Speed Limits for Macroscopic Transitions

Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control. While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive. Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones. We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the “gradient” of the observable, in contrast with conventional speed limits depending on the entire range of the observable. This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics. Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new trade-off relation between time and the quantum phase difference. Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound. We also find that, beyond expectation values of macroscopic observables, the speed of macroscopic quantum coherence can be bounded from above by our general approach. The newly obtained bounds are verified in transport phenomena in particle systems and nonequilibrium dynamics in many-body spin systems. We also demonstrate that our strategy can be applied for finding new speed limits for macroscopic transitions in stochastic systems, including quantum ones, where the bounds are expressed by the entropy production rate. Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.

Popular Summary

How fast a physical state changes in time is a crucial concept for nonequilibrium physics. The first breakthrough was made in 1945 by Mandelstam and Tamm, who showed that the speed of state transition for unitary quantum dynamics is bounded using the energy fluctuation of the system. Since then, rigorous bounds on transition speed, i.e., speed limits, have played an essential role in fundamental physics as well as various applications such as quantum control. Despite the above success, when we consider dynamics involving macroscopic transitions, many conventional speed limits diverge and fail to predict physically reasonable timescales. Physically relevant speed limits in such situations are urgently required since macroscopic transitions can naturally appear in many fields, e.g., a foundation of nonequilibrium statistical mechanics and quantum engineering.

We develop a general framework to derive qualitatively tighter speed limits of macroscopic transitions than many conventional ones by employing the local conservation law of probability, the fundamental principle in physics. Mapping general systems onto graphs, we formulate valuable speed limits applicable to a wide range of dynamics. When applied to unitary quantum dynamics, the bound indicates the novel trade-off relation between time and the quantum phase difference for physically important quantities, such as the expectation value of observables or macroscopic coherence. We also apply our framework to classical and quantum macroscopic stochastic systems and derive useful speed limits using the entropy production rate.

We believe that our work paves the way toward discovering novel speed limits based on continuity equations, which can provide unprecedented fundamental limits to various nonequilibrium macroscopic phenomena.

Figure 1

Schematic illustration of our achievements. We establish a general framework for deriving qualitatively tighter speed limits of a quantity $A$ than many conventional ones, which depend on the entire range of $A$, such as $\mathrm{\Delta }A$ or $\Vert A{\Vert }_{\mathrm{op}}$. Our strategy is to map general dynamics of our interest to dynamics on a graph, where we use the local conservation of probability. In contrast with conventional bounds, our speed limits involve the gradient $\mathrm{\nabla }A$ of $A$ on the graph, which can significantly tighten the bound when $\mathrm{\nabla }A\ll \mathrm{\Delta }A\text{ or }\Vert A{\Vert }_{\mathrm{op}}$. When applied to macroscopic quantum systems (such as macroscopic transport of atoms or relaxation of a locally perturbed spin chain), our theory indicates a novel trade-off relation between time and the quantum phase difference. When applied to macroscopic stochastic dynamics, including the quantum one, our theory indicates a trade-off relation between time and quantities such as the entropy production.

Figure 2

A simple example of a process with a macroscopic transition. For a single particle at the left end ($⟨\hat{x}⟩=1$) to relax to the extended region [$⟨\hat{x}⟩=\mathcal{O}(L)$] on a one-dimensional lattice with size $L$, it takes at least time $t=\mathcal{O}(L^a)(a\ge 1)$. On the other hand, the bound in inequality (1) is loose and cannot capture this timescale, since the time-evolved state can be nearly orthogonal to the initial state even for short times with $t=\mathcal{O}(L^0)$.

Figure 3

(a) Schematic illustration of a graph composed of vertices $\mathcal{V}$ and edges $\mathcal{E}$. Each of the vertices satisfies the continuity equation as shown in Eq. (6). (b) An example of a graph, where each vertex indicates the microscopic states $i,j,\dots$ such as the basis of the Hilbert space. (c) Another example of a graph, which is one dimensional and has edges $\mathcal{E}=\{(1,2),\dots ,(n,n+1),\dots ,(N-1,N)\}$. We are interested in a situation where the observable $A$ is smooth on a graph, i.e., $|a_n-a_m|$ with $(n,m)\in \mathcal{E}$ are much smaller than the maximum range of $A$, denoted by $D_A$.

Figure 4

Schematic illustration of the trade-off relation between the current and the expectation value of the transition energy. The local current $|J_{nm}^q|$ and the expectation value of the local transition Hamiltonian $\propto Y_{nm}$ cannot be simultaneously large when $2|\mathrm{Tr}[{\hat{H}}_{nm}{\hat{\rho }}_{mn}]|$ is fixed, since $|J_{nm}^q|=2|\mathrm{Tr}[{\hat{H}}_{nm}{\hat{\rho }}_{mn}]|\sin {\theta }_{nm}$ and $Y_{nm}=2|\mathrm{Tr}[{\hat{H}}_{nm}{\hat{\rho }}_{mn}]|\cos {\theta }_{nm}$, where ${\theta }_{nm}$ plays the role of the difference in quantum phases. Summation over all of the edges $n{\sim }_{A}m$ leads to the trade-off relation between $\sum _{n{\sim }_{A}m}|J_{nm}^q|$, $E_{\mathrm{trans}}$, and $R_p$, as detailed in Appendix pp3.

Figure 5

(a)–(c) Speed of the position, $⟨\dot{\hat{x}}⟩$ (black), and the speed limits for the initial states (a) $|{\psi }_1⟩$ (the right panel shows the short-time regime), (b) $|{\psi }_2⟩$, and (c) $|{\psi }_3⟩$ in a single-particle system. The bounds are given by ${\mathcal{B}}_{\mathrm{Lip}}$ in Eq. (42) (blue), ${\mathcal{B}}_p$ in Eq. (45) (red), and ${\mathcal{B}}_H$ in Eq. (48) (green). In the inset of the right panel of (a), we also show the bound ${\mathcal{B}}_{\mathrm{UR}}$ based on the uncertainty relation in Eq. (2) (gray). Our bounds are not divergent with $L$ and $t$ even for long times and become much tighter than ${\mathcal{B}}_{\mathrm{UR}}$. Furthermore, the bounds typically become tighter in (c) than in (a) and (b), since $E_{\mathrm{trans}}$ is larger. We note that the data for ${\mathcal{B}}_{\mathrm{Lip}}$ and ${\mathcal{B}}_p$ overlap for (a) and (b). (d)–(f) Speed of the macroscopic coherence measure $|\dot{\mathcal{C}}|$ defined in Eq. (80) (black), and the speed limits for the initial states (d) $|{\psi }_1⟩$ (the right panel is for short times), (e) $|{\psi }_2⟩$, and (f) $|{\psi }_3⟩$. The bounds are given by ${\stackrel{\mathcal{~}}{\mathcal{B}}}_{\mathrm{Lip}}$ in Eq. (81) (blue), and ${\stackrel{\mathcal{~}}{\mathcal{B}}}_p$ (red) and ${\stackrel{\mathcal{~}}{\mathcal{B}}}_H$ (green) in Eq. (54). We again find that each speed limit provides a physically reasonable bound, which does not diverge with $L$ and $t$. We use $K=1,W_l=0,L=100$, and $M=1$.

Figure 6

Three initial states for a single-particle system: $|{\psi }_1⟩$ in Eq. (77), $|{\psi }_2⟩$ in Eq. (78), and $|{\psi }_3⟩$ in Eq. (79) with $L=12$. The state $|{\psi }_3⟩$ has a larger expectation value of the transition Hamiltonian than $|{\psi }_1⟩$ and $|{\psi }_2⟩$.

Figure 7

The speed of the macroscopic coherence measured by the variance of $\hat{x}$, $|\dot{\mathbb{V}}[\hat{x}]|$, and the speed limit ${\mathcal{B}}_{\mathrm{var}}$ in inequality (55) for the initial states (a) $|{\psi }_2⟩$ and (b) $|{\psi }_3⟩$ for a single-particle system. The speed limit can provide a good bound, especially for $t\lesssim L/(4\cdot 2K)$ from $|{\psi }_2⟩$ [see (a)]. Moreover, it becomes smaller when the expectation value of the transition Hamiltonian (relative to $R_p$) becomes larger [see (b)]. We use $K=1,W_l=0,L=100$, and $M=1$.

Figure 8

An example of the graph $\mathcal{E}$ (or, equivalently, ${\mathcal{E}}_X$) for the system with many hardcore bosons in Eq. (82). We can calculate $X_i$ for each Fock state $|i⟩$. We have $\Vert \mathrm{\nabla }X{\Vert }_{\mathrm{\infty }}=1$ and $C_H=2MK$. The case with $L=6$ and $M=2$ is shown.

Figure 9

Verification of the speed limits for systems with multiple hardcore bosons from the initial state (a),(c),(e) $|{\mathrm{\Psi }}_1⟩$ or (b),(d),(f) $|{\mathrm{\Psi }}_3⟩$. (a),(b) Speed of the sum of the positions of the particles, $⟨\dot{\hat{X}}⟩$ (black), and the speed limits given by ${\mathcal{B}}_{\mathrm{Lip}}$ in Eq. (42) (blue), ${\mathcal{B}}_p$ in Eq. (45) (red), and ${\mathcal{B}}_H$ in Eq. (48) (green). We find that our speed limits are valid even for many-body systems. (c),(d) Speed of the position of the rightmost particle, $⟨\dot{\hat{\mu }}⟩$ (black), and our speed limits. Remarkably, the bounds ${\mathcal{B}}_{\mathrm{Lip}}$ and ${\mathcal{B}}_p$ can provide good bounds for $t\lesssim 12$ from $|{\mathrm{\Psi }}_1⟩$, for which $|⟨\dot{\hat{\mu }}⟩|$ exhibits nonmonotonic behavior. (e),(f) Speed of the macroscopic coherence $|\dot{\mathcal{C}}|$ (black) and the speed limits given by ${\stackrel{\mathcal{~}}{\mathcal{B}}}_{\mathrm{Lip}}$ in Eq. (81) (blue), and ${\stackrel{\mathcal{~}}{\mathcal{B}}}_p$ (red) and ${\stackrel{\mathcal{~}}{\mathcal{B}}}_H$ (green) in Eq. (54). We use $K=1,V=0.1,W_l=0,L=24$, and $M=3$.

Figure 10

(a) Schematic figure of the many-body spin system in Eq. (88). The spins consist of the spin-conserving interaction for the entire spins and the transverse magnetic field only at the first site. (b),(c) Speed of the magnetization, $⟨\dot{\hat{M}}⟩$ (black), and the speed limits from the initial states (b) $|{\mathrm{\Phi }}_1⟩$ and (c) $|{\mathrm{\Phi }}_2⟩$. The bounds are given by ${\mathcal{B}}_{\mathrm{Lip}}$ (blue), ${\mathcal{B}}_p$ (red), ${\mathcal{B}}_H$ (green), and ${\mathcal{B}}_{\mathrm{UR}}^{\prime }$ (gray). The speed limits can qualitatively capture (part of) the oscillations of $|⟨\dot{\hat{M}}⟩|$ and provide nice bounds for certain times, such as $t\simeq p\pi /2(p=1,2,\dots )$ for ${\mathcal{B}}_{\mathrm{Lip}}$ and ${\mathcal{B}}_p$ from $|{\mathrm{\Phi }}_1⟩$. Note that some of the curves are almost overlapped (${\mathcal{B}}_{\mathrm{Lip}}\simeq {\mathcal{B}}_p$ for $|{\mathrm{\Phi }}_1⟩$ and ${\mathcal{B}}_H\simeq {\mathcal{B}}_{\mathrm{UR}}^{\prime }$ for $|{\mathrm{\Phi }}_2⟩$). (d) Time evolution of the macroscopic coherence measured by the variance $\mathbb{V}[\hat{M}]$ (black) and its bound ${\mathcal{B}}_{\mathrm{var}}$ (red) from the GHZ state $|{\mathrm{\Phi }}_3⟩$. We find that the bound works well especially for $t\simeq p\pi /2(p=1,2,\dots )$. We consider the case with $L=10,J_x=J_y=0.1,J_z=0.05,$ and $h=1$ for all of the calculations.

Figure 11

(a) Schematic illustration for our model, which describes the symmetric and asymmetric hopping models. We consider $L$ sites and $M$ particles ($M=3$ is shown), which hop between neighboring sites at right and left with rates $q$ and $1-q$, respectively. We consider the distance ${\xi }_l$ of particles from the first site in Eq. (119). (b) Speed of the sum of ${\xi }_l$, $|⟨\dot{d}(t)⟩|$ (black), and the speed limits for the symmetric simple exclusion process ($q=0.5$). The main and inset panels show results for the long and short times, respectively. The speed is well bounded by the current bound ${\mathcal{B}}_{\mathrm{cur}}^c$ (blue), which is further bounded by the entropic bound ${\mathcal{B}}_{\mathrm{\Sigma }}^c$ (red) and the bound ${\mathcal{B}}_{\mathrm{\Theta }}^c$ (green). (c) Speed limits of $|⟨\dot{d}(t)⟩|$ for the asymmetric simple exclusion process ($q=0.7$). While ${\mathcal{B}}_{\mathrm{cur}}^c$ works well for short times, none of the bounds become tight for long times due to the stationary current. (d) Modified speed limit of $|⟨\dot{d}(t)⟩-V_d|$ (black) for the asymmetric simple exclusion process ($q=0.7$). We find that the bound ${\tilde{\mathcal{B}}}_{{\mathrm{\Sigma }}^{\mathrm{HS}}}^c$ (green) is useful to evaluate the speed of the transition both for (left) short and (right) long times. (e) Speed of the maximum distance, $|⟨\dot{\mu }(t)⟩|$ (black), and the speed limits for the symmetric simple exclusion process ($q=0.5$). The main and inset panels show results for the long and short times, respectively. As seen from the almost collapse of the curves, the speed is tightly bounded by the bounds ${\mathcal{B}}_{\mathrm{cur}}^c$ (blue), ${\mathcal{B}}_{\mathrm{\Sigma }}^c$ (red), and ${\mathcal{B}}_{\mathrm{\Theta }}^c$ (green), which are obtained from the coarse-grained variables. We use $L=18$ and $M=3$ for all the cases.

Figure 12

Time dependence of the speed of the sum of the positions of the particles, $|⟨\dot{\hat{X}}⟩|$ (black), from the initial state $|{\mathrm{\Psi }}_1⟩$ [the same situation discussed in Fig. 9]. In contrast with the bounds ${\mathcal{B}}_p$ in Eq. (45) (red) and ${\mathcal{B}}_H$ in Eq. (48) (green), the quantity $\mathcal{D}$ (purple) in Eq. (130) with $c=2$ cannot bound $|⟨\dot{\hat{X}}⟩|$. This indicates that the trade-off relation between time and energy fluctuation does not hold in our formalism. We note that the bound ${\mathcal{B}}_{\mathrm{UR}}$ in inequality (2) (gray), which is based on the standard trade-off relation between time and energy fluctuation, provides a rather loose upper bound for $|⟨\dot{\hat{X}}⟩|$.

Figure 13

Schematic illustration of the exponential suppression of the tail (shaded in cyan) of the probability distribution $p(t)$ given in Eq. (134), where $p_n(0)={\delta }_{0n}$ is assumed.