Untangling Topology and Loss: A New Frontier for Quantum Control

Author: Denis Avetisyan

New research reveals how entanglement can be harnessed to characterize and manipulate dynamic phases in non-Hermitian topological systems, paving the way for advanced quantum information processing.

Entanglement entropy and transport currents provide a powerful means to characterize dynamic phase transitions in non-Hermitian topological systems with non-reciprocal coupling and skin effects.

Conventional understandings of entanglement dynamics struggle to account for systems governed by both topology and non-Hermiticity. This research, detailed in ‘Entanglement dynamics driven by topology and non-Hermiticity’, demonstrates that entanglement entropy and transport currents robustly characterize distinct dynamic phases-bulk, edge, and skin-like-arising in non-Hermitian topological systems. Specifically, the work reveals a programmable approach to steering entanglement via tailored non-Hermitian couplings, enabling control over information shuttling and localization. Could this framework pave the way for novel quantum information processing architectures leveraging synthetic materials and beyond?

Beyond Equilibrium: Embracing the Dynamics of Non-Hermitian Systems

For generations, the foundation of physics has rested upon the principle of Hermitian Hamiltonians – mathematical operators ensuring that physical observables, like energy, remain real-valued. However, a growing body of research reveals that this constraint doesn’t universally hold true for all physical systems. Many real-world scenarios, from open quantum systems interacting with their environment to driven-dissipative systems and even certain effective descriptions of disordered materials, inherently exhibit non-Hermiticity. This arises when energy is not conserved – for example, through gain and loss, or via coupling to external reservoirs. Consequently, the traditional framework needs adaptation, as non-Hermiticity introduces complex energy spectra and fundamentally alters the behavior of quantum states, paving the way for previously unattainable physical effects and technologies.

Unlike conventional quantum systems described by Hermitian Hamiltonians – which guarantee real-valued energy spectra – non-Hermitian systems permit complex energies, fundamentally altering their behavior. This allowance isn’t merely a mathematical curiosity; it manifests in observable physical consequences. For example, phenomena like exceptional points – where eigenvalues and eigenvectors coalesce – become possible, leading to enhanced sensitivity to perturbations and drastically different responses to external stimuli. Furthermore, non-Hermiticity gives rise to unidirectional propagation of waves, where energy flows preferentially in one direction, and the paradoxical existence of states with gain and loss simultaneously. These unique characteristics move beyond the limitations of Hermitian physics, offering potential advancements in areas like laser design, topological insulators, and sensing technologies by enabling control over wave dynamics in ways previously unattainable.

The departure from Hermitian physics unlocks possibilities for designing entirely new states of matter with properties not found in conventional materials. Researchers are actively investigating non-Hermitian systems to create and control wave phenomena – including light, sound, and matter waves – in ways previously considered impossible. This includes manipulating wave propagation to achieve enhanced sensing capabilities, developing novel lasers with unique emission characteristics, and engineering topological insulators that are robust against defects. Furthermore, the ability to control gain and loss within these systems allows for the creation of parity-time ( $\mathcal{PT}$ ) symmetric structures, offering potential advancements in areas like optical switching and signal amplification, and even hinting at the possibility of realizing unconventional superconductivity.

The Emergence of Boundaries: Witnessing the Non-Hermitian Skin Effect

The non-Hermitian skin effect (NHSE) manifests as an exponential localization of wavefunctions towards the boundaries of a system. This means that, unlike typical wave behavior, the probability amplitude of a particle described by the wavefunction decays rapidly as one moves away from the edge of the system. The degree of localization is determined by the strength of the non-Hermiticity and the system’s geometry; larger non-Hermiticity generally leads to more pronounced boundary accumulation. Crucially, this localization is not due to disorder, as in Anderson localization, but arises from the asymmetric hopping between lattice sites, which is inherent in non-Hermitian systems. The effect is observed across a variety of physical realizations, including photonic lattices, metamaterials, and driven dissipative systems.

The non-Hermitian skin effect and Anderson localization, while both resulting in wavefunction localization, originate from distinct physical mechanisms. Anderson localization arises from disorder-induced scattering, leading to a suppression of transmission and localization in the bulk of the system, even with orthogonal eigenstates. Conversely, the non-Hermitian skin effect stems from the non-orthogonal nature of the system’s eigenstates-specifically, the breakdown of biorthogonality-which forces states to accumulate at the boundaries. This accumulation isn’t due to scattering from disorder, but rather a fundamental property of non-Hermitian systems where the left and right eigenvectors are distinct. The degree of non-orthogonality, quantified by the difference between the left and right eigenvectors, directly correlates with the strength of the skin effect and the extent of boundary localization; a perfectly Hermitian system exhibits full orthogonality and lacks this phenomenon.

Open boundary conditions are crucial for observing and characterizing the non-Hermitian skin effect because they allow the boundary-localized states to fully manifest. Unlike periodic boundary conditions which can mask the effect through state mixing, open boundaries prevent reflections and ensure that the eigenstates associated with the skin effect – those exponentially decaying or growing away from the bulk – are clearly defined at the system edges. This unambiguous localization is essential for experimental verification and theoretical analysis, as it allows direct measurement of the wavefunction decay or amplification and facilitates the identification of the non-trivial topological properties associated with the skin effect. The contrast between open and closed/periodic boundary conditions directly demonstrates the system’s sensitivity to boundary effects and confirms the non-Hermitian nature of the Hamiltonian.

Beyond the Standard Model: A Generalized SSH Framework

The Su-Schrieffer-Heeger (SSH) model, originally developed to describe polyacetylene, provides a foundational understanding of topological insulators by demonstrating the emergence of topologically protected edge states. However, the standard SSH model relies on Hermitian Hamiltonians, limiting its ability to represent systems with non-Hermitian characteristics such as gain and loss, or effective long-range interactions. Non-Hermitian systems exhibit unique features like exceptional points and non-Bloch band theory, which are absent in Hermitian systems. Consequently, extensions to the SSH model are necessary to accurately simulate and analyze a broader range of physical phenomena, particularly in areas like metamaterials, optoelectronics, and open quantum systems where non-Hermiticity is inherent or intentionally engineered. These extensions typically involve modifying the Hamiltonian to include complex potentials or asymmetric hopping terms, enabling the investigation of topological properties in the presence of gain, loss, or dissipation.

The Generalized Su-Schrieffer-Heeger (SSH) model extends the original Hermitian framework by introducing non-Hermitian terms and allowing for tunable hopping amplitudes between lattice sites. Specifically, the model utilizes complex-valued hopping parameters $t_{1,2}$ , where $t_1$ and $t_2$ represent the intra- and inter-cell hopping integrals, respectively. By varying the magnitude and phase of these parameters, researchers can simulate systems exhibiting parity-time (PT) symmetry, exceptional points, and other non-Hermitian phenomena. This tunability, combined with the inclusion of gain and loss, allows for the emulation of open quantum systems and the investigation of non-unitary dynamics not captured by the standard SSH model, enabling the study of complex topological phases in dissipative environments.

The Generalized SSH model facilitates the investigation of how non-Hermitian physics modifies topological properties and associated dynamics. By incorporating non-Hermiticity – typically through gain and loss terms – into the standard SSH Hamiltonian, the model allows researchers to observe phenomena such as exceptional points and their influence on energy band topology. Specifically, the interplay between non-Hermiticity and topology can lead to novel dynamic behaviors, including non-reciprocal transport, asymmetric mode hopping, and sensitivity to initial conditions, which are not present in conventional Hermitian topological systems. Analysis within this framework utilizes techniques like Floquet engineering and time-dependent perturbation theory to characterize the evolution of wave packets and the stability of topological edge states under the influence of gain and loss, offering insights into potential applications in areas like unidirectional amplification and robust waveguiding.

From Theory to Observation: An Acoustic Realization of Non-Hermitian Physics

The Generalized Su-Schrieffer-Heeger (SSH) model, typically explored within condensed matter physics to describe topological insulators and soliton behavior, is effectively simulated using an acoustic analog platform. This approach leverages the mathematical equivalence between the SSH model Hamiltonian and the equations governing sound wave propagation in specifically designed structures. By fabricating acoustic metamaterials with spatially varying sound velocities and on-site potentials, the system mimics the hopping terms and potential differences present in the SSH model. This allows for experimental investigation of phenomena predicted by the theoretical model, offering a readily tunable and observable analog for exploring non-Hermitian physics and topological effects in a controllable physical system.

The simulation of non-Hermitian systems is achieved through the manipulation of acoustic waves in a specifically designed medium. By controlling parameters such as the sound velocity and dissipation within the material, an effective non-Hermitian Hamiltonian can be realized. This allows for the observation of phenomena characteristic of non-Hermitian physics, notably the skin effect, where acoustic waves exponentially localize at the boundaries of the system. The acoustic analog effectively maps the mathematical properties of non-Hermitian quantum mechanics onto the propagation of sound, enabling experimental verification of theoretical predictions regarding energy localization and anomalous transport behavior in these systems.

The acoustic analog platform facilitates detailed study of non-Hermitian dynamics and transport through precise manipulation of system parameters such as gain and loss, and hopping amplitudes. Investigations across all dynamic regimes demonstrate initial entanglement entropy (EE) values consistently measured using the base-2 logarithm $log_2$ . This quantitative measurement of EE allows for characterization of the system’s topological properties and the degree of entanglement present at the initial time step, providing a robust metric for assessing non-Hermitian behavior and validating theoretical predictions.

Beyond Conventional Boundaries: Implications for a New Era of Physics

Conventional physics relies heavily on the principle of Hermitian symmetry, which guarantees that a system’s energy remains constant over time and that measurable physical properties are real numbers. However, the emerging field of non-Hermitian physics deliberately relaxes this requirement, investigating systems where Hermitian symmetry is broken. This seemingly subtle shift has profound implications, suggesting that energy is not necessarily conserved in the traditional sense and that complex, rather than purely real, energy eigenvalues are possible. These non-Hermitian systems, often realized through engineered dissipation or gain, exhibit unusual behaviors like exceptional points – singularities in parameter space where properties diverge – and allow for the exploration of physics beyond the standard framework, potentially unlocking new ways to control and manipulate wave phenomena and matter itself. The deliberate departure from Hermitian constraints offers a unique lens through which to examine fundamental concepts like symmetry, stability, and the very nature of quantum states.

The non-Hermitian skin effect (NHSE) presents a pathway toward innovative device technologies predicated on exquisitely sensitive and controllable systems. This effect, characterized by the accumulation of wave functions at the edges of a material, doesn’t just alter wave behavior – it fundamentally reshapes how information is processed. Crucially, research demonstrates that under certain conditions, these edge-dominated dynamics can achieve complete suppression of entanglement entropy, effectively isolating quantum information and minimizing decoherence. This level of control over entanglement is particularly promising for the development of robust quantum sensors and information storage devices, as it allows for the creation of systems less susceptible to environmental noise. The potential extends to novel control mechanisms, where even subtle changes in external conditions can be amplified through the edge-localized states, offering unprecedented sensitivity and precision in device operation.

Investigations into non-Hermitian physics suggest the potential for discovering entirely new states of matter, radically altering current understandings of wave behavior. Recent studies demonstrate that these systems, characterized by “skin-like” dynamics, aren’t simply static but exhibit periodic information shuttling – a rhythmic transfer of data within the material. Remarkably, entanglement entropy, a measure of quantum connectedness, is confined to a limited spatial region, rather than spreading throughout the system, indicating strong localization. Observations of negative total current within ‘skin-beat’ and ‘skin-oscillation’ regimes further confirm the dynamic nature of the non-Hermitian skin effect, revealing unidirectional wave transport. Crucially, scaling these systems to larger sizes consistently demonstrates a suppression of entanglement entropy, providing compelling evidence that these non-Hermitian systems exhibit increasingly robust localization as their dimensions grow.

The research illuminates how entanglement entropy serves as a crucial indicator of dynamic phase transitions within non-Hermitian topological systems. This mirrors Stephen Hawking’s insight: “Look up at the stars and not down at your feet.” The study doesn’t impose control through overarching design, but rather observes the emergent order arising from the interplay of topology and non-Hermiticity. Just as Hawking suggested finding wonder in the vastness above, this work finds governance not in dictating quantum behavior, but in recognizing how self-organization-manifesting as measurable entanglement-naturally steers the system’s evolution. Every connection carries influence, and the research demonstrates this principle by showing how entanglement entropy can characterize and even engineer quantum information dynamics.

Beyond the Horizon

The demonstrated interplay between entanglement entropy, transport currents, and dynamic phases in non-Hermitian topological systems suggests a shift in perspective. Control, in the traditional sense, proves elusive; the system doesn’t yield to direct manipulation. Instead, localized rules governing entanglement and non-reciprocal coupling appear sufficient to generate emergent, global behaviors. This research hints that robust quantum information dynamics arise not from imposed order, but from the system’s inherent capacity for self-organization.

Unresolved questions remain, particularly regarding the scalability of these findings. Acoustic analogs, while insightful, possess inherent limitations. The true test lies in realizing and characterizing these dynamic phases in genuinely quantum systems. Further investigation should focus on the interplay between the skin effect and entanglement degradation, exploring whether controlled dissipation can be leveraged to steer the system towards desired states – or if such steering is merely an illusion of observation.

The expectation of complete control should be tempered. A more fruitful path likely involves designing systems where local rules are sufficient to produce the desired global patterns. Weak top-down influence, allowing for exploration of the phase space, will almost certainly prove more effective than attempts at rigid command. The challenge, then, isn’t to dictate behavior, but to cultivate an environment where robust, emergent dynamics can flourish.

Original article: https://arxiv.org/pdf/2512.24107.pdf

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