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Physical Review A

Generation of Bures-Hall mixed states using coupled kicked tops

Ayana Sarkar, Santosh Kumar

Published Mar 18, 2021• 9 citations
Publisher

Abstract

We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer $G$ functions.

DOI
10.1103/physreva.103.032423

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We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer GGG functions.
The entanglement of eigenstates in two coupled, classically chaotic kicked tops is studied in dependence of their interaction strength. The transition from the noninteracting and unentangled system toward full random matrix behavior is governed by a universal scaling parameter. Using suitable random matrix transition ensembles we express this transition parameter as a function of the subsystem sizes and the coupling strength for both unitary and orthogonal symmetry classes. The universality is confirmed for the level spacing statistics of the coupled kicked tops and a perturbative description is in good agreement with numerical results. The statistics of Schmidt eigenvalues and entanglement entropies of eigenstates is found to follow a universal scaling as well. Remarkably, this is not only the case for large subsystems of equal size but also if one of them is much smaller. For the entanglement entropies a perturbative description is obtained, which can be extended to large couplings and provides very good agreement with numerical results. Furthermore, the transition of the statistics of the entanglement spectrum toward the random matrix limit is demonstrated for different ratios of the subsystem sizes.
The Bures distance holds a special place among various distance measures due to its several distinguishing features and has applications in diverse problems in quantum information theory. It is related to fidelity, and among other things, it serves as a bona fide measure for quantifying the separability of quantum states. In this work, we calculate exact analytical results for the mean root fidelity and mean-square Bures distance between a fixed density matrix and a random density matrix and also between two random density matrices. In the course of derivation, we also obtain the spectral density for the product of the above pairs of density matrices. We corroborate our analytical results using Monte Carlo simulations. Moreover, we compare these results with the mean-square Bures distance between reduced density matrices generated using coupled kicked tops and find very good agreement.
One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.
We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify "information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into "regular" and "chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.
In quantum information geometry, the curvature of von-Neumann entropy and relative entropy induce a natural metric on the space of mixed quantum states. Here we use this information metric to construct a random matrix ensemble for states and investigate its key statistical properties such as the asymptotic eigenvalue density and mean entropy. We present an algorithm for generating these entropy-based random density matrices, thus providing a new recipe for random state generation that differs from the well established Hilbert-Schmidt and Bures-Hall ensemble approaches. We also prove a duality between the entropy-based state ensemble and a random Hamiltonian model constructed from the thermodynamic length over the set of Gibbs states. This Hamiltonian model is found to display Wigner level repulsion, implying that the dual state ensemble can be realised as a random Gibbs state with respect to a class of chaotic Hamiltonians. As an application we use our model to compute the survival probability of a randomly evolved thermofield double state, predicting a ramp and plateau over time that is characteristic of quantum chaos. For other applications, the entropy-based ensemble can be used as an uninformative prior for Bayesian quantum state or Hamiltonian tomography.
Mixed state ensembles such as the Bures-Hall and Hilbert-Schmidt measure are probability distributions that characterise the statistical properties of random density matrices and can be used to determine the typical features of mixed quantum states. Here we extend this framework by considering the properties of random states with fixed average energy, and the ensemble-averaged density matrix is derived under this additional physical constraint. This gives rise to a type of microcanonical ensemble for random mixed states and we connect its properties to a statistical mechanical entropy and temperature. Our results are illustrated using a variety of simple spin systems, and we find that they can exhibit exotic features such as phase transitions in the absence of energetic interactions and finite relative energy fluctuations in the thermodynamic limit.