Ergodicity and Born's rule in an entangled two-qubit Bohmian system

Area of Science:

• Quantum mechanics
• Quantum information theory
• Statistical mechanics

Background:

• Born's rule is a fundamental principle in quantum mechanics, relating probabilities to wave function amplitudes.
• Bohmian mechanics offers a deterministic interpretation of quantum phenomena, utilizing particle trajectories.
• Ergodicity describes systems where time averages equal ensemble averages, relevant for statistical properties.

Purpose of the Study:

• Investigate the connection between ergodicity and the dynamical emergence of Born's rule in entangled two-qubit systems.
• Analyze how entanglement strength influences the validity of Born's rule for arbitrary initial conditions.
• Examine the role of Bohmian trajectories in establishing quantum probabilities.

Main Methods:

• Simulation of Bohmian trajectories for a two-qubit system composed of harmonic oscillators.
• Analysis of ergodicity properties of these trajectories for varying initial conditions and entanglement levels.
• Comparison of trajectory distributions with the predictions of Born's rule (|Ψ|²).

Main Results:

• Most Bohmian trajectories in entangled systems are ergodic, yielding invariant distributions.
• Strong entanglement leads to chaotic-ergodic trajectories that dynamically establish Born's rule for any initial distribution.
• Weak entanglement results in ordered, non-ergodic trajectories; Born's rule is only guaranteed for specific initial conditions.

Conclusions:

• The dynamical establishment of Born's rule is not universally guaranteed by the presence of chaotic and ergodic Bohmian trajectories.
• Entanglement plays a crucial role, with strong entanglement facilitating the emergence of Born's rule.
• The validity of Born's rule depends on both entanglement and the initial state distribution in certain regimes.