> For the complete documentation index, see [llms.txt](https://docs.quantaria.xyz/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.quantaria.xyz/deep-dives/bond-dimension-and-entanglement.md). # Bond dimension and entanglement entropy The bond dimension χ is the main setting in a Quantaria layer. This page explains why it is the right knob, using the link between bond dimension and entanglement. ## The cut Take a tensor network and cut one bond. The network splits into a left part and a right part. The bond is the only connection between them, so its dimension limits how much information can pass across the cut. ## The bound For a state written as an MPS, the entanglement entropy across a cut is at most log₂χ, where χ is the bond dimension at that cut. This is the sentence in the docs that links tensor networks to quantum systems. The bound is a ceiling. A bond of dimension 64 can carry at most 6 bits across its cut, and actual entanglement may be less. ## Area law Ground states of many physical systems have entanglement that grows with the boundary of a region, not with its volume. This is the area law. For one-dimensional systems, it means entanglement stays roughly constant across cuts, so a modest χ can describe the state well. Quantaria's research direction, area-law rank allocation, applies this idea to neural layers: measure how much each bond carries and set χ to match. This is planned research, not part of Hilbert-1. See [Roadmap](https://github.com/bhavy2908/dataai_docs/tree/main/overview/roadmap.md). ## What this means for a layer * A larger χ lets the layer carry more correlation across its internal cuts. * A smaller χ is cheaper, but the layer may lose structure the task needs. * Different bonds may need different χ values. Uniform χ is a simplification. ## Quantaria's use Bond dimensions are set per layer during training under rank-adaptive training ([Rank-adaptive training](/deep-dives/rank-adaptive-training.md)). The entanglement bound is the physics reason the bond is a meaningful capacity knob. ## Related * [Schmidt decomposition](/deep-dives/schmidt-decomposition.md) * [SVD and truncation](/deep-dives/svd-and-truncation.md) * [Quantum Path](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/quantum-path.md) --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation by asking a question. Perform an HTTP GET request on the following URL with the `ask` and `goal` query parameters: ``` GET https://docs.quantaria.xyz/deep-dives/bond-dimension-and-entanglement.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is what the user is ultimately trying to achieve, the reason they need the answer. Sharing it helps GitBook give you a better, more relevant answer. A goal is most helpful when it describes the outcome the user wants rather than restating the question. For example, with `ask=how do I create an API token`, a goal like `automate deployments from our CI pipeline` lets GitBook tailor the answer to that use case. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.