Universal fluctuations around typicality for quantum ergodic systems

Abstract

For a quantum system in a macroscopically large volume V , prepared in a pure state and subject to maximally noisy or ergodic unitary dynamics, the reduced density matrix of any sub-system v ≪ V is almost surely totally mixed. We show that the fluctuations around this limiting value, evaluated according to the invariant measure of these unitary flows, are captured by the Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this statement, applicable when the unitary transformations conserve the energy but are maximally noisy or ergodic on any energy shell, allows to decipher the fluctuations around canonical typicality. According to typicality, if the large system is prepared in a generic pure state in a given energy shell, the reduced density matrix of the sub-system is almost surely the canonical Gibbs state of that sub-system. We show that the fluctuations around the Gibbs state are encoded in a deformation of the GUE whose covariance is specified by the Gibbs state. Contact with the eigenstate thermalisation hypothesis (ETH) is discussed.Introduction.-Understanding how isolated quantum systems relax towards thermal equilibrium is a central question in quantum statistical physics. There are essentially two lines of thought in the literature: an ensemblist point of view, according to which a system in thermal equilibrium is represented by a statistical ensemble of states close to the canonical or micro-canonical mixed state, and a purist perspective, according to which a large system in a single pure state may behave as thermal for any local measurement. See [1][2][3] for thorough discussions. These understandings led to the related notions of typicality [4][5][6], partially present in Von Neumanns work [7], and of eigen-state thermal hypothesis (ETH) [8][9][10], formalising under which conditions a pure state can describe thermal equilibrium locally.Thermalisation occurs via entanglement which allows information to spread over the system. Recent studies of random quantum circuits [11][12][13][14] gave detailed hints on this mechanism. Thermalisation may be quantified by looking at properties of the system state reduced to small sub-systems, such as the entanglement spectrum [15]. It is now believed that there is a link between thermalisation and the distribution of the entanglement spectrum [16,17], e.g. whether it is Poisson or Wigner-Dyson like. For instance, entanglement spectrum statistics have been used as diagnosis for thermalisation, distinguishing Clifford circuits from universal quantum circuits [18]. This establishes a link between thermalisation, which is a statement about typical behaviour, and fluctuations, which goes beyond typical behaviours.This letter aims at making this link more precise, and possibly rigorous. Under the assumption of some form of ergodicity, we present a simple derivation and characterisation of the fluctuations of density matrices reduced to small sub-systems embedded in a large system prepared in a pure state. The form of these fluctuations makes contac...

Equilibration of quantum cat states

“…It would be great to have an equivalent statement here. A possible way for obtaining such simplification in our case already discussed in [25] would be the following : We can suppose that the "actual" group whose action leaves invariant the stationary state is not given by the set of all unitaries that commutes with H but rather with an Hamiltonian H = H + δH with δH a small perturbation which ¡¡mixes¿¿ the different energy sectors separated by energy ≈ δH. The microcanonical ensemble is recovered in the case where the spectrum of H is fully degenerate in the energy window of interest [E − δE, E + δE].…”

Equilibration of quantum cat states

“…It would be great to have an equivalent statement here. A possible way for obtaining such simplification in our case already discussed in [26] would be the following : We can suppose that the "actual" group whose action leaves invariant the stationary state is not given by the set of all unitaries that commutes with H but rather with an Hamiltonian H = H+δH with δH a small perturbation which "mixes" the different energy sectors separated by energy ≈ δH. The microcanonical ensemble is recovered in the case where the spectrum of H is fully degenerate in the energy window of interest [E − δE, E + δE].…”

Rank one HCIZ at high temperature: interpolating between classical and free convolutions

“…The HCIZ integral has applications in problems directly linked to random matrix theory (RMT) such as the study of the sum of invariant ensembles [22][23][24], the development of large deviation principles [25], the study of the so-called orbital beta processes [26]. It is also linked to the enumeration of Hurwitz numbers in algebraic geometry [27,28] and to quantum ergodic transport ( [29]), to cite a few recent results. It is then tempting to try to generalize this formula for arbitrary positive β, just like one can study the eigenvalue distribution of β ensembles in RMT for general β [1].…”