Exact emergent quantum state designs from quantum chaotic dynamics

Abstract

We present exact results on a novel kind of emergent random matrix universality that quantum many-body systems at infinite temperature can exhibit. Specifically, we consider an ensemble of pure states supported on a small subsystem, generated from projective measurements of the remainder of the system in a local basis. We rigorously show that the ensemble, derived for a class of quantum chaotic systems undergoing quench dynamics, approaches a universal form completely independent of system details: it becomes uniformly distributed in Hilbert space. This goes beyond the standard paradigm of quantum thermalization, which dictates that the subsystem relaxes to an ensemble of quantum states that reproduces the expectation values of local observables in a thermal mixed state. Our results imply more generally that the distribution of quantum states themselves becomes indistinguishable from those of uniformly random ones, i.e. the ensemble forms a quantum state-design in the parlance of quantum information theory. Our work establishes bridges between quantum many-body physics, quantum information and random matrix theory, by showing that pseudo-random states can arise from isolated quantum dynamics, opening up new ways to design applications for quantum state tomography and benchmarking.

High-temperature thermalization implies the emergence of quantum state designs

“…This shows that the emergence of quantum state designs reported in Refs. [14][15][16] is generic for such thermalizing systems if we consider general measurement bases. Interesting open problems that remain are: Can one show that this emergence of quantum state designs is true for generic, local Hamiltonians in the long-time limit even for atypical measurement bases such as the computational basis?…”

“…Recently, Refs. [14][15][16] asked whether additional information may be obtained when, instead of tracing out the remainder of the system Ā, we measure it in an orthonormal basis {|z Ā} and study the ensemble of post-measurement states…”

“…Interestingly, Refs. [14][15][16], found that for specific systems thermalizing to infinite temperature (i.e., ω A = 1 A /d A ), the ensembles {(p z , |ψ z )} z closely resemble ensembles of quantum states that are chosen uniformly at random from the set of pure states on the Hilbert space H A of A.…”

“…However, Refs. [14][15][16] found that, for specific measurement bases, in particular the computational basis constructed from tensor-product states, the postmeasurement ensemble are approximate k-designs for k > 1 after sufficiently long time. This indicates that these postmeasurement ensembles contain information that is not contained in the reduced density matrix alone and that thermalization in terms of the reduced density matrix is a weaker property than the emergence of quantum state designs as post-measurement ensembles.…”

“…Indeed, Refs. [14,16] found that in concrete models, even measuring the computational basis induces approximate quantum state k-designs after sufficiently long time. Moreover the accuracy of the designs continued to increase even long after the thermalization time.…”