Relative Entropy of Random States and Black Holes

Abstract

We study the relative entropy of highly excited quantum states. First, we sample states from the Wishart ensemble and develop a large-N diagrammatic technique for the relative entropy. The solution is exactly expressed in terms of elementary functions. We compare the analytic results to small-N numerics, finding precise agreement. Furthermore, the random matrix theory results accurately match the behavior of chaotic many-body eigenstates, a manifestation of eigenstate thermalization. We apply this formalism to the AdS/CFT correspondence where the relative entropy measures the distinguishability between different black hole microstates. We find that black hole microstates are distinguishable even when the observer has arbitrarily small access to the quantum state, though the distinguishability is nonperturbatively small in Newton's constant. Finally, we interpret these results in the context of the subsystem Eigenstate Thermalization Hypothesis (sETH), concluding that holographic systems obey sETH up to subsystems half the size of the total system.

5+3 pages; v2: two appendices, discussion of implications for quantum hypothesis testing, and references added

References in corpus (4)

• Black holes as mirrors: quantum information in random subsystems
• Strong and weak thermalization of infinite non-integrable quantum systems
• Relative entropy and the Bekenstein bound
• Relative entropy of excited states in conformal field theories of arbitrary dimensions

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• A Modified Cosmic Brane Proposal for Holographic Renyi Entropy
• The refined quantum extremal surface prescription from the asymptotic equipartition property
• Subsystem distances between quasiparticle excited states
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• Large and Small Corrections to the JLMS Formula from Replica Wormholes
• Symmetry protected entanglement in random mixed states
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• Perturbative distinguishability of black hole microstates from AdS/CFT correspondence
• Construction of Quantum Target Space from World-Sheet States using Quantum State Tomography
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