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Physical review. E

Distribution of the entanglement entropies of nonergodic quantum states

Devanshu Shekhar, Pragya Shukla

Published May 21, 2025• 1 citation
Publisher

Abstract

The increasing relevance of Haar ensembles of pure ergodic quantum states for quantum information processes has been eclipsed by the challenges in their experimental realization. As a result, alternative ensembles are needed that can approach them approximately. This motivates us to begin from a multiparametric ensemble of pure bipartite states, nonuniformly distributed in Hilbert space, and to seek a route to Haar ensembles by changing the ensemble parameters and thereby the entanglement entropy distribution. We show that the latter's variation is not sensitive to individual details of the ensemble parameters, rather it is governed collectively, i.e., it is a function of all of them and can thus be described by a common mathematical formulation for a wide range of nonergodic states. This has important implications for quantum state engineering: access from an arbitrary initial state ensemble to Haar ensembles can be controlled just by a single function representing collective information about the ensemble parameters.

DOI
10.1103/n16c-45rh

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The increasing relevance of Haar ensembles of pure ergodic quantum states for quantum information processes has been eclipsed by the challenges in their experimental realization. As a result, alternative ensembles are needed that can approach them approximately. This motivates us to begin from a multiparametric ensemble of pure bipartite states, nonuniformly distributed in Hilbert space, and to seek a route to Haar ensembles by changing the ensemble parameters and thereby the entanglement entropy distribution. We show that the latter's variation is not sensitive to individual details of the ensemble parameters, rather it is governed collectively, i.e., it is a function of all of them and can thus be described by a common mathematical formulation for a wide range of nonergodic states. This has important implications for quantum state engineering: access from an arbitrary initial state ensemble to Haar ensembles can be controlled just by a single function representing collective information about the ensemble parameters.
We consider physical Hamiltonians that can be represented by the multiparametric Gaussian ensembles, theoretically derive the state ensembles for its eigenstates and analyze the effect of varying system conditions on its bipartite entanglement entropy. Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics of different states of a given Hamiltonian or different Hamiltonians subjected to same symmetry constraints. The parameter turns out to be a single functional of the system parameters and thereby reveals a deep web of connection hidden underneath different quantum states.
We analyze the subsystem size scaling of the entanglement entropy of a nonergodic pure state that can be described by a multiparametric Gaussian ensemble of complex matrices in a bipartite basis. Our analysis indicates, for a given set of global constraints, the existence of an infinite number of universality classes of local complexity, characterized by the complexity parameter, for which the entanglement entropy reveals a universal scaling with subsystem size. A rescaling of the complexity parameter helps us to identify the critical regime for the entanglement entropy of a broad range of pure nonergodic states.
Abstract A non-ergodic quantum state of a complex system is in general random as well as multi-parametric, former due to a lack of exact information due to complexity and latter reflecting its varied behavior in different parts of the Hilbert space. An appropriate representation for the reduced density matrix of such a state is a generalized, multi-parametric Wishart ensemble with unit trace. Our theoretical analysis of these ensembles not only resolves the controversy about the growth rates of the average information entropies of the generic states but also leads to new insights in their entanglement dynamics. While the state itself is multi-parametric, we find that the growth of the average measures can be described in terms of an information-theoretic function, referred as the complexity parameter. The latter in turn leads to a common mathematical formulation of the measures for a wide range of states; it could also act as a possible tool for hierarchical arrangement of the entangled states of different systems.
We investigate multipartite entanglement via the statistical properties of pure quantum states of n-qubits. By analyzing the distribution of purity among balanced bipartitions, we compare Haar-typical states, uniformly distributed on the unit sphere of states, with Hadamard states, being characterized by equal weights in the computational basis. We analyze different ensembles of Hadamard states characterized by their phase distributions. Through analytical and numerical calculations, we show that Hadamard states exhibit, on average, a higher degree of entanglement than Haar-typical states. In addition, we show that a particular class of Hadamard states, characterized by real coefficients with alternating signs, known as hypergraph states, appears especially relevant in the search for maximally multipartite entangled states, both for their structural simplicity and the increased likelihood of sampling highly entangled states. These results identify Hadamard states as a tractable yet promising class for exploring multipartite entanglement structures and advancing the characterization of maximally multipartite entangled quantum states.
In this work, we establish a relation between entanglement entropy and fractal dimension D of generic many-body wave functions, by generalizing the result of Page [Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291] to the case of sparse random pure states (SRPS). These SRPS living in a Hilbert space of size N are defined as normalized vectors with only N^{D} (0≤D≤1) random nonzero elements. For D=1, these states used by Page represent ergodic states at an infinite temperature. However, for 0<D<1, the SRPS are nonergodic and fractal, as they are confined in a vanishing ratio N^{D}/N of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy S_{1}(A) of a subsystem A, with Hilbert space dimension N_{A}, scales as S_{1}[over ¯](A)∼DlnN for small fractal dimensions D, N^{D}<N_{A}. Remarkably, S_{1}[over ¯](A) saturates at its thermal (Page) value at an infinite temperature, S_{1}[over ¯](A)∼lnN_{A} at larger D. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly nonergodic. Finally, we generalize our results to Renyi entropies S_{q}(A) with q>1 and to genuine multifractal states and also show that their fluctuations have ergodic behavior in a narrower vicinity of the ergodic state D=1.
We theoretically derive the probability densities of the entanglement measures of a pure non-ergodic many-body state, represented in a bipartite product basis and with its reduced density matrix described by a generalized, multi-parametric Wishart ensemble with unit trace. Our results indicate significant fluctuations of the measures around their average behavior (specifically for the states away from separability and maximum entanglement limits). The information is relevant not only for hierarchical arrangement of entangled states (e.g., revealing the flaws in their characterization based on average behavior) but also for phase transition studies of many body systems.