Quantum Speed Limit for States with a Bounded Energy Spectrum

Quantum speed limits set the maximal pace of state evolution. Two well-known limits exist for a unitary time-independent Hamiltonian: the Mandelstam-Tamm and Margolus-Levitin bounds. The former restricts the rate according to the state energy uncertainty, while the latter depends on the mean energy relative to the ground state. Here we report on an additional bound that exists for states with a bounded energy spectrum. This bound is dual to the Margolus-Levitin one in the sense that it depends on the difference between the state’s mean energy and the energy of the highest occupied eigenstate. Each of the three bounds can become the most restrictive one, depending on the spread and mean of the energy, forming three dynamical regimes which are accessible in a multilevel system. The new bound is relevant for quantum information applications, since in most of them, information is stored and manipulated in a Hilbert space with a bounded energy spectrum.

Figure 1

Dynamical regimes of quantum state evolution. The dynamical regimes are displayed on the energy uncertainty ($\mathrm{\Delta }E$) vs mean energy ($E$) graph. Three regimes are identified depending on which of three terms in Eq. (7) is most constraining: the yellow area is the MT regime, the blue area is the (original) ML regime, and the red area is the regime associated with new dual ML bound. The solid black line denotes the maximal energy uncertainty as a function of mean energy when the spectrum is bounded; see Eq. (8). The circles and triangles are followed by labels referring to the examples presented in Figs. 2 and 3, respectively. Note that the point 2(a) is a singular point where all three regimes touch.

Figure 2

Qubit system evolution in different dynamical regimes. The amplitude of the two-time state overlap and its limits are plotted as a function of time. The black line represents the state evolution, whereas the yellow, blue, and red lines denote the MT, ML, and dual ML bounds. The region excluded by the unified bound in Eq. (7) is indicated by the gray area. (a) The case of $E=\mathrm{\Delta }E$ is the only one saturating both bounds. (b) For a qubit state with $E<\mathrm{\Delta }E$, the evolution is initially limited by the MT bound and at later times by the ML one. (c) A state with $E>E_{\max }-\mathrm{\Delta }E$ has its evolution restricted by the dual ML bound for times $t>{\tau }_c^{\star }$.

Figure 3

Qutrit system evolution in different dynamical regimes. (a) For a qutrit state with $E<\mathrm{\Delta }E<(E_{\max }-E)$, the evolution is initially limited by the MT bound and at later times by the ML one. (b) A state with $\mathrm{\Delta }E<E$ is exclusively restricted by the MT bound. (c) For a state with $(E_{\max }-E)<\mathrm{\Delta }E<E$, the dual ML bound is the one constraining the dynamics for times $t>{\tau }_c^{\star }$.