Typicality in random matrix product states

TL;DR

This paper investigates whether typicality, a property often associated with random states in Hilbert space, also emerges in physically relevant matrix product states, showing it occurs under certain scaling conditions.

Contribution

The study analytically and numerically demonstrates that typicality appears in matrix product states when their rank scales polynomially with the system size, extending the concept to more realistic states.

Findings

Typicality occurs when the MPS rank scales polynomially with system size.

Numerical evidence suggests typicality may emerge even with linear rank scaling.

Analytical results confirm typicality for rank scaling greater than quadratic.

Abstract

Recent results suggest that the use of ensembles in Statistical Mechanics may not be necessary for isolated systems, since typically the states of the Hilbert space would have properties similar to the ones of the ensemble. Nevertheless, it is often argued that most of the states of the Hilbert space are non-physical and not good descriptions of realistic systems. Therefore, to better understand the actual power of typicality it is important to ask if it is also a property of a set of physically relevant states. Here we address this issue, studying if and how typicality emerges in the set of matrix product states. We show analytically that typicality occurs for the expectation value of subsystems' observables when the rank of the matrix product state scales polynomially with the size of the system with a power greater than two. We illustrate this result numerically and present some indications that typicality may appear already for a linear scaling of the rank of the matrix product state.