Universal energy equalization under Haar-random unitary operations

Area of Science:

• Quantum thermodynamics
• Statistical mechanics
• Quantum information theory

Background:

• Finite quantum systems can be prepared in specific initial states, such as energy-diagonal ensembles.
• Unitary operations, particularly Haar-random ones, are fundamental tools in quantum information and can drive system evolution.
• Understanding energy dynamics and work extraction in quantum systems is crucial for developing quantum technologies.

Purpose of the Study:

• To investigate the work done on finite quantum systems under Haar-random unitary operations.
• To determine if energy distribution equalization occurs and its implications for work extraction.
• To explore the conditions under which similar energy equalization phenomena arise, such as during unitary quenches.

Main Methods:

• Theoretical analysis of work done on finite quantum systems subjected to Haar-random unitary operations.
• Investigating the shift in energy distribution towards the infinite-temperature value.
• Numerical simulations using a spin model to illustrate the predicted behavior.
• Analyzing unitary quenches with Hamiltonians satisfying the Eigenstate Thermalization Hypothesis and spectral conditions.

Main Results:

• On average, the system's energy shifts towards the infinite-temperature value, leading to energy distribution equalization.
• States with energy above the infinite-temperature value are identified as a resource for work extraction.
• The energy equalization trend becomes typical in large systems due to vanishing fluctuations.
• Similar energy equalization is observed in unitary quenches when specific spectral conditions are met.

Conclusions:

• Haar-random unitary operations drive finite quantum systems towards an infinite-temperature-like energy distribution.
• This energy equalization process indicates that high-energy states can be utilized as a resource for work extraction.
• The Eigenstate Thermalization Hypothesis, along with nondegeneracy and flatness conditions, is sufficient to reproduce Haar-typical energy equalization during unitary quenches.