Emergent quantum state designs from individual many-body wavefunctions

Abstract

Quantum chaos in many-body systems provides a bridge between statistical and quantum physics with strong predictive power. This framework is valuable for analyzing properties of complex quantum systems such as energy spectra and the dynamics of thermalization. While contemporary methods in quantum chaos often rely on random ensembles of quantum states and Hamiltonians, this is not reflective of most real-world systems. In this paper, we introduce a new perspective: across a wide range of examples, a single non-random quantum state is shown to encode universal and highly random quantum state ensembles. We characterize these ensembles using the notion of quantum state k-designs from quantum information theory and investigate their universality using a combination of analytic and numerical techniques. In particular, we establish that k-designs arise naturally from generic states as well as individual states associated with strongly interacting, timeindependent Hamiltonian dynamics. Our results offer a new approach for studying quantum chaos and provide a practical method for sampling approximately uniformly random states; the latter has wide-ranging applications in quantum information science from tomography to benchmarking.

Exact emergent quantum state designs from quantum chaotic dynamics

“…In other words, there is no protocol performable on A which can information-theoretically differentiate the projected states from uniformly random ones. Theoretical and ex-perimental evidence have been given conjecturing the appearance of such universality in physical systems [9,10]; our results complement these by furnishing an exactlysolvable model where this conjecture can be proven.…”

“…Finally, Lemma 4 of [10] specifies that random variables (|Ψ+ Ψ+|) ⊗k Ψ+|Ψ+ k and 2 t−N A Ψ + |Ψ + are independent, allowing us to distribute the integral: the former equals (3) while the latter evaluates to 1, giving our claimed result.…”

“…To this end we consider here the projected ensemble, introduced in Refs. [9,10]. This is a collection of pure states supported on a local subsystem A, each of which is associated with the outcome of a projective measurement of the complementary subsystem B in a fixed local basis.…”

“…Projected ensembles and quantum state-designs. The projected ensemble is defined as follows [9,10]. Consider a single generator state |Ψ of a large system of N qubits (generalization to a qudit system is immediate), and a bipartition into subsystems A and B with N A and N B qubits respectively.…”

“…We focus in this paper on generator states arising from quench dynamics of systems without explicit conservation laws. As quantum thermalization dictates that the first moment should acquire a universal form ρ (1) E ∝ I A over time, it is natural to conjecture that higher moments become similarly 'maximally-mixed' [9,10]. To quantify this, we appeal to the notion of quantum state-designs in QIT [11][12][13][14], which measures the similarity of E to an ensemble of uniformly (i.e.…”