Quantum Speed Limits Based on the Sharma-Mittal Entropy

TL;DR

The paper develops a new class of quantum speed limits (QSLs) grounded in the Sharma-Mittal entropy $S_{q,z}$. By bounding the time derivative of $S_{q,z}$ in terms of the minimal eigenvalue $\lambda_{\min}(\rho_t)$ and the Schatten speed, it derives SME-based QSLs for general nonunitary dynamics, including CPTP channels and non-Hermitian evolutions. Extending to many-body systems, the authors show the SME-based QSL for reduced states scales with the Hamiltonian variance $\Delta H^2$ and demonstrate the XXZ model as a concrete example with oscillatory QSLs. The results offer a spectrum-only, entropic perspective that complements distance- and thermodynamics-based QSLs, with potential applications in entropic uncertainty relations, metrology, and quantum control.

Abstract

Quantum speed limits (QSLs) establish intrinsic bounds on the minimum time required for the evolution of quantum systems. We present a class of QSLs formulated in terms of the two-parameter Sharma-Mittal entropy (SME), applicable to finite-dimensional systems evolving under general nonunitary dynamics. In the single-qubit case, the QSLs for both quantum channels and non-Hermitian dynamics are analyzed in detail. For many-body systems, we explore the role of SME-based bounds in characterizing the reduced dynamics and apply the results to the XXZ spin chain model. These entropy-based QSLs characterize fundamental limits on quantum evolution speeds and may be employed in contexts including entropic uncertainty relations, quantum metrology, coherent control and quantum sensing.

Figures (6)

• Figure 1: The phase structure of the function $[{h_q}(\rho)]^{\frac{q-z}{1-q}}$ exhibits distinct behaviors depending on the parameters $q$ and $z$. For $q \in (0,1)$, the $q$-purity satisfies ${h_q}(\rho) \geq 1$. In this range, the inequality $[{h_q}(\rho)]^{\frac{q-z}{1-q}} \leq 1$ holds for $0 < q < z < 1$ as well as for $z > 1$. When $0 < z < q < 1$, the inequality $[{h_q}(\rho)]^{\frac{q-z}{1-q}} \geq 1$ holds. In contrast, for $q \geq 1$, the $q$-purity fulfills ${h_q}(\rho) \leq 1$. Under this condition, the relation $[{h_q}(\rho)]^{\frac{q-z}{1-q}} \geq 1$ holds when $q > z > 1$, while $[{h_q}(\rho)]^{\frac{q-z}{1-q}} \leq 1$ holds for $z > q > 1$.
• Figure 2: Phase diagram for the quantum speed limit $\tau_{q,z}^{\text{QSL}}$ given in \ref{['QSL1']}. The green shaded area indicates the region ($0 < q < z < 1$) in which the QSL applies.
• Figure 3: Density plots illustrate the behavior of the QSL time $\tau^{\text{QSL}}_{q,z}$ [as defined in \ref{['channel5']}] and the normalized relative error ${\widetilde{\varsigma}_{q,z}}(\tau)$ [specified by Eq. \ref{['channel3']}] as functions of $\gamma\tau$. The initial state is given by \ref{['ini_state']} with Bloch vector $\{r,\theta,\phi\}=\{\tfrac{1}{2},\tfrac{\pi}{4},\tfrac{\pi}{4}\}$.
• Figure 4: Physical signatures for the amplitude-damping dynamics (same parameters as Fig. \ref{['figchannel1']}). (a) The longitudinal polarization $\langle\sigma_z\rangle_t$ tends monotonically to $+1$, (b) the transverse polarization $\langle\sigma_x\rangle_t$ decays to $0$, and (c) the fidelity $F(\rho_0,\rho_t)$ stays strictly positive for all finite times.
• Figure 5: Density plots of the QSL time $\tau^{\text{QSL}}_{q,z}$\ref{['NonHermitian3']} for $z = 3/4$, $1/2$ and $1/4$, for a two-level quantum system evolving under the non-Hermitian Hamiltonian $H{\text{sys}} = \omega \sigma_x + i \eta \sigma_z$. The initial state is given by \ref{['ini_state']} with the Bloch vector $\{r,\theta,\phi\}=\{\frac{1}{2},\frac{\pi}{4},\frac{\pi}{4}\}$. $\eta/\omega = 2$ for panels (a)-(c) and $\eta/\omega = 1/2$ for panels (d)-(f).

Theorems & Definitions (9)

• Definition 1: SharmaMittalJMathSci1975S.Mazumdar2019
• Theorem 1
• Definition 2
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Lemma 2
• Theorem 5