Unifying Quantum and Classical Speed Limits on Observables

The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system. We derive a bound on the speed with which observables of open quantum systems evolve. This speed limit is divided into Mandelstam and Tamm’s original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, and both are generalized to open quantum systems. By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds on the speed of evolution. We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits and that a preferred basis of speed operators serves to completely characterize the observables that saturate the speed limits. We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.

Popular Summary

How quickly can an observable change as a quantum system evolves in the presence of an environment? What features of the system dictate the speed at which physical quantities can evolve? How sensitive is the speed of an observable’s evolution to effects from the environment? We probe these questions by deriving speed limits on observables, that is, uncertainty relations that bound their rate of change.

Our main results rely on discriminating the “quantumlike” coherent contributions to the evolution of an open quantum system from the “classical-like” incoherent contributions. This division allows us to derive both upper and lower bounds on the rate of change of an observable that are provably tighter than previous results. In this way, we also unify—and improve upon—previously known quantum and classical speed limits on observables. By characterizing the conditions under which such bounds saturate, we construct tailored Hamiltonians (the mathematical operators that specify how quantum systems evolve) that drive an observable at its maximum allowed speed.

Our results provide a framework to study the dynamics of observables in open quantum and classical systems by characterizing the effect that coherent and incoherent dynamics have on the speed of observables. This allows us to quantify the influence from the open dynamics of a quantum system and to quantify the speedup that coherent dynamics can provide to incoherent processes.

Figure 1

Speed limits for observables on a qubit. The speed limit (3), derivable from the Cramér-Rao bound, and the coherent-incoherent speed limits (12) and (11) impose constraints on the rate $\dot{a}$ at which observables evolve. We illustrate this on a qubit with a state parametrized by $\rho =\frac{1}{2}(\mathbb{1}+x{\sigma }_x+y{\sigma }_y+z{\sigma }_z)$ and driven by the Hamiltonian $H=(\omega /2){\sigma }_y$ and dephasing along ${\sigma }_z$ with a rate $\kappa$. Black lines with circles denote the speeds $|\dot{x}|$ and $|\dot{z}|$ of observables ${\sigma }_x$ and ${\sigma }_z$, and blue (red) lines denote the coherent-incoherent (lower) speed limits. The grayed areas denote rates forbidden by the coherent-incoherent speed limits. In green, the looser speed limit (3) appears inside the forbidden region. (a,c) The qubit is initialized in state $z_0=1$ and thus evolves in the $y=0$ plane on and inside the Bloch sphere, which in turn implies that both observables ${\sigma }_z$ and ${\sigma }_x$ have coherent-incoherent decompositions that satisfy conditions for the saturation of Eq. (10), with ${\sigma }_{\{z,x\}}={\alpha }_C^{\{z,x\}}{L}_C+{\alpha }_I^{\{z,x\}}{L}_I$. For unitary dynamics [$\kappa =0$, (a)], this also means that the speed limit (3) saturates, and the three curves coincide. For open-system dynamics [$\kappa \ne 0$, (c)], the red and black curves coincide, but the speed limit (3) is looser except when ${\alpha }_C^{\{z,x\}}={\alpha }_I^{\{z,x\}}$. Alternatively, whenever ${\alpha }_C^{\{z,x\}}$ and ${\alpha }_I^{\{z,x\}}$ have the same (different) sign, the coherent-incoherent (lower) speed limits saturate. (b,d) The coherent-incoherent speed limits (12) and (11) are not tight for a qubit initialized in $x_0=y_0=z_0=1/\sqrt{3}$, when the state does not evolve within the $y=0$ plane, and the observables no longer have a decomposition solely in terms of $L_C$ and $L_I$. Nevertheless, these bounds serve to constrain observables’ dynamics more than the speed limit (3).

Figure 2

Constraints on evolution of observables. The integrated speed limits (19) and (20) place constraints on the rate of change of an observable, which depend on the path taken by the state of the system in state space. The former bound says that the area under the curve $|\dot{a}|/\mathrm{\Delta }A$ of an arbitrary path is upper bounded by the path length $\mathcal{L}$, as $\int |\dot{a}|/\mathrm{\Delta }Adt\le 2\mathcal{L}$. Geodesics put a more stringent constraint on the dynamics of an observable: The area under the curve $|\dot{a}|/\mathrm{\Delta }A$ must satisfy $\int |\dot{a}|/\mathrm{\Delta }Adt\le 2{\mathcal{L}}_{\min }$. Note that $A$ need not be related to the Hamiltonian or the generator of the dynamics but is rather an arbitrary observable of interest.