Microstate Distinguishability, Quantum Complexity, and the Eigenstate Thermalization Hypothesis

TL;DR

This paper uses quantum complexity theory to analyze how difficult it is to distinguish eigenstates satisfying ETH, revealing a deep connection between quantum complexity and thermalization properties.

Contribution

It establishes a quantitative link between quantum complexity and the distinguishability of ETH eigenstates, providing new insights into thermalization and quantum information.

Findings

Trace distance between eigenstates is exponentially suppressed under ETH.

Exponential hardness of state discrimination implies ETH-like matrix elements.

Quantum complexity bounds the difficulty of distinguishing thermal eigenstates.

Abstract

In this work, we use quantum complexity theory to quantify the difficulty of distinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis (ETH). After identifying simple operators with an algebra of low-energy observables and tracing out the complementary high-energy Hilbert space, the ETH leads to an exponential suppression of trace distance between the coarse-grained eigenstates. Conversely, we show that an exponential hardness of distinguishing between states implies ETH-like matrix elements. The BBBV lower bound on the query complexity of Grover search then translates directly into a complexity-theoretic statement lower bounding the hardness of distinguishing these reduced states.