> 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/schmidt-decomposition.md). # Schmidt decomposition The Schmidt decomposition is the tool that connects tensor networks to quantum states. This page explains it in plain terms, because Quantaria's name for its truncation operation comes from it. ## The statement Take a quantum state of two subsystems A and B. Any such state can be written as |ψ⟩ = Σᵢ λᵢ |aᵢ⟩ |bᵢ⟩ where the |aᵢ⟩ are orthonormal states of A, the |bᵢ⟩ are orthonormal states of B, and the λᵢ are non-negative Schmidt coefficients. The number of nonzero λᵢ is the Schmidt rank. The entanglement entropy is computed from the λᵢ squared. ## Why it matches tensor networks Cutting a bond of a tensor network into a left and a right part gives exactly this form. The Schmidt coefficients are the singular values of the matrix formed by the two sides. This is why SVD and Schmidt decomposition are the same operation. ## Truncation Keeping only the largest λᵢ and discarding the rest is Schmidt-rank truncation. The retained weight is the sum of the kept squared coefficients. The discarded weight measures the error. Quantaria uses this operation to control how much of each bond is retained. See [Tensor-Network Weights](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/tensor-network-weights.md#schmidt-truncation). ## Limits of the analogy * The bond in a neural layer is not a physical system's cut. The analogy is mathematical: the same decomposition applies, but the physical meaning does not carry over automatically. * A low Schmidt rank in a layer means the layer can be compressed across that cut. Whether that helps the model's accuracy is a separate question, answered by evaluation. ## Related * [SVD and truncation](/deep-dives/svd-and-truncation.md) * [Bond dimension and entanglement entropy](/deep-dives/bond-dimension-and-entanglement.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/schmidt-decomposition.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.