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arXiv:2312.11635 · Journal of High Energy Physics

Spread complexity for measurement-induced non-unitary dynamics and Zeno effect

Aranya Bhattacharya, Rathindra Nath Das, Bidyut Dey, Johanna Erdmenger B

Published Mar 29, 2024• 29 citations
arXivHTMLPDF

Abstract

A bstract Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schrödinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.

DOI
10.1007/jhep03(2024)179

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A bstract Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schrödinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.
A bstract We consider the statistics of the results of a measurement of the spreading operator in the Krylov basis generated by the Hamiltonian of a quantum system starting from a specified initial pure state. We first obtain the probability distribution of the results of measurements of this spreading operator at a certain instant of time, and compute the characteristic function of this distribution. We show that the moments of this characteristic function are related to the so-called generalised spread complexities, and obtain expressions for them in several cases when the Hamiltonian is an element of a Lie algebra. Furthermore, by considering a continuum limit of the Krylov basis, we show that the generalised spread complexities of higher orders have a peak in the time evolution for a random matrix Hamiltonian belonging to the Gaussian unitary ensemble. We also obtain an upper bound in the change in generalised spread complexity at an arbitrary time in terms of the operator norm of the Hamiltonian and discuss the significance of these results.
Krylov complexity is a measure of operator complexity that exhibits universal behavior and bounds a large class of other measures. In this paper, we generalize Krylov complexity from a closed system to an open system coupled to a Markovian bath, where Lindbladian evolution replaces Hamiltonian evolution. We show that Krylov complexity in open systems can be mapped to a non-Hermitian tight-binding model in a half-infinite chain. We discuss the properties of the non-Hermitian terms and show that the strengths of the non-Hermitian terms increase linearly with the increase of the Krylov basis index nnn. Such a non-Hermitian tight-binding model can exhibit localized edge modes that determine the long-time behavior of Krylov complexity. Hence, the growth of Krylov complexity is suppressed by dissipation, and at long times, Krylov complexity saturates at a finite value much smaller than that of a closed system with the same Hamiltonian. Our conclusions are supported by numerical results on several models, such as the Sachdev-Ye-Kitaev model and the interacting fermion model. Our work provides insights for discussing complexity, chaos, and holography for open quantum systems.
We show that the characteristic function of the probability distribution associated with the change of an observable in a two-point measurement protocol with a perturbation can be written as an autocorrelation function between an initial state and a certain unitary evolved state by an effective unitary operator. Using this identification, we probe how the evolved state spreads in the corresponding conjugate space, by defining a notion of the complexity of the spread of this evolved state. For a sudden quench scenario, where the parameters of an initial Hamiltonian (taken as the observable measured in the two-point measurement protocol) are suddenly changed to a new set of values, we first obtain the corresponding Krylov basis vectors and the associated Lanczos coefficients for an initial pure state, and define the associated spread complexity. Interestingly, we find that in such a protocol, the Lanczos coefficients can be related to various cost functions used in the geometric formulation of circuit complexity, for example, the one used to define Fubini-Study complexity. We illustrate the evolution of spread complexity both analytically, by using Lie algebraic techniques, and also by performing numerical computations. This is done for cases when the Hamiltonian before and after the quench are taken as different combinations of chaotic and integrable spin chains. We show that the spread complexity saturates for large values of the parameter only when the prequench Hamiltonian is chaotic. Furthermore, in these examples, we also discuss the importance of the role played by the initial state, which is determined by the time-evolved perturbation operator.
A bstract In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the Krylov subspace . The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position 〈 n 〉 defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments 〈 n p 〉 for p > 1, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.
We investigate phase transitions in the extended Su-Schrieffer-Heeger (SSH) model with next-nearest-neighbor hoppings and an imaginary staggered chemical potential. In the presence of small non-Hermiticity, exceptional points emerge in pairs from the gap-closing momenta near the topological phase boundaries of the Hermitian limit. Utilizing the Krylov spread complexity and entanglement entropy, we analyze two dynamical protocols: (i) preparing the non-Hermitian ground state via a unitary transformation, and (ii) evolving the system under the non-Hermitian Hamiltonian. We show that the spread complexity, and long-time spread complexity as well as entanglement entropy can effectively signal phase transitions in the first and second protocols, respectively. To unravel the detailed structure of the transitions, we introduce the momentum-resolved complexity that identifies the characteristic modes and tracks their evolution with the driving parameter. In the regime where the system possesses a purely imaginary spectrum, we further identify dynamical phases based on the saturation behavior of the spread complexity. The entanglement entropy is also found to exhibit similar saturation behavior, thereby providing a more experimentally accessible probe of the dynamical phases.
The complexity of quantum states under dynamical evolution can be investigated by studying the spread with time of the state over a predefined basis. It is known that this complexity is minimized by choosing the Krylov basis, thus defining the spread complexity. We study the dynamics of spread complexity for quantum maps using the Arnoldi iterative procedure. The main illustrative quantum many-body model we use is the periodically kicked Ising spinchain with nonintegrable deformations, a chaotic system where we look at both local and nonlocal interactions. In the various cases, we find distinctive behavior of the Arnoldi coefficients and spread complexity for regular versus chaotic dynamics: suppressed fluctuations in the Arnoldi coefficients as well as larger saturation value in spread complexity in the chaotic case. We compare the behavior of the Krylov measures with that of standard spectral diagnostics of chaos. We also study the effect of changing the driving frequency on the complexity saturation.