Quantum speed limit theorems are quantum mechanics theorems concerning the orthogonalization interval, the minimum time for a quantum system to evolve between two orthogonal states.
Consider an initial pure quantum state expressed as a superposition of its energy eigenstates . If the state is let to evolve for an interval by the Schrödinger equation it becomes , where is the reduced Planck constant. If the initial state is orthogonal to the evolved state then and the minimum interval required to achieve this condition is called the orthogonalization interval or time.
Mandelstam-Tamm theorem
The Mandelstam-Tamm theorem states that
, where
is the variance of the system's energy and is the Hamiltonian operator. The theorem is named after Leonid Mandelstam and Igor Tamm. In this case, quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space; it is the distance along this curve measured by the Fubini-Study metric.
Proof
We want to find the smallest interval such that
.
We note that
using Euler's formula and noting that the sine function is odd. Then
, since , . We note that . Thus . Since then if . So the second term vanishes for and .
For this bound to become an equality we demand , that is or . Thus
, which holds for only two energy eigenstates and . Thus, the only state that attains this bound is a two-level pure quantum state (qubit) in an equal superposition
of energy eigenstates and , unique up to degeneracy of the energy level and arbitrary phase factors , of the eigenstates.
Margolus–Levitin theorem
The Margolus–Levitin theorem states that
, where , is the system's average energy and is the Hamiltonian operator, such that does not depend on time; has zero ground state energy.
The theorem is named after Norman Margolus and Lev B. Levitin.
Proof
We want to find the smallest interval such that
. We note that , as . Since requires then .
For this bound to become an equality we demand , that is or . Thus
, which holds for only two energy eigenstates and . Thus, the only state that attains this bound is a two-level pure quantum state (qubit) in an equal superposition
of energy eigenstates and , unique up to degeneracy of the energy level and arbitrary phase factors , of the eigenstates.
Time-varying Hamiltonian
The Margolus-Levitin theorem generalizes to the case with time-varying Hamiltonian and mixed states.
Let be the Hamiltonian at time interval , such that still has zero energy in the ground state. Let the system start at some mixed state with density operator and evolve by the Schrödinger equation over time. Then , where is the Bures distance between the starting state and the ending state.
To obtain the original theorem, set to be independent of time, and , then since pure states evolve to pure states, , and so by the formula for the Bures distance between pure states, , and when the starting and ending states are orthogonal, we obtain . However, the Margolus–Levitin theorem has not yet been established in time-dependent quantum systems, whose Hamiltonians are driven by arbitrary time-dependent parameters, except for the adiabatic case.
Other relevant theorems
Relevant theorems concerning the Margolus–Levitin and the Mandelstam-Tamm theorems were proved in 2009 by Lev B. Levitin and Tommaso Toffoli.
Theorem
In the case the orthogonalization interval satisfies
Theorem
For any state
, where is the maximum energy eigenvalue of and , wherein for the qubit state with .
Proof
Let
. Assume a contrario that . We can define . But then . Thus, replacing with does not change and therefore the set of energy eigenvalues is bounded from above. To prove the existence of the lower bound on , let the average energy be . We note that replacing energy levels in with will not affect its validity. But after such a replacement, the average energy is , and we can chose . Thus . Using the bound on from the Margolus–Levitin theorem completes the proof.
Furthermore if then
, which is satisfied iff , , and .
See also
Bekenstein bound
Bremermann's limit
Landauer's principle
Kolmogorov complexity
Koomey's law
Limits to computation
Moore's law
Uncertainty principle
References
Lloyd, Seth; Ng, Y. Jack, "Black Hole Computers", Scientific American (April 2007), p. 53–61