Quantum Speed Limits For Open System Dynamics Based On A Representation-Basis-Dependent $\boldsymbol{\ell^{p}_{w}}$-Seminorm

Abstract

We report a family of quantum speed limits (QSLs) that give evolution time lower bounds between an initial and a final state whose separation is described by a certain representation basis dependent norm derived from the weighted $\ell^{p}_{w}$-seminorm. These QSLs are applicable to open, closed, time-dependent, or time-independent systems in finite-dimensional Hilbert spaces whose density matrices are piecewise time differentiable. They can be extended to systems over separable Hilbert spaces as well. Crucially, these QSLs are valid for arbitrary operators, not just density matrices, provided that a modest technical condition is fulfilled. When compared to the existing QSLs applied to pure state time-independent Hamiltonian evolution, qubit spontaneous emission, high-fidelity gate implementation, coherent state photon loss and operator coherence or dephasing, ours consistently show improved sharpness in most cases, along with greater universality and still retaining computational efficiency.

Figures (5)

• Figure 1: Various QSLs for a qubit undergoing spontaneous emission against the decay rate $\gamma$ when $\tau = 1$. Panel (a) compares $\tau_{1,\mathbb{1}_1,\mathcal{B}_E}^{\text{sup}}$, $\tau_{1,\mathbb{1}_1,\mathcal{B}_E}^{\text{int}}$ with the MT and DL bounds. Note that $\tau_{1,\mathbb{1}_1,\mathcal{B}_E}^{\text{int}}$ equals the actual evolution time $\tau$. Panel (b) compares $\tau_{2,\mathbb{1}_4,\mathcal{B}_E}^{\text{sup}}$ and $\tau_{2,\mathbb{1}_4,\mathcal{B}_E}^{\text{int}}$ with the MT and DL bounds. Panels (c) and (d) vary $p$ and $w$ while keeping the other parameters fixed to those used in Panel (b), respectively.
• Figure 2: Various QSLs for the gate operation defined in Sec. \ref{['Subsec:Appl_NV']} on an NV-center spin against the magnetic field ratio $B_0/B_1$. For simplicity, we set $\hbar = 1$ in all plots. Panel (a) shows the fully optimized $\tau_{\text{opt}}^{\text{int}}$ bound. It uses a wider $y$-axis scale from the rest for better illustration. For Panel (b), (c) and (d), we vary the parameters $p$, $w$ and $\mathcal{B}$ by fixing the remaining parameters to $p=2$, $w=\mathbb{1}_9$ and $\mathcal{B}$ as the computational basis $\mathcal{B}_c$, respectively.
• Figure 3: Various QSLs for a bosonic mode undergoing photon loss plotted against the decay rate $\kappa$, with fixed evolution time $\tau=1$ and initial amplitude $\alpha_0=2$. In Fig. \ref{['fig:photon_loss_p2']}, which uses $p = 2$ and $w = \mathbb{1}$, thin solid lines are the analytical results while markers are numerical ones computed using the truncated Fock orthonormal set $\{ \ket{n} \}_{n=0}^{19}$. In Fig. \ref{['fig:photon_loss_p3']}, which uses $p = 3$ and $w = \mathbb{1}$, all results are numerically computed in the same truncated Fock orthonormal set. The thin dashed and thick solid lines are for the basic bounds and the partially optimized ones defined in the text, respectively.
• Figure 4: $\tau_{\text{opt}}^{\text{sup}}$ and $T_Q$ for the qubit dephasing process.
• Figure 5: $\tau_{\text{opt}}^{\text{sup}}$ and $T_C$ for the qubit coherence generation.

Theorems & Definitions (3)

• Theorem 1
• proof
• Theorem 2