac{\pi}{3}$ and $\frac{\pi}{4}$ of the initial state, respectively. All times are in units of $(\gamma_t)^{-1}$ (the horizontal axis shows the dimensionless time $\gamma_t t$).
• Figure 3: The evolution time $T$ under the dissipative channel is compared with the speed limit time $T_{\mathrm{ISL}}$ [as defined in \ref{['isl']}] and the dynamics comes from the master equation specified by \ref{['equa']} with $\gamma_t = 2$. The black solid, blue dot and green dash lines represent the cases of $\theta=\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{\pi}{4}$ of the initial state, respectively. All times are in units of $(\gamma_t)^{-1}$ (the horizontal axis shows the dimensionless time $\gamma_t t$).
• Figure 4: The actual evolution time $T$ under the dephasing channel is compared to the speed limit time $T_{\mathrm{ISL}}$ defined in \ref{['geometricisl']} and the dynamics comes from the master equation specified by \ref{['dephasing']}, with the parameters set as $\gamma_t = 2$ and $\omega_0 = 0$. The black solid, blue dotted, and pink dashed lines correspond to initial state parameters of $\theta = \frac{\pi}{2}$, $\theta = \frac{\pi}{3}$, and $\theta = \frac{\pi}{4}$, respectively. All times are in units of $(\gamma_t)^{-1}$ (the horizontal axis shows the dimensionless time $\gamma_t t$).
• Figure 5: The actual evolution time $T$ under the dissipative channel is compared to the speed limit time $T_{\mathrm{ISL}}$ defined in \ref{['geometricisl']} and the dynamics comes from the master equation specified by \ref{['equa']} with $\gamma_t = 2$. The black solid, blue dotted, and green dashed lines represent the initial state parameters $\theta = \frac{\pi}{2}$, $\theta = \frac{\pi}{3}$, and $\theta = \frac{\pi}{4}$, respectively. All times are in units of $(\gamma_t)^{-1}$ (the horizontal axis shows the dimensionless time $\gamma_t t$).

Theorems & Definitions (7)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Lemma 2