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arXiv:2406.18234 · Physical Review Letters

Measurement-Induced Spectral Transition

Ken Mochizuki, Ryusuke Hamazaki

Published Jan 8, 2025• 7 citations
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Abstract

We show that noisy quantum dynamics exposed to generalized measurements exhibit a spectral transition between gapless and gapped phases. To this end, we employ the Lyapunov spectrum obtained through singular values of a nonunitary matrix describing the dynamics. We discover that the gapless and gapped phases, respectively, correspond to the volume-law and area-law phases of the entanglement entropy for the dominant Lyapunov vector. This correspondence between the spectral gap and the scaling of entanglement offers an intriguing common structure with ground-state phase transitions. We also discuss some crucial differences from ground-state transitions, such as the extensive entanglement and the exponentially small gaps. Furthermore, we show that the spectral transition leads to the transition of the timescale for the memory loss and purification of initial states.

DOI
10.1103/physrevlett.134.010410

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We show that noisy quantum dynamics exposed to generalized measurements exhibit a spectral transition between gapless and gapped phases. To this end, we employ the Lyapunov spectrum obtained through singular values of a nonunitary matrix describing the dynamics. We discover that the gapless and gapped phases, respectively, correspond to the volume-law and area-law phases of the entanglement entropy for the dominant Lyapunov vector. This correspondence between the spectral gap and the scaling of entanglement offers an intriguing common structure with ground-state phase transitions. We also discuss some crucial differences from ground-state transitions, such as the extensive entanglement and the exponentially small gaps. Furthermore, we show that the spectral transition leads to the transition of the timescale for the memory loss and purification of initial states.
Competition among repetitive measurements of noncommuting observables and unitary dynamics can give rise to a wide variety of entanglement phases. Here, we propose a general framework based on Lyapunov analysis to characterize topological properties in monitored quantum systems through their spectrum and many-body topological invariants. We illustrate this framework by analyzing (1+1)-dimensional monitored circuits for Majorana fermions, which are known to exhibit topological and trivial area-law entangled phases as well as a critical phase with sub-volume-law entanglement. Through the Lyapunov analysis, we identify the presence (absence) of edge-localized zero modes inside the bulk gap in the topological (trivial) phase and a bulk gapless spectrum in the critical phase. Furthermore, by suitably exploiting the fermion parity with twisted measurement outcomes at the boundary, we construct a topological invariant that distinguishes the two area-law phases and dynamically characterizes the critical phase. Our framework thus provides a general route to extend the notion of bulk-edge correspondence to monitored quantum dynamics.
We investigate the structure of effective Hamiltonians governing monitored dynamics of a one-dimensional Majorana chain using Lyapunov spectral analysis. We focus on the gapless phase of the Lyapunov spectrum, where the effective Hamiltonians behave differently from conventional critical Hamiltonians in isolated systems; the spectral gap closes faster than 1/L1/L1/L but slower than 1/L21/L^21/L2 and the entanglement entropy grows as [ln⁡(L)]2[\ln(L)]^2[ln(L)]2 where LLL is the system size. We find that the effective Hamiltonians have random long-range power-law hoppings with nontrivial magnitude correlations, rather than being independently and identically distributed. To elucidate the role of these non-Gaussian correlations, we construct random power-law hopping models that capture the essential features of the effective Hamiltonians. The spectral gaps of the constructed models decay faster than 1/L1/L1/L but slower than 1/L21/L^21/L2. We find that, in the absence of hopping correlations, the ground-state entanglement exhibits ln⁡(L)\ln(L)ln(L) scaling. In the presence of correlations, by contrast, the entanglement entropy is enhanced and its system-size dependence is consistent with [ln⁡(L)]2[\ln(L)]^2[ln(L)]2 scaling over the system sizes studied. These results suggest that correlations among long-range hopping magnitudes are responsible for the entanglement scaling rarely encountered in the ground states of conventional isolated quantum systems.
Abstract We explore the Lyapunov spectrum and entanglement entropy in quantum trajectories evolved by quantum measurements and spatially homogeneous unitary gates. In models with temporally random and Floquet unitary gates, we find that the Lyapunov exponents typically converge to values independent of measurement outcomes and that spectral transitions of the Lyapunov spectrum and entanglement transitions of ground states occur at the same measurement thresholds. Our results indicate that (i) randomness due to quantum measurements is enough, or equivalently, randomness of unitary gates is not essential, for the typical convergence of the Lyapunov spectrum and that (ii) the correspondence between the spectral transition and entanglement transition is ubiquitous in a wide range of systems. We also discuss why the typical convergence occurs, in light of the uniqueness and positive definiteness of stationary states in the completely-positive trace-preserving (CPTP) dynamics averaged over random measurement outcomes. Our discussions indicate that properties of the averaged CPTP dynamics play a key role in characterizing measurement-induced transitions in quantum trajectories.
Tensor networks have emerged as a powerful theoretical and computational framework for understanding and simulating quantum many-body systems. Their efficiency in representing quantum states, particularly those satisfying area laws of entanglement, is intimately linked to the structure of their constituent tensors. A fundamental aspect of these networks is their spectral properties, which manifest in various forms, such as the singular values of bond tensors or the eigenvalues of associated transfer matrices. Concurrently, entanglement measures provide quantitative insights into the quantum correlations present within these systems. This paper explores the intricate relationship between the spectra arising from tensor network representations and the different entanglement measures used to characterize quantum states. We delve into how the decay rates of singular values directly impact entanglement entropy, how entanglement spectra derived from Schmidt decompositions can reveal topological order, and how the spectral gaps of transfer matrices are connected to correlation lengths. By analyzing various tensor network architectures, including Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), we elucidate the mechanisms through which their spectral features encode profound information about quantum entanglement. The study highlights how a deeper understanding of these spectral properties can not only guide the construction of more efficient tensor network algorithms but also provide fundamental insights into the nature of quantum phases and phase transitions.
We study an entanglement phase transition in a class of chaotic non-Hermitian spin chains whose spin-spin coupling terms commute with the non-Hermitian contributions. Two representative models are investigated: the transverse-field Ising model with a complex longitudinal field and the non-Hermitian XX model with a transverse field. By analyzing their complex spectra, we find that both models undergo a dissipation-induced gapless-gapped phase transition when the transverse field exceeds a model-dependent threshold. Interestingly, the complex gap does not vary monotonically with the dissipation rate; instead, it exhibits pronounced oscillations before entering the gapped phase. By simulating their non-unitary dynamics, we show that the steady-state entanglement entropy undergoes a transition from volume-law to area-law scaling as the dissipation rate increases. Moreover, several unexpected features emerge within the volume-law regime: a larger complex gap or dissipation rate may lead to a more entangled steady state. We trace these unusual behaviors of the complex gap and the steady-state entanglement to level crossings between the maximal imaginary level and other spectral levels. Our work uncovers an exotic entanglement transition in chaotic non-Hermitian many-body systems.
We investigate entanglement phase transitions from volume-law to area-law entanglement in a quantum many-body state under continuous position measurement on the basis of the quantum trajectory approach. We find the signatures of the transitions as peak structures in the mutual information as a function of measurement strength, as previously reported for random unitary circuits with projective measurements. At the transition points, the entanglement entropy scales logarithmically and various physical quantities scale algebraically, implying emergent conformal criticality, for both integrable and nonintegrable one-dimensional interacting Hamiltonians; however, such transitions have been argued to be absent in noninteracting regimes in some previous studies. With the aid of U(1)U(1)U(1) symmetry in our model, the measurement-induced criticality exhibits a spectral signature resembling a Tomonaga-Luttinger liquid theory from symmetry-resolved entanglement. These intriguing critical phenomena are unique to steady-state regimes of the conditional dynamics at the single-trajectory level and are absent in the unconditional dynamics obeying the Lindblad master equation, in which the system ends up with the featureless, infinite-temperature mixed state. We also propose a possible experimental setup to test the predicted entanglement transition based on the subsystem particle-number fluctuations. This quantity should readily be measured by the current techniques of quantum gas microscopy and is in practice easier to obtain than the entanglement entropy itself.