Quantum systems of many particles have always fascinated researchers, but understanding them is no easy task. The sheer number of quantum states in these systems increases exponentially with size, making them challenging to explore and characterize. Furthermore, interactions with the environment make these systems even more complex. However, a recent study by Kohei Kawabata of Princeton University and colleagues has made significant progress towards developing a theoretical framework for open, many-body quantum systems [1].

Kawabata and his team have proposed a classification system based on symmetry principles, which will provide a roadmap for researchers to explore the vast range of phenomena that can arise in these systems, including quantum chaos. This achievement builds upon earlier breakthroughs in random matrix theory, which was introduced by Eugene Wigner in the 1950s to describe the energy spectra of heavy nuclei [2,3]. Random matrices have since become a valuable tool in various fields, including physics, finance, and neuroscience [5]. They also hold promise for studying quantum chaos, which investigates how classical chaos emerges from quantum principles [6].

The classification system developed by Kawabata and colleagues is based on Lindbladians, which are mathematical objects that describe the dynamics of open quantum systems. In 1976, physicist Göran Lindblad introduced the concept of a Lindblad master equation, which governs the evolution of a forgetful or Markovian open quantum system [7]. In this equation, the Hamiltonian is replaced by a more complex mathematical object called the Lindbladian superoperator.

The researchers’ classification framework extends the work done on random matrices and Lindbladians. In 2019, Kawabata and colleagues generalized the Altland-Zirnbauer classification—originally applied to Hamiltonians—to non-Hermitian matrices [9]. However, Lindbladians possess additional constraints due to their interaction with the environment, making the classification more challenging. Despite this obstacle, the researchers were able to overcome it and generalize the classification of non-Hermitian systems to arbitrary Lindbladians.

This classification framework provides researchers with a powerful tool for understanding and exploring open, many-body quantum systems. It opens up new avenues for studying phenomena like quantum chaos and will aid in the discovery of novel properties and behaviors in these systems. With this new framework, researchers can navigate the complex landscape of quantum systems and uncover deeper insights into their nature.

### FAQs

Question 1: What is quantum chaos?

Answer: Quantum chaos investigates how classical chaos emerges from quantum principles. It explores the exponential sensitivity of a system to its initial conditions, often referred to as the butterfly effect.

Question 2: What is a Lindbladian?

Answer: A Lindbladian is a mathematical object that describes the dynamics of an open quantum system. It replaces the role of the Hamiltonian in the Lindblad master equation, which governs the evolution of a Markovian open quantum system.

Question 3: How does the classification framework help researchers?

Answer: The classification framework based on symmetry principles provides researchers with a roadmap for exploring the vast range of phenomena in open, many-body quantum systems. It aids in understanding quantum chaos and discovering new properties and behaviors in these systems.

Sources:

[1] https://physics.aps.org/articles/v16/149

[2] E. Wigner, Phys. Rev. 98, 145 (1955).

[3] E. Wigner, J. Math. Phys. 2, 273 (1961).

[4] F. J. Dyson, J. Math. Phys. 3, 140 (1962).

[5] M. L. Mehta, Random Matrices (Elsevier, 2004).

[6] G. Casati and B. V. Chirikov, Quantum Chaos: Between Order and Disorder (Cambridge University Press, 1995).

[7] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).

[8] A. Altland and M. Zirnbauer, Phys. Rev. B 55, 1142(R) (1997).

[9] Y. Ashida et al., Phys. Rev. X 9, 021053 (2019).