Mapping Quantum Complexity with Information Lattices

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Author: Denis Avetisyan

A novel framework decomposes quantum states by scale and position to reveal hidden structure and redundancies in complex systems.

A higher-dimensional system is reduced to a quasi-one-dimensional information lattice by evaluating local information - determined by both position $n$ and scale $\ell$ - along a chosen direction while effectively merging data from other directions into an enlarged computational space, as demonstrated by constructing chains from subsystems or concentric rings $\mathcal{C}^{l}_{n}$ defined by radius $n$ and thickness $\ell$.
A higher-dimensional system is reduced to a quasi-one-dimensional information lattice by evaluating local information – determined by both position $n$ and scale $\ell$ – along a chosen direction while effectively merging data from other directions into an enlarged computational space, as demonstrated by constructing chains from subsystems or concentric rings $\mathcal{C}^{l}_{n}$ defined by radius $n$ and thickness $\ell$.

This work introduces a higher-dimensional information lattice based on the inclusion-exclusion principle for robust quantum state characterization and analysis of topological order.

Characterizing the intricate organization of quantum many-body states remains a central challenge in modern physics, particularly as system dimensionality increases. This limitation motivates the work presented in ‘Higher-Dimensional Information Lattice: Quantum State Characterization through Inclusion-Exclusion Local Information’, which generalizes the information lattice framework to higher dimensions by resolving information redundancies via an inclusion-exclusion principle. This approach provides a position- and scale-resolved decomposition of quantum information, allowing for the extraction of universal features like localization lengths and signatures of topological order. Can this framework ultimately offer a complete, scale-dependent map of quantum entanglement in complex systems?

Beyond Simple Correlation: Mapping the Architecture of Quantum Information

While metrics like Von Neumann Entropy effectively quantify the total amount of quantum correlation within a system, they offer limited insight into where that information resides. This presents a significant challenge in characterizing complex quantum states, as knowing the overall correlation isn’t enough to understand the system’s structure or dynamics. Consider a spatially extended quantum system: a single entropy value collapses all spatial distribution details into a single number, obscuring potentially crucial local correlations. This is akin to knowing the total weight of an object without knowing how that weight is distributed – a vital detail for understanding its stability or function. Consequently, researchers require tools capable of dissecting quantum correlations to reveal their spatial arrangement, paving the way for a more nuanced comprehension of quantum phenomena and potentially unlocking new technologies.

The characterization of complex quantum states and their evolution hinges not simply on how much information exists, but on how that information is spatially organized and distributed. Unlike traditional metrics that provide a total measure – such as Von Neumann Entropy – a detailed understanding of information structure reveals critical insights into a system’s behavior. This is particularly relevant when studying phenomena like topological order, where information isn’t localized in a conventional sense but is instead encoded in the relationships between distant quantum components. By dissecting the informational landscape, researchers can move beyond merely quantifying correlation to mapping its architecture, allowing for a more nuanced and predictive understanding of quantum dynamics and potentially unlocking new avenues for quantum technologies. The ability to resolve information at multiple scales and positions is therefore paramount for unraveling the intricacies of complex quantum systems.

Current approaches to quantifying quantum information often fall short when attempting to pinpoint where and at what scale correlations exist within a complex system, a limitation particularly problematic when investigating exotic states like those exhibiting topological order. The newly presented framework addresses this challenge by dissecting quantum information into contributions resolved both spatially and across different scales of observation. Crucially, this decomposition isn’t simply additive; it actively accounts for redundancies arising from overlapping contributions, providing a more accurate and nuanced picture of information distribution than previously possible. This refined analysis allows researchers to move beyond simply knowing that quantum correlations exist, and begin to understand their precise structural organization – a vital step towards unlocking the potential of these intricate quantum systems and harnessing their unique properties.

The 1D chain of three qubits exists in a state where all information is localized to individual subsystems, specifically qubits 'ab' and 'bc', resulting in a total information content of 2 bits.
The 1D chain of three qubits exists in a state where all information is localized to individual subsystems, specifically qubits ‘ab’ and ‘bc’, resulting in a total information content of 2 bits.

A Hierarchical Framework: The Higher-Dimensional Information Lattice

The Higher-Dimensional Information Lattice builds upon the foundational Information Lattice by incorporating positional and scale information to characterize quantum information with greater granularity. Traditional Information Lattices define relationships between subsystems without specifying their location or size within a larger system. This extended framework introduces dimensions representing spatial location and scale, allowing for a detailed, position- and scale-resolved description of information distribution. This capability enables the analysis of how information is organized and accessed at different levels of magnification or within specific regions of a quantum system, moving beyond a purely topological understanding to include geometric and dimensional aspects of information encoding.

The Higher-Dimensional Information Lattice utilizes two primary labeling schemes to facilitate hierarchical decomposition of quantum information: Multiscale Labels and Positional Labels. Multiscale Labels denote the scale at which a subsystem is defined, effectively categorizing subsystems based on their size or resolution. Positional Labels, conversely, specify the location of a subsystem within the overall lattice structure. The combination of these two label types provides a unique identifier for each subsystem, allowing for unambiguous tracking and organization of information across different scales and positions. This dual-labeling system is fundamental to the framework’s ability to represent and manipulate complex quantum states in a hierarchical manner, enabling analysis of information flow and relationships between subsystems.

The Higher-Dimensional Information Lattice necessitates the use of convex subsystems to facilitate unambiguous information organization; intersections of these subsystems must be uniquely labeled. This constraint is fundamental to the framework’s ability to consistently decompose quantum information across scales. Empirical analysis demonstrates that, in specific instances, this decomposition exhibits a scaling behavior of $α ℓx^{-2}$, where α represents a constant, ℓ denotes the scale, and x represents a positional variable. This scaling suggests an inverse square relationship between the informational content and positional distance at a given scale, providing a quantitative measure of information distribution within the lattice.

The information lattice of the cat state reveals overlapping subsystems with dependent density matrices-indicated by negative values-and demonstrates that a single bit of information requires access to the entire system, as local measurements are insufficient.
The information lattice of the cat state reveals overlapping subsystems with dependent density matrices-indicated by negative values-and demonstrates that a single bit of information requires access to the entire system, as local measurements are insufficient.

From Disorder to Topology: Validating the Lattice with Physical Systems

The Higher-Dimensional Information Lattice provides a quantitative framework for characterizing quantum states within the 2D Anderson Model, a paradigmatic system used to study the effects of disorder on electron localization. This lattice, constructed from RĂ©nyi entropies of subsystems, allows for the precise determination of the entanglement structure of these disordered states. Analysis using this framework confirms that the entanglement entropy scales with the area of the subsystem boundary, indicating a volume-law scaling characteristic of localized phases. Specifically, the lattice dimensions correlate with the RĂ©nyi entropy, enabling the differentiation of various quantum states and providing a means to quantify the degree of localization induced by disorder. Observed correlation decay lengths in these localized states exhibit exponential decay, a key feature captured by the lattice representation.

The Higher-Dimensional Information Lattice facilitates the analysis of topological order within the Toric Code model by providing a framework to quantify and understand topological entanglement entropy. The Toric Code is a specific example of a quantum system exhibiting non-local correlations arising from topological properties, and the lattice allows for the decomposition of the system into subsystems. By examining the information accessible from these subsystems and their boundaries, the framework can reveal the presence and magnitude of topological order, directly relating to calculations of $S = -Tr(\rho \log \rho)$ where $\rho$ is the reduced density matrix and $S$ is the entanglement entropy. Crucially, topological entanglement entropy, unlike conventional entanglement entropy, remains finite even in the thermodynamic limit and is a hallmark of topologically ordered phases.

The Higher-Dimensional Information Lattice demonstrates Overlap Redundancy, where information pertaining to a quantum state is accessible through multiple, overlapping subsystems. This redundancy is specifically illustrated by Cat States, which exhibit non-classical correlations distributed across these subsystems. In the context of the 2D Anderson Model, analysis of localized states reveals that correlation decay lengths exhibit exponential decay; this indicates that correlations diminish rapidly with distance, but the redundancy inherent in the lattice allows information to persist despite localized correlations. This characteristic is crucial for understanding how information is encoded and retrieved within disordered quantum systems and provides insight into the robustness of quantum states.

Analysis of the Anderson model's ground state on a 40x40 lattice reveals that disorder (W=10) induces anisotropic localization with distinct correlation lengths, while the absence of disorder (W=0) results in a critical, scale-invariant state characterized by a power-law decay of information.
Analysis of the Anderson model’s ground state on a 40×40 lattice reveals that disorder (W=10) induces anisotropic localization with distinct correlation lengths, while the absence of disorder (W=0) results in a critical, scale-invariant state characterized by a power-law decay of information.

Reshaping Quantum Information Analysis: Implications and Future Directions

The Higher-Dimensional Information Lattice presents a novel approach to characterizing quantum states by moving beyond traditional, holistic descriptions. This framework decomposes complex quantum information into a hierarchical structure, revealing relationships across multiple spatial scales and positions within the system. By spatially resolving information, researchers gain insights into how quantum properties are distributed and interconnected, enabling more efficient analysis than methods that treat the entire state as a single entity. This detailed view is particularly valuable when investigating entanglement, decoherence, and other nuanced quantum phenomena, as it pinpoints the origins and propagation pathways of information within the system. Ultimately, the lattice facilitates a deeper understanding of quantum behavior and provides a powerful tool for tackling previously intractable problems in quantum information science.

The Higher-Dimensional Information Lattice offers a fundamentally new approach to characterizing quantum systems by moving beyond traditional, holistic views. This framework doesn’t simply assess a system as a whole, but rather dissects it into subsystems analyzed at multiple scales and positions, revealing intricate relationships previously obscured. By pinpointing how information resides and propagates within these localized components, researchers gain an unprecedented ability to understand complex quantum behaviors. This detailed characterization isn’t merely descriptive; it paves the way for precise control, allowing manipulation of specific subsystems to steer the overall system’s evolution. Consequently, the lattice promises advancements in fields demanding delicate quantum control, such as the development of more robust quantum computing architectures and the design of novel quantum materials with tailored properties.

Investigations are now directed towards leveraging the Higher-Dimensional Information Lattice to address longstanding challenges within quantum many-body physics. Recent analysis of the 2D Anderson model demonstrates a compelling correlation: the direction of information propagation consistently aligns with the average Fermi velocity orientation, suggesting a fundamental link between information flow and electronic behavior. Furthermore, in quasi-one-dimensional systems exhibiting clean conditions, information scaling follows a $1/\ell_x^{-2}$ relationship, where $\ell_x$ represents the system length. These findings not only deepen the understanding of information dynamics within complex quantum systems but also hold promise for the design and optimization of novel quantum materials and technologies, potentially enabling more efficient quantum computation and communication.

By decomposing the information of a spin-up configuration using inclusion-exclusion principles and successively summing over spatial dimensions, the analysis reveals a quasi-one-dimensional information lattice that quantifies the state's information content at different scales.
By decomposing the information of a spin-up configuration using inclusion-exclusion principles and successively summing over spatial dimensions, the analysis reveals a quasi-one-dimensional information lattice that quantifies the state’s information content at different scales.

The pursuit of characterizing quantum states, as detailed in this work through the higher-dimensional information lattice, mirrors a fundamental tenet of empirical inquiry. The lattice’s decomposition by position and scale, particularly its accounting for overlap redundancies, acknowledges that information isn’t simply ‘found’ but actively constructed through observation and refinement. As Max Planck observed, “A new scientific truth does not conquer by convincing old scientists, but because the old scientists die.” This principle applies directly to the study of quantum states; each iteration of the lattice, each failure to perfectly model a state, refines the understanding, pushing beyond existing paradigms. Data isn’t the goal – it’s a mirror of human error, and even what can’t be measured still matters – it’s just harder to model.

Where Do We Go From Here?

The construction of this information lattice, while offering a structured view of quantum state decomposition, merely shifts the burden of proof. One can neatly arrange data, account for redundancies-and the authors have done so with commendable rigor-but the truly interesting questions remain stubbornly unanswered. Does this lattice representation reveal anything fundamentally new about the emergence of topological order, or is it simply a more elaborate taxonomy of what was already implicitly known? The significance level of any ‘insight’ derived from this formalism requires further scrutiny.

A critical limitation lies in the scalability of this approach. While the paper demonstrates the lattice’s utility on relatively simple systems, the computational cost of constructing and analyzing such structures for large, highly entangled states is likely to be prohibitive. Future work must address this challenge, perhaps by exploring approximations or identifying specific classes of quantum states where the lattice representation offers a genuine advantage. One wonders if the pursuit of ever-more-detailed decompositions isn’t a distraction from the core issue: understanding the minimal information required to characterize a quantum state.

Ultimately, a model isn’t a mirror of reality-it’s a mirror of its maker. This lattice, with its inherent assumptions about locality and scale, reflects a particular way of thinking about quantum information. The true test will be whether it can inspire new theoretical frameworks or facilitate the development of novel quantum technologies. Or whether it will simply become another elegantly constructed, yet ultimately descriptive, tool in the quantum information scientist’s toolbox.

Original article: https://arxiv.org/pdf/2512.20793.pdf

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

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2025-12-25 11:02