Chisa Hotta : Typicality of Quantum States

Speaker : Chisa Hotta, Professor, University of Tokyo

Abstract:

Quantum many-body systems in thermal equilibrium are conventionally described by the Gibbs state, which involves an exponentially large number of basis states for a system of size (N). Its direct construction and manipulation therefore become practically infeasible for sufficiently large systems.

However, studies since the early 2000s have established that one can often bypass the Gibbs ensemble and instead use a single pure state randomly chosen from a microcanonical energy shell. Such a state can reproduce the expectation values of local observables with high accuracy, equivalent to those obtained from the Gibbs ensemble. This remarkable property is known as canonical typicality or, more broadly, thermal typicality.

We have been developing a method to construct such thermal typical states using matrix product states (MPS), which we call thermal pure quantum MPS (TPQ-MPS). The basic idea is to start from a random state representing the infinite-temperature limit and cool it down to a target temperature through imaginary-time evolution. In principle, a sufficiently entangled pure state can encode the full thermal entropy, with its entanglement entropy following a volume law. In practice, however, the entanglement that can be stored in an MPS is limited by the available computational resources. We compensate for this limitation by introducing random sampling and averaging, which effectively incorporates the missing entropy and leads to states that are neither fully pure nor fully mixed.

This provides a framework to classify thermal-state representations according to their degree of purity, ranging from the Gibbs ensemble to a fully developed thermal pure quantum (TPQ) state. We will discuss how this classification can be quantified, how it depends on the available computational resources, and how TPQ-MPS can be applied to thermal states of two-dimensional quantum systems, including the Kitaev model.

The concept of typicality also extends to nonequilibrium dynamics. By evolving a single TPQ state in real time, one can reproduce thermal ensemble dynamics without explicitly handling the exponentially large Gibbs state. This dynamical typicality offers a powerful route to long-time quantum dynamics, beyond the time scales accessible to many conventional numerical approaches.

If time permits, we will also discuss ongoing developments toward variational approaches for constructing thermal states.

Related papers:

A Iwaki, A Shimizu, C Hotta, Phys. Rev. Research 3, L022015 (2021)

A Iwaki, C Hotta, Phys. Rev. B 106, 094409 (2022)

A Iwaki, C Hotta, Journal of Physics: Conference Series 2207, 012031 (2022)

M Gohlke, A Iwaki, C Hotta, SciPost Physics 15, 206 (2023)

A. Iwaki, C. Hotta,Phys. Rev.B 109, 224410 (2024)