Quantum information dynamics is an emerging field that connects several topics, including non-equilibrium and driven quantum systems, many-body localization and thermalization, quantum chaos and black holes, and tensor network holography. Traditionally, physics has focused on the dynamics of matter and spacetime. However, considering the dynamics of quantum information is a new trend that deepens our understanding of the dynamics of matter and spacetime.

Entanglement entropy is a tool used to quantify quantum information, as well as various information measures constructed from it. One of the most prominent differences between quantum and classical information is the phenomenon of quantum entanglement. In quantum systems, information is not stored in any particular entity that makes up the system, but rather in the relationships among the entities. Quantum entanglement is a non-local information storage scheme that is unique to quantum systems. Any attempt to separate a subsystem from an entangled quantum system would cut off its quantum entanglement with the rest of the system, leading to an information loss or an increase in entropy. The amount of entropy associated with separating the subsystem is the entanglement entropy, which also quantifies the entanglement between the subsystem and its counterpart.

For a given quantum many-body state, each choice of the subsystem (which may be disconnected) is associated with an entanglement entropy. Several quantum information measures, such as mutual or tripartite information, can be constructed by adding and subtracting entanglement entropies over different subsystems. However, the amount of data is enormous. For a system of \(N\) qubits, there are \(2^N\) different choices of subsystems, and thus potentially \(2^N\)different entanglement entropies. One approach to arranging this data efficiently is to encode the entanglement entropies in the energy function of a statistical mechanics model.

The idea is to consider each choice of the subsystem as an Ising configuration, where the qubits within (outside) the subsystem are marked with a down (up) spin. In this way, the entanglement entropy of a subsystem can be thought of as the energy associated with the corresponding Ising configuration. Once the energy function is defined, the principles of statistical mechanics can be used to define a statistical ensemble of subsystems. The probability of each choice of subsystem is proportional to the Boltzmann weight \(W=e^{-S}\) set by the entanglement entropy \(S\). This Boltzmann weight was first proposed and named as the entanglement feature in arXiv:1709.01223. The entire statistical ensemble encodes the entanglement entropies over all possible subsystems of a given quantum many-body state. For instance, a disentangled product state corresponds to the paramagnetic limit of the Ising ensemble, where all spin configurations appear with equal probability. On the other hand, a strongly entangled state corresponds to an Ising ensemble with strong ferromagnetic correlation.

As the many-body state evolves with time, the entanglement features also evolve. Thus, the dynamics of quantum information can be formulated as the dynamics of the entanglement features that describe the entanglement Ising ensemble. Our group is working to develop and apply this idea to understand quantum information dynamics in quantum chaotic arXiv:1803.10425 or periodically driven systems. This Ising formulation of quantum many-body entanglement is closely related to tensor network holography, which allows us to explore the corresponding dynamics in the holographic bulk as well.