In quantum mechanics, a quantum speed limit (QSL) is a limitation on the minimum time for a quantum system to evolve between two distinguishable states.^[1] QSL are closely related to time-energy uncertainty relations. In 1945, Leonid Mandelstam and Igor Tamm derived a time-energy uncertainty relation that bounds the speed of evolution in terms of the energy dispersion.^[2] Over half a century later, Norman Margolus and Lev Levitin showed that the speed of evolution cannot exceed the mean energy,^[3] a result known as the Margolus–Levitin theorem. Realistic physical systems in contact with an environment are known as open quantum systems and their evolution is also subject to QSL.^[4][5] Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes,^[6] which was verified in a cavity QED experiment.^[7]

QSL have been used to explore the limits of computation^[8][9] and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature. ^[10] In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems. ^[11][12] In 2021, both the Mandelstam-Tamm and the Margolus-Levitin QSL bounds were concurrently tested in a single experiment^[13] which indicated there are "two different regimes: one where the Mandelstam-Tamm limit constrains the evolution at all times, and a second where a crossover to the Margolus-Levitin limit occurs at longer times."

Mandelstam-Tamm limit

Let ${\displaystyle ds^{2))$ be the Bures metric, defined by

$${\displaystyle ds(\rho ,\rho +d\rho )^{2}={\frac {1}{2))\sum _{jk}{\frac {|\langle j|d\rho |k\rangle |^{2)){p_{j}+p_{k))))$$

${\displaystyle H}$

$${\displaystyle ds\leq {\frac {\sigma _{H}(t)}{\hbar ))dt}$$

${\displaystyle \sigma _{H}(t)}$

${\displaystyle t}$

Two corollaries:

• The time taken to evolve from ${\displaystyle \rho }$ to ${\displaystyle \rho '}$ is ${\displaystyle \tau \geq {\frac {\hbar }((\bar {\sigma ))_{H))}dist(\rho ,\rho ')}$, where is ${\displaystyle {\bar {\sigma ))_{H}:={\frac {1}{\tau ))\int _{0}^{\tau }\sigma _{H}(t)dt}$ the time-averaged uncertainty in energy.
• The time taken to evolve from one pure state to another pure state orthogonal to it is ${\displaystyle \tau \geq {\frac {\pi }{2)){\frac {\hbar }((\bar {\sigma ))_{H))))$. ^[14]

Applications

Computation machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. Suppose the computing machinery is a physical system evolving under Hamiltonian that does not change with time. Then according to the Margolus–Levitin theorem, its operations per time per energy obey

$${\displaystyle {\frac {1/\delta t_{\perp )){E))\leq {\frac {1}{\hbar {\frac {\pi }{2))))=6\times 10^{33}\mathrm {s} ^{-1}\cdot \mathrm {J} ^{-1}.}$$