Beyond Hermiticity: Entanglement’s Strange New Phase Transitions

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Author: Denis Avetisyan

New research reveals unexpected entanglement behavior in non-Hermitian quantum systems, challenging conventional understandings of phase transitions.

The Non-Hermitian Tight-Binding Floquet-Impurity (NHTFI) model demonstrates entanglement dynamics, with steady-state entanglement entropy exhibiting sensitivity to driving frequency—specifically, differing behaviors are observed at $\Omega = 0.9$ and $\Omega = 2$, suggesting a nuanced relationship between driving parameters and system entanglement.
The Non-Hermitian Tight-Binding Floquet-Impurity (NHTFI) model demonstrates entanglement dynamics, with steady-state entanglement entropy exhibiting sensitivity to driving frequency—specifically, differing behaviors are observed at $\Omega = 0.9$ and $\Omega = 2$, suggesting a nuanced relationship between driving parameters and system entanglement.

This study investigates entanglement phase transitions in one-dimensional non-Hermitian spin chains, demonstrating their occurrence even when the non-Hermitian term and coupling terms do not commute.

While conventional studies of quantum entanglement largely focus on Hermitian systems, the emergence of non-Hermitian physics necessitates a re-evaluation of entanglement transitions in open quantum systems. This work, ‘Entanglement Phase Transition in Chaotic non-Hermitian Systems’, investigates entanglement behavior in one-dimensional non-Hermitian spin chains, revealing a dissipation-driven transition from volume-law to area-law scaling of entanglement entropy alongside unusual oscillations in the complex energy spectrum. Specifically, we demonstrate that such transitions can occur even when the non-Hermitian and coupling terms do not directly interact, linking these features to level crossings in the complex energy landscape. Could these findings illuminate novel pathways for controlling and harnessing entanglement in dissipative quantum systems?

The Illusion of Equilibrium: Why Systems Resist Stasis

Conventional condensed matter physics often operates under the assumption of equilibrium, envisioning systems settling into stable, predictable states. However, this framework struggles to accurately describe the behavior of many physical systems encountered in reality. These systems are frequently open – constantly exchanging energy and information with their surroundings – and, crucially, are subject to continuous measurement. This ongoing observation doesn’t simply reveal a pre-existing state; instead, the act of measurement fundamentally alters the system’s dynamics, driving it away from equilibrium. Consider a quantum system probed repeatedly; each measurement collapses the wave function, preventing the system from relaxing into a ground state. This relentless probing, rather than passive observation, is increasingly recognized as a primary driver of complex behavior, necessitating a shift in focus towards understanding non-equilibrium phenomena and the emergent phases they give rise to.

Conventional condensed matter physics traditionally centers on systems reaching a stable, equilibrium state. However, a growing body of research reveals that many physical systems are inherently open and constantly interacting with their environment, often through the process of measurement. This realization necessitates a shift in perspective, moving beyond equilibrium assumptions to explore non-equilibrium dynamics. Continuous monitoring—repeatedly measuring a quantum system—doesn’t simply reveal information; it actively shapes the system’s behavior, driving it towards entirely new phases of matter not observed in isolation. These measurement-induced phases aren’t defined by minimizing energy, but by maximizing entanglement in response to the ongoing observation, representing a fundamentally different way to categorize and understand the organization of matter and potentially opening doors to novel quantum technologies.

Measurement-Induced Entanglement Phase Transitions (MIPTs) represent a fundamentally new way of categorizing and understanding quantum behavior, moving beyond the traditional focus on systems defined by their inherent properties. These transitions aren’t driven by changes in external parameters like temperature or pressure, but rather by the act of measurement itself. As a quantum system is continuously monitored, entanglement—a key resource for quantum technologies—can dramatically restructure, leading to distinct phases characterized by differing levels of entanglement and correlation. This isn’t merely a perturbation of an existing state; it’s the emergence of qualitatively new behavior, where the very process of gaining information dictates the system’s organization. Consequently, MIPTs offer a powerful framework for interpreting complex quantum phenomena—from the behavior of many-body systems to the dynamics of quantum information—and suggest that observation isn’t a passive act, but an active force shaping the quantum world, potentially leading to novel applications in quantum error correction and state preparation.

The potential to harness quantum systems for advanced information processing hinges on a detailed understanding of Measurement-Induced Phase Transitions (MIPTs). These transitions, arising from the very act of observing a quantum system, fundamentally alter its entanglement structure and computational capabilities. Unlike traditional approaches focused on isolated, equilibrium systems, recognizing and controlling MIPTs allows for the design of quantum devices resilient to environmental noise and capable of performing complex computations. Specifically, the emergence of an “entanglement cliff” during a MIPT – where entanglement vanishes abruptly – presents both a challenge and an opportunity; mitigating this cliff ensures robust information storage, while precisely controlling its location offers a novel pathway for creating and manipulating quantum states. Current research focuses on tailoring measurement strategies and system parameters to navigate these transitions, effectively turning a potential fragility into a powerful tool for quantum information technologies, potentially unlocking breakthroughs in areas like quantum error correction and scalable quantum computing.

Modeling the Open System: A Necessary Distortion

Open quantum systems, unlike isolated systems, exchange energy and information with their surrounding environment. This interaction causes the system’s state to evolve in a manner that does not preserve the total probability, a process termed non-unitary evolution. The standard time-dependent Schrödinger equation, $i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle$, guarantees unitary time evolution and thus conservation of probability, making it inadequate for describing open systems. Consequently, a modified mathematical formalism is required to accurately represent the dynamics of systems subject to environmental influences, where the Hamiltonian is no longer necessarily Hermitian and the state vector $|\psi(t)\rangle$ may not remain normalized under time evolution.

Non-Hermitian Hamiltonians extend the standard Hermitian formalism by allowing for complex eigenvalues, representing the rates of decay or growth of quantum states. In the context of open quantum systems, these complex eigenvalues directly correspond to the loss or gain of probability amplitude due to interactions with the environment, effectively modelling processes like spontaneous emission or absorption. Mathematically, a non-Hermitian Hamiltonian $H = H_0 – i\frac{\Gamma}{2}$ can be used, where $H_0$ is the Hermitian part representing the system’s intrinsic energy and $\Gamma$ represents the decay rate. The eigenvectors of such a Hamiltonian are not necessarily orthonormal, requiring the introduction of right and left eigenvectors and a bi-orthogonal decomposition to ensure proper normalization and probabilistic interpretation of the system’s evolution.

No-quantum jump trajectories represent a post-selection technique used in the analysis of open quantum systems. This method involves conditionally analyzing system evolution only when a measurement does not trigger a quantum jump – a sudden change in the system state due to environmental interaction. By effectively discarding trajectories where jumps occur, the analysis focuses on the subset of system evolutions that remain coherent for a longer duration. This post-selection process simplifies the mathematical description by reducing the complexity associated with stochastic wave function collapse and allows for the identification of underlying deterministic dynamics that would otherwise be obscured by environmental noise. The resulting trajectories are not physically realistic in the sense that they represent a specific, filtered subset of possible evolutions, but they provide valuable insights into the system’s behavior under idealized conditions and facilitate the calculation of relevant physical quantities, such as transition probabilities and expectation values, as if the system were isolated.

The standard time-dependent Schrödinger equation, $i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle$, forms the basis for modelling open quantum systems by incorporating non-Hermitian terms into the Hamiltonian, $H$. This extension allows for the description of systems interacting with their environment, where information and energy can be exchanged. Specifically, non-Hermitian Hamiltonians introduce complex energy eigenvalues, with the imaginary parts representing the rates of decay or growth of quantum states. By utilizing a non-Hermitian $H$, the time evolution of the system’s wavefunction is no longer unitary, accurately reflecting the loss or gain of probability associated with interactions and measurements, and providing a mathematically rigorous framework for analyzing open quantum dynamics.

The complex gap and corresponding eigenenergy spectra of the NHTFI model vary with both frequency (Ω) and chain length (N), revealing the model's sensitivity to these parameters.
The complex gap and corresponding eigenenergy spectra of the NHTFI model vary with both frequency (Ω) and chain length (N), revealing the model’s sensitivity to these parameters.

Spin Chains as Laboratories for Entanglement’s Demise

One-dimensional spin chains, including the paradigmatic Ising model and its variants, are extensively utilized in the study of measurement-induced phase transitions (MIPT) due to their tractable mathematical formulation and the emergence of complex quantum phenomena. These chains, described by a discrete lattice of interacting spins – each with a value of +1 or -1 in the Ising case – allow for analytical and numerical investigations that are often intractable in higher-dimensional systems. The relative simplicity of the Hamiltonian, typically involving nearest-neighbor interactions and an external field, enables the calculation of correlation functions and entanglement measures crucial for characterizing MIPT. Furthermore, the low dimensionality facilitates the visualization of entanglement spreading and the identification of critical behavior associated with the transition from a featureless paramagnetic phase to a many-body localized phase induced by projective measurements.

The Non-Hermitian Transverse Field Ising Model is derived from the standard $Ising$ model by introducing non-Hermitian terms, specifically imaginary components within the Hamiltonian. This modification allows for the simulation of measurement processes as an inherent part of the system’s dynamics; the imaginary terms represent the loss of quantum information due to measurement, effectively modelling the decoherence experienced by qubits. By analyzing the entanglement properties—such as concurrence or entanglement entropy—of the Non-Hermitian model, researchers can quantitatively assess how measurement impacts quantum correlations and ultimately influences the stability of entangled states within the spin chain. This approach provides a direct link between theoretical modeling and the practical limitations imposed by measurement in quantum information processing.

The Non-Hermitian XX model represents a refinement of investigations into measurement-induced phase transitions (MIPT) by providing an alternative Hamiltonian for studying non-Hermitian dynamics. Unlike the Non-Hermitian Transverse Field Ising Model which includes a $Z$ component, the XX model focuses solely on interactions between spin components in the $x$ and $y$ directions. This simplification allows for analytical tractability in certain regimes and facilitates the examination of how non-Hermitian terms – representing the effects of measurement or dissipation – influence entanglement and the emergence of novel phases distinct from their Hermitian counterparts. Specifically, the model’s Hamiltonian, typically expressed as $H = \sum_{i} (J \sigma_x^i \sigma_x^{i+1} + J \sigma_y^i \sigma_y^{i+1} – i\gamma \sigma_x^i)$, allows investigation of measurement effects without the complexities introduced by the $Z$ component, offering a complementary approach to understanding MIPT.

The Lindblad Master Equation offers a complementary approach to modeling open quantum spin chains by directly describing the time evolution of the system’s density matrix, $ \rho $, under the influence of decoherence and dissipation. Unlike the non-Hermitian approach which utilizes a pseudo-Hermitian Hamiltonian to simulate measurement-induced decay, the Lindblad equation employs a completely positive trace-preserving map to account for the interaction with the environment. Specifically, the equation takes the form $ \frac{d\rho}{dt} = -i[H, \rho] + \sum_{k} L_k \rho L_k^\dagger – \frac{1}{2} \{L_k^\dagger L_k, \rho \} $, where $H$ is the system Hamiltonian, and the $L_k$ are Lindblad operators representing the interaction with the environment. Crucially, calculations using the Lindblad formalism have corroborated findings obtained through the non-Hermitian Hamiltonian method regarding entanglement dynamics and measurement effects, providing independent validation of the observed phenomena in these open spin systems.

The complex gap and steady-state entanglement entropy, as functions of system size, exhibit distinct behaviors depending on the parameter Ω, transitioning from a smaller gap and lower entropy at Ω=0.9 to a larger gap and higher entropy at Ω=2.
The complex gap and steady-state entanglement entropy, as functions of system size, exhibit distinct behaviors depending on the parameter Ω, transitioning from a smaller gap and lower entropy at Ω=0.9 to a larger gap and higher entropy at Ω=2.

Witnessing the Inevitable: Experiments Confirm Theoretical Collapse

The Faber Polynomial Method presents a sophisticated numerical approach to modeling Many-Body Interactions with Periodic Time-dependent driving (MIPT), a system characterized by non-unitary dynamics where energy is not conserved. Unlike traditional methods struggling with these dissipative systems, the Faber Polynomial technique effectively maps the complex time evolution operator into a series representation, enabling highly accurate calculations of key physical observables. This allows researchers to precisely determine entanglement properties, such as entanglement entropy, which serve as sensitive indicators of quantum phase transitions. By efficiently handling the non-Hermitian nature of MIPT, the method facilitates detailed investigations into the system’s behavior and confirms theoretical predictions regarding the emergence of complex energy spectra and the transition between different entanglement phases – crucial for understanding the fundamental limits of quantum information processing in driven-dissipative systems and offering insights into novel quantum materials.

The quantification of entanglement through Entanglement Entropy serves as a sensitive probe of the Measurement-Induced Phase Transition (MIPT). As the system undergoes this transition, characterized by increasing measurement rates, the energy spectrum exhibits a notable feature: the emergence of a complex gap. This gap isn’t simply an absence of energy levels, but a region where energy levels become complex, signifying the decay of quasi-particle excitations and a fundamental shift in the system’s behavior. The size of this complex gap directly correlates with the critical dissipation rate – specifically, $4\sqrt{J^2 – \Omega^2}$ – and its appearance signals the breakdown of the initial, gapless phase and the onset of a new, gapped phase characterized by diminished entanglement. Monitoring the behavior of this gap, therefore, provides a powerful method for identifying and characterizing the MIPT and understanding the underlying mechanisms driving the transition from a volume-law entangled state to an area-law entangled state.

In certain models of Measurement-Induced Phase Transition (MIPT), the transition between phases exhibiting differing entanglement properties is sharply defined by a critical dissipation rate, calculated as $4\sqrt{J^2 – \Omega^2}$. This value represents the point at which the system’s susceptibility to measurement-driven decoherence overcomes the inherent resilience provided by the system’s Hamiltonian, characterized by the coupling strength, $J$, and the transverse field, $\Omega$. Below this critical dissipation rate, entanglement persists, scaling with system volume; however, exceeding it induces a dramatic shift towards a phase where entanglement is suppressed, exhibiting a scaling independent of system size. Consequently, this critical dissipation rate serves as a precise benchmark for identifying the entanglement phase transition and understanding the interplay between Hamiltonian dynamics and measurement-induced decoherence in these quantum systems.

Investigations into measurement-induced phase transitions reveal a critical interplay between the transverse field strength, denoted as $Ω$, and the system’s coupling constant, $J$. When $Ω$ reaches or exceeds $J$, a distinct shift in the system’s behavior is observed – a transition from a gapless to a gapped phase. This change is fundamentally linked to the entanglement structure; as the system enters the gapped phase, entanglement, previously extensive and scaling with system volume, diminishes and transitions towards an area law, becoming independent of system size. This entanglement transition serves as a robust indicator of the broader phase transition, signifying a change in the system’s fundamental properties and confirming the theoretical predictions regarding the critical behavior at this threshold.

The behavior of entanglement, quantified by entanglement entropy, provides a crucial window into the nature of many-body phase transitions in measurement-induced phase transitions (MIPT). Research indicates that the scaling of this entropy shifts dramatically depending on the rate of dissipation within the system. In systems experiencing low dissipation, entanglement scales according to a Volume Law, meaning the entropy grows proportionally to the system’s volume – a characteristic of highly entangled states. However, as the dissipation rate increases, a transition occurs, and the entanglement entropy begins to scale with an Area Law, becoming independent of system size. This change signifies a loss of long-range entanglement and the emergence of a fundamentally different phase, where information is localized and the system’s properties are dictated by its boundaries rather than its bulk. This shift from extensive to non-extensive scaling serves as a key signature identifying the entanglement phase transition and providing insight into the underlying physics of information scrambling and many-body localization.

Recent advancements in quantum technology have enabled the experimental realization of Measurement-Induced Phase Transitions (MIPT) through the construction of hybrid quantum circuits. Researchers are leveraging diverse physical platforms – including bosonic, fermionic, and systems utilizing Majorana fermions – to simulate and observe the predicted phenomena. These circuits are meticulously designed to introduce controlled measurements that drive the system towards the critical point, allowing for direct validation of theoretical predictions concerning entanglement scaling and the emergence of distinct phases. Specifically, observations confirm the shift from volume law entanglement – where entropy scales with system size – to area law behavior, a key signature of the phase transition. This experimental confirmation, across multiple physical realizations, solidifies the understanding of MIPT as a robust and observable phenomenon, paving the way for exploring its potential applications in quantum information processing and materials science.

The study of entanglement phase transitions in these non-Hermitian systems reveals a truth long understood by those who’ve watched complex systems evolve. It isn’t about imposing order, but about observing the inevitable shifts in connection. The researchers note transitions even when the non-Hermitian term commutes with the coupling – a seeming contradiction, yet perfectly natural. As the system ‘grows up,’ its entanglement behavior changes, mirroring the unpredictable flourishing and decay seen in all living things. It recalls a sentiment shared by Richard Feynman: “The first principle is that you must not fool yourself – and you are the easiest person to fool.” This work doesn’t build a system, it simply observes the patterns emerging from inherent instability, accepting the complex spectra as a symptom of growth, not a flaw.

What Shadows Will Fall?

The exploration of entanglement phase transitions within non-Hermitian systems reveals not so much a destination, but a widening of the labyrinth. The finding that transitions persist even with commuting non-Hermitian terms isn’t a resolution, but a sharpening of the question: what symmetries truly constrain the evolution of complex spectra? Long stability in these systems—a neatly defined phase transition—is the sign of a hidden disaster, a predictable point of qualitative change masked by apparent order. The current work illuminates a specific instance, but the landscape of possible non-Hermitian interactions remains largely uncharted.

The reliance on one-dimensional spin chains, while providing analytical tractability, introduces a strong constraint. True systems don’t conform to such elegant geometries. The next evolution will demand a confrontation with higher dimensions, and with the inevitable disorder that accompanies them. Moreover, the Lindblad master equation, despite its utility, is a simplification. It presumes a Markovian evolution, a memoryless dissipation. But dissipation, like all things, accumulates history.

The study of quantum trajectories offers a glimpse beyond the averaged descriptions, but it is a computationally expensive path. The real challenge isn’t simply to observe these transitions, but to anticipate them, to understand the subtle pre-transitional signals embedded within the noise. Systems don’t fail; they evolve into unexpected shapes. The task, then, is not to build robust systems, but to cultivate the capacity to read the portents of their change.

Original article: https://arxiv.org/pdf/2511.10410.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-15 16:50