Tightening Quantum Speed Limits for Almost All States

Conventional quantum speed limits perform poorly for mixed quantum states: They are generally not tight and often significantly underestimate the fastest possible evolution speed. To remedy this, for unitary driving, we derive two quantum speed limits that outperform the traditional bounds for almost all quantum states. Moreover, our bounds are significantly simpler to compute as well as experimentally more accessible. Our bounds have a clear geometric interpretation; they arise from the evaluation of the angle between generalized Bloch vectors.

Figure 1

Let $\rho$ and $\sigma$ be two mixed qubit states with the same spectrum, $\rho =\lambda |r_1⟩⟨r_1|+(1-\lambda )|r_2⟩⟨r_2|$, and $\sigma =\lambda |s_1⟩⟨s_1|+(1-\lambda )|s_2⟩⟨s_2|$, $\lambda \in (0,1)$, $\lambda \ne 1/2$, where $\{|r_1⟩,|r_2⟩\}$ and $\{|s_1⟩,|s_2⟩\}$ are two orthonormal bases. The problem of unitarily evolving $\rho$ to $\sigma$ can be mapped to evolving $|r_1⟩$ to $|s_1⟩$ (or equivalently, $|r_2⟩$ to $|s_2⟩$). Equation (1) is tight for pure states; thus, any Hamiltonian that takes $|r_1⟩$ to $|s_1⟩$ will also take $\rho$ to $\sigma$ in the same time. For any Hamiltonian with bounded standard deviation $\mathrm{\Delta }E\le \mathcal{E}$, this time is bounded from below by $\theta /\mathcal{E}$, where $\theta =d(|r_1⟩,|s_1⟩)$ is the distance between $|r_1⟩$ and $|s_1⟩$, i.e., half of the angle between the vectors associated with $|r_1⟩$ and $|s_1⟩$. However, Eq. (1) for the same constraint on the Hamiltonian suggests that $T_{\mathcal{L}}=\mathcal{L}(\rho ,\sigma )/\mathcal{E}$, with $\mathcal{L}(\rho ,\sigma )<\theta$ for every choice of $\lambda \ne 0$, 1 [see Eq. (9)], making the QSL unattainable for all mixed states.

Figure 2

(a) Bounds $T_{\mathcal{L}}$ [Eq. (1)], $T_{\mathrm{\Phi }}$ [Eq. (6)], and $T_{\mathrm{\Theta }}$ [Eq. (4)], as a function of the eigenvalue $\lambda$, for two mixed and antipodal qubit states $\rho =\lambda |r_1⟩⟨r_1|+(1-\lambda )|r_2⟩⟨r_2|$ and $\sigma =\lambda |r_2⟩⟨r_2|+(1-\lambda )|r_1⟩⟨r_1|$. The unitary evolution is generated by the Hamiltonian $H=e^{i\phi }|r_1⟩⟨r_2|+\mathrm{H}.\mathrm{c}.$ Bounds are symmetric with respect to $\lambda =1/2$. The same hierarchy holds for nonantipodal mixed qubit states. Bound $T_{\mathrm{\Theta }}$ is always attainable. (b) For $N=3$ (qutrits), the hierarchy between the three bounds expressed with three regions of the polytope defined by the spectrum $\{{\lambda }_1,{\lambda }_2,{\lambda }_3\}$ of states $\rho$ and $\sigma$, as indicated in the legend. The corners represent pure states, while the center represents the maximally mixed state. The shape of the regions reflects a specific choice of $H$, $\rho$, and $\sigma$; similar features are common to those of any pair of states. For the case of qutrits, $T_{\mathcal{L}}$ is never larger than $\max [T_{\mathrm{\Theta }},T_{\mathrm{\Phi }}]$ (see Sec. IV of the Supplemental Material [55]). (c) Evaluation of $1-T_{\mathcal{L}}/\max [T_{\mathrm{\Theta }},T_{\mathrm{\Phi }}]$ as a measure of the tightness of the new bounds, for $3\le N\le 10$, with a sample size of $10^6$. For every run, a different Haar-random state and a different Hamiltonian are generated. $T_{\mathcal{L}}$ can be larger than $\max [T_{\mathrm{\Theta }},T_{\mathrm{\Phi }}]$, but only for 0.1% of the sampled states, and only with a difference of 1% with respect to the largest of the new bounds. (d) Density plot of $10^5$ qutrit states, sampled approximately uniformly in terms of purity. The axes show numerical estimations of $1-T_{\mathcal{L}}/\max [T_{\mathrm{\Theta }},T_{\mathrm{\Phi }}]$ (horizontal) and $1-\mathrm{tr}[{\rho }^2]$ (vertical). We obtained the Pearson correlation coefficient $r=0.8$ for the linear model between $1-\mathrm{tr}[{\rho }^2]$ and $1-T_{\mathcal{L}}/\max [T_{\mathrm{\Theta }},T_{\mathrm{\Phi }}]$. Bounds $T_{\mathrm{\Theta }}$ and $T_{\mathcal{L}}$ coincide for pure states (bottom left), as shown analytically, and differ for increasingly mixed states (top right). This behavior qualitatively extends to $N$-dimensional systems.