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.
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.
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 analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2) symmetry, constrained to fixed total spin J and magnetization Jz=0. Focusing on spin-21 lattices and subsystem fractions f<21, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume V. In addition to the expected leading volume law term, we prove the existence of a 21logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2) symmetry shapes average entanglement.
We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.
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.
Quantum many-body systems display an extraordinary degree of complexity, yet many of their features are universal: they depend not on microscopic details, but on a few fundamental physical aspects such as symmetries. A central challenge is to distill these universal characteristics from model-specific ones. Random quantum states sampled from a uniform distribution, the Haar measure, provide a powerful framework for capturing this typicality. Here, we experimentally study the entanglement and symmetries of random many-body quantum states generated by evolving simple product states under ergodic Floquet models. We find excellent agreement with the predictions from the Haar-random state ensemble. First, we measure the Rényi-2 entanglement entropy as a function of the subsystem size, observing the Page curve. Second, we probe the subsystem symmetries using entanglement asymmetry. Finally, we measure the moments of partially transposed reduced density matrices obtained by tracing out part of the system in the generated ensembles, thereby revealing distinct entanglement phases. Our results offer an experimental perspective on the typical entanglement and symmetries of many-body quantum systems.