Microstate Distinguishability, Quantum Complexity, and the Eigenstate Thermalization Hypothesis

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.

19 pages, 4 figures; minor changes

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Cited by in corpus (4)

• Quantum Computational Complexity -- From Quantum Information to Black Holes and Back
• Fixing detailed balance in ancilla-based dissipative state engineering
• Wavefunction branching: when you can't tell pure states from mixed states
• A Strategy for Proving the Strong Eigenstate Thermalization Hypothesis : Chaotic Systems and Holography