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Entropy for quantum pure states and quantum H theorem
1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
Summary
Researchers developed a new quantum entropy measure using localized Wannier functions. This quantum phase space entropy can dynamically reach a maximum value, similar to the quantum H theorem, even for pure states.
Area of Science:
- Quantum mechanics
- Statistical physics
- Condensed matter physics
Background:
- Standard von Neumann entropy is zero for quantum pure states.
- Quantum phase space representations are crucial for understanding quantum states.
- Wannier functions provide localized basis sets in condensed matter physics.
Purpose of the Study:
- To introduce a novel quantum entropy definition for pure states.
- To explore the dynamical behavior and long-time properties of this new entropy.
- To establish a connection between this new entropy and fundamental theorems in statistical mechanics.
Main Methods:
- Construction of a complete set of Wannier functions localized in both position and momentum.
- Unitary mapping of quantum pure states onto the quantum phase space.
- Definition of a new entropy based on the probability distribution in quantum phase space.
- Proof of an inequality concerning the entropy's fluctuation over time.
Main Results:
- A novel quantum entropy for pure states is defined using quantum phase space probability distributions.
- An inequality regarding the long-time fluctuation of this entropy is proven.
- For typical initial states, the entropy dynamically relaxes to a maximized value with small fluctuations, echoing the quantum H theorem.
- The new entropy is non-zero for pure states and is always greater than the entropy of its subsystems.
Conclusions:
- The developed quantum entropy offers a new perspective on the statistical properties of pure quantum states.
- This entropy provides a measure that reflects the irreversibility and approach to equilibrium in quantum systems.
- The construction and numerical implementation of Wannier functions facilitate the study of this entropy's dynamics.
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