Abstract We probe the quantum speed limit (QSL) of an electron when it is trapped in a non-uniform magnetic field. We show that the QSL increases to a large value, but within the regime of causality, by choosing a proper variation in magnetic fields. We also probe the dependence of QSL on spin of electron and find that it is higher for spin-down electron in the relativistic regime. This can be useful in achieving faster speed of transmission of quantum information. Further, we use the Bremermann–Bekenstein bound to find a critical magnetic field that bridges the gap between non-relativistic and relativistic treatments and relates to the stability of matter. An analytical framework is developed. We also provide a plausible experimental design to supplement our theory.
Quantum speed limit (QSL) defines the theoretical upper bound on how fast a quantum system can evolve between states. It imposes a fundamental constraint on the rate of quantum information processing. For a relativistic spin-up electron in a uniform magnetic field, QSL increased with the magnetic field strength till around
1015 Gauss, before saturating at a saturated QSL (SQSL) of 0.2407c, where c is the speed of light. We show that by using variable magnetic fields, it is possible to surpass this limit, achieving SQSL upto 0.4-0.6c. To attain this quantum phenomenon, we solve the evolution equation of relativistic electron in spatially varying magnetic fields and find that the energies of various electron states become non-degenerate as opposed to the constant magnetic field case. This redistribution of energy is the key ingredient to accomplish higher QSL and, thus, a high information processing speed. We further explore how QSL can serve as a bridge between relativistic and non-relativistic quantum dynamics, providing insights via the Bremermann-Bekenstein bound, a quantity which constrains the maximal rate of information production. We also propose a practical experimental setup to realize these advancements. These results hold immense potential for propelling fields of quantum computation, thermodynamics and metrology.
We analyze the influence of relativistic effects on the minimum evolution time between two orthogonal states of a quantum system. Defining the initial state as a homogeneous superposition between two Hamiltonian eigenstates of an electron in a uniform magnetic field, we obtain a relation between the minimum evolution time and the displacement of the mean radial position of the electron wave packet. The quantum speed limit time is calculated for an electron dynamics described by Dirac and Schr\"odinger-Pauli equations considering different parameters, such as the strength of magnetic field and the linear momentum of the electron in the axial direction. We highlight that when the electron undergoes a region with extremely strong magnetic field the relativistic and nonrelativistic dynamics differ substantially, so that the description given by the Schr\"odinger-Pauli equation enables the electron to travel faster than
c, which is prohibited by Einstein's theory of relativity. This approach allows a connection between the abstract Hilbert space and the space-time coordinates, besides the identification of the most appropriate quantum dynamics used to describe the electron motion.