Random matrix theory of the isospectral twirling

Abstract

We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref. [1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs, Entanglement, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CP-maps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones.

Quantum Chaos is Quantum

“…Here, we want to make some more general considerations. Given a probe to quantum chaos defined as P t (U ) := tr(T (t) O 1 U ⊗t O 2 U †⊗t ), see [7,10], we can establish the following Proposition 1. Let P t (U ) a probe of quantum chaos of order t. If the number k of non Clifford gates in the doped Clifford circuit U ∈ C k is k = O((α + t)N t 4 log 2 (t)), then:…”

“…Quantum chaos is a certain type of complex quantum behavior that results in the exponential decay of outof-time-order correlation functions (OTOC) [2][3][4] efficient operator spreading [5,6], small fluctuations of the purity [7] and information scrambling [8,9]. All these quantities can be unified in a single framework [10] which shows that, in order to simulate quantum chaos, one needs at least a unitary 4-design, that is, a set of unitary operators that reproduces up to the four moments of the Haar distribution over the unitary group U(d) in an d-dimensional Hilbert space.…”

Energy storage and coherence in closed and open quantum batteries

“…Since in an open quantum system coherence is typically exponentially suppressed, we are interested in showing how decoherence free subspaces [62,72,100] can be used to obtain more efficient quantum batteries. In the spirit of the typicality arguments used both in [33,81,102] we can ask how typical quantum maps can be used to exchange energy. Finally, an important generalization would be to take in consideration the entropy change in open quantum systems and extend these results to the free energy available to a quantum battery.…”

“…We derive upper bounds to the energy, and thus also find maximum bounds on the ergotropy, which is the maximum energy change when maximizing over unitary operations [31], and in particular of the isospectral twirling of the work, e.g. the spectral-preserving unitary average over the time evolution [33,70,81], or entangling power [52].…”

Isospectral Twirling and Quantum Chaos

“…Such behavior is ruled by spectral properties, while the eigenvectors are being chosen to be Haar-like, that is, we use the Haar-isospectral twirling (1). We will see that, insofar only the properties of the spectrum of the Hamiltonian H are concerned, different ensembles of spectra associated with different RMT distinguish the temporal profile of the chaos probes in the transient before the onset of the asymptotic behavior, which is the same for all the ensembles of spectra with a Schwartzian probability distribution [67]. By averaging over the unitary group in Equation ( 1), we have, on the one hand, effectively erased any information coming from the eigenstates of the Hamiltonian, and on the other hand, already introduced some of the properties of chaotic or ergodic Hamiltonians.…”