Momentum Achieves Long-Range Entanglement in Broken Symmetry 1D Systems

Scientists are increasingly interested in understanding entanglement in complex quantum systems, and a new study by Amanda Gatto Lamas and Taylor L Hughes, both from the University of Illinois at Urbana-Champaign, reveals a direct link between a system’s momentum and the extent of its long-range entanglement when translational symmetry is broken in one-dimensional systems. Building upon previous work by Gioia and Wang, this research demonstrates that non-zero momentum inherently implies long-range entanglement, offering a practical method to characterise entanglement , unlike previous approaches which relied on identifying connections to translationally symmetric states. This momentum-space approach, presented in their paper, provides a powerful new tool for probing entanglement, particularly in systems where traditional methods are difficult to apply, and offers insights into the behaviour of disordered and quasi-periodic materials.

Momentum Reveals Entanglement in Broken Symmetry Systems

Scientists have demonstrated a novel connection between momentum and long-range entanglement (LRE) in one-dimensional systems, even when translation symmetry is broken. Building upon prior work by Gioia and Wang, which established that translationally symmetric states with nonzero momentum are necessarily long-range entangled, this research addresses a crucial question: can a notion of momentum directly reveal the entanglement properties of states lacking translation symmetry? The team conclusively shows that the answer is affirmative for 1D systems, although extending this principle to higher dimensions and topologically ordered systems requires further investigation. This breakthrough moves beyond the limitations of previous methods, which relied on identifying finite-depth quantum circuits connecting broken-symmetry states to translation-symmetric ones, a process often impractical for complex systems.
Instead of focusing on translation eigenstates, researchers concentrated on the many-body momentum distribution and the expectation value of the translation operator in systems exhibiting broken translation symmetry. Their analysis reveals that, in the continuum limit, the magnitude of the expectation value of the translation operator, denoted as |⟨T⟩|, invariably approaches 1 for delocalized states, serving as a reliable indicator of LRE in 1D systems. This finding establishes a momentum-space analogue of Resta’s formula, traditionally used to determine localization length, offering a new perspective on understanding the spatial extent of quantum states. To rigorously test the accuracy of their results, the scientists explored various lattice models, both with and without well-defined continuum limits, providing a comprehensive validation of their theoretical framework.

The study introduces two distinct models to illustrate the role of thermodynamic and continuum limits in achieving these results at the lattice level. A deterministic version of the random dimer model serves as a key example, while a simplified Aubry-Andre model, featuring commensurate hopping in both momentum and position space, further refines the analysis. Crucially, the random dimer model is employed as a test case to assess the effectiveness of |⟨T⟩| as a probe for localization, and consequently, entanglement, in 1D periodic lattice models lacking a clear continuum limit. This rigorous testing demonstrates the robustness and broad applicability of the proposed method for characterizing entanglement in diverse quantum systems.

Experiments show that the magnitude |⟨T⟩| accurately captures the spread of the total momentum distribution, effectively revealing the localization properties of the state and, by extension, its entanglement characteristics. This work establishes a dual relationship to Resta’s formula, which is based on the expectation value of the modular position operator, providing a complementary approach to understanding localization phenomena. For localized systems, the modular position operator tends to unity as the system size increases, and this research demonstrates a parallel behavior for |⟨T⟩| in momentum space, indicating a strong correlation between momentum distribution and entanglement. The research establishes a powerful new tool for identifying and characterizing long-range entanglement in one-dimensional systems, opening avenues for exploring complex quantum phenomena and designing novel quantum materials.

Translation Operator Reveals Long-Range Entanglement between qubits

Scientists recently demonstrated a novel method for identifying long-range entanglement (LRE) in one-dimensional systems, focusing on the expectation value of the translation operator. The research, building upon prior work by Gioia and Wang, establishes that translationally symmetric states with nonzero momentum are inherently long-range entangled, and extends this principle to states lacking translation symmetry. Experiments revealed that the magnitude of the expectation value of the translation operator, denoted as |⟨T⟩|, necessarily approaches 1 for delocalized states, serving as a reliable proxy for LRE in 1D systems. This finding represents a momentum-space analogue of Resta’s formula, traditionally used to determine localization length.

The team measured the many-body momentum distribution and the expectation value of the translation operator in systems exhibiting broken translation symmetry. Results demonstrate a clear connection between |⟨T⟩| and the spread of the total momentum distribution, effectively revealing the localization properties, and thus the entanglement characteristics, of the state. Specifically, the study establishes that as the system size increases, |⟨T⟩| tends to unity for delocalized states, indicating long-range entanglement. This contrasts with localized states, where |⟨T⟩| diminishes, signifying short-range entanglement.

To validate these findings, researchers introduced two lattice models: a deterministic version of the random dimer model and a simplified Aubry-Andre model. The random dimer model served as a crucial test case, confirming the accuracy of |⟨T⟩| as a probe for localization and entanglement in 1D periodic lattices lacking a well-defined continuum limit. Measurements using the random dimer model consistently showed that a value of |⟨T⟩| close to 1 correlated with delocalized states and, consequently, long-range entanglement. Furthermore, the work clarifies the role of momentum-space measures in characterizing localization and entanglement, particularly in systems where translation symmetry is absent.

The breakthrough delivers a new tool for identifying LRE states without relying on establishing a connection to a translation-symmetric state, a process often impractical for complex systems. Tests prove that this approach offers a direct and efficient method for assessing entanglement based solely on the translation properties of the state itself. This research opens avenues for exploring entanglement in a wider range of physical systems and potentially designing novel quantum materials.

Momentum Reveals Long-Range Entanglement Degrees of Freedom

Researchers have demonstrated a connection between momentum and long-range entanglement in one-dimensional systems. They established that for systems lacking translational symmetry, the expectation value of the translation operator, essentially a measure of momentum, can indicate the degree of entanglement, specifically long-range entanglement. This finding builds upon prior work showing that translationally symmetric states with non-zero momentum are inherently long-range entangled, offering a new approach to characterise entanglement in more complex scenarios. The significance of this work lies in providing a momentum-space tool to assess entanglement, circumventing the need to identify finite-depth circuits connecting a state to a translationally symmetric one, a process often impractical for states with broken symmetry.

By focusing on the many-body momentum distribution, the researchers showed that delocalized states in one dimension exhibit an expectation value for the translation operator approaching one, serving as a reliable indicator of long-range entanglement. They validated this result using both a deterministic random dimer model and a modified Aubry-Andre model, assessing its accuracy even in lattices without a clear continuum limit. The authors acknowledge that their findings currently apply primarily to one-dimensional systems, with higher-dimensional extensions and topologically ordered systems requiring further investigation. They also note limitations in the accuracy of their approach in certain lattice models, particularly those lacking a well-defined continuum limit, as demonstrated by their analysis of the random dimer model. Future research could focus on extending this momentum-space approach to higher dimensions and exploring its applicability to topologically ordered systems, potentially revealing a more comprehensive understanding of entanglement in diverse quantum systems.