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.
[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.
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."
For example, in applications like Ramsey interferometry, the QSL determines the minimum time required for phase accumulation during control sequences, directly impacting the sensor's temporal resolution and sensitivity.
An arbitrary pure state can be written as a linear combination of energy eigenstates: The task is to provide a lower bound for the time interval
The quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space; the distance along this curve is measured by the Fubini–Study metric.
[16] This is sometimes called the quantum angle, as it can be understood as the arccos of the inner product of the initial and final states.
A mixed state can be understood as a sum over pure states, weighted by classical probabilities; likewise, the Bures metric is a weighted sum of the Fubini–Study metric.
understood as an infinitessimal path length along a curve parametrized by
For a pure state evolving under a time-varying Hamiltonian, the time
For the case of a pure state, Margolus and Levitin[3] obtain a different limit, that where
This form applies when the Hamiltonian is not time-dependent, and the ground-state energy is defined to be zero.
The Margolus–Levitin theorem can also be generalized to the case where the Hamiltonian varies with time, and the system is described by a mixed state.
[18] The Margolus–Levitin theorem has not yet been experimentally established in time-dependent quantum systems, whose Hamiltonians
are driven by arbitrary time-dependent parameters, except for the adiabatic case.
[15] This is a two-level state in an equal superposition for energy eigenstates
This result establishes that the combined limits are strict: Levitin and Toffoli also provide a bound for the average energy in terms of the maximum.
(This is the quarter-pinched sphere theorem in disguise, transported to complex projective space.)
Suppose the computing machinery is a physical system evolving under Hamiltonian that does not change with time.
Then, according to the Margolus–Levitin theorem, the number of operations per unit time per unit energy is bounded above by This establishes a strict upper limit on the number of calculations that can be performed by physical matter.
The processing rate of all forms of computation cannot be higher than about 6 × 1033 operations per second per joule of energy.
[21][22] This bound is not merely a fanciful limit: it has practical ramifications for quantum-resistant cryptography.
Imagining a computer operating at this limit, a brute-force search to break a 128-bit encryption key requires only modest resources.
Brute-forcing a 256-bit key requires planetary-scale computers, while a brute-force search of 512-bit keys is effectively unattainable within the lifetime of the universe, even if galactic-sized computers were applied to the problem.
The Bekenstein bound limits the amount of information that can be stored within a volume of space.
The maximal rate of change of information within that volume of space is given by the quantum speed limit.
[1] That is, Hawking radiation is emitted at the maximal allowed rate set by these bounds.