This note is part of [[quantum/Practical Quantum Information System]]. > [!info] Course code > Use the companion repository for this lecture's runnable lab, helper functions, and regression checks: > - [notebooks/by_concept/schmidt_decomposition.ipynb](https://github.com/montekkundan/quantum-code/blob/main/notebooks/by_concept/schmidt_decomposition.ipynb) > - [qcourse/entanglement.py](https://github.com/montekkundan/quantum-code/blob/main/qcourse/entanglement.py) > - [tests/test_states_density_entanglement.py](https://github.com/montekkundan/quantum-code/blob/main/tests/test_states_density_entanglement.py) [TODO: add video - Schmidt Decomposition] ## What This Concept Is If you want the cleanest mathematical tool for analyzing pure bipartite entanglement, this is it. The [[quantum/Glossary#Schmidt decomposition|Schmidt decomposition]] rewrites a pure state in a basis where its entanglement structure becomes transparent. For many students, this is the moment entanglement stops being a fuzzy physical idea and becomes a tractable linear-algebra object. The decomposition gives you a direct test for separability, a route to reduced-state spectra, and the data needed for entanglement entropy. ## Foundation Terms You Need First A [[quantum/Glossary#Schmidt decomposition|Schmidt decomposition]] expresses a pure bipartite state in paired basis vectors with nonnegative [[quantum/Glossary#Schmidt coefficient|Schmidt coefficients]]. [[quantum/Glossary#Schmidt coefficient|Schmidt coefficients]] determine the reduced-state eigenvalues. A [[quantum/Glossary#Separable state|separable pure state]] has Schmidt rank 1. It also helps to remember the singular value decomposition from linear algebra. In practical two-qubit work, the Schmidt decomposition is often computed exactly that way by reshaping amplitudes into a matrix and taking its SVD. ## How The Idea Actually Works Any pure bipartite state can be written as $ |\psi\rangle = \sum_i \sqrt{\lambda_i}\, |\alpha_i\rangle |\beta_i\rangle, $ where the $\lambda_i$ are nonnegative and sum to one. The number of nonzero coefficients is the Schmidt rank. This form does two important jobs at once. First, it tests separability: if only one coefficient is nonzero, the state factors and is separable. Second, it gives the reduced density matrix spectrum immediately: the $\lambda_i$ are the eigenvalues of the reduced state. That means the Schmidt decomposition is not only a convenient rewriting. It is the hinge that connects pure-state entanglement to density matrices and entropy. Once you know the coefficients, several later quantities become almost automatic to compute. This is one reason the topic matters so much pedagogically. It is one of the rare places where the algebra, the physics, and the information-theoretic interpretation line up cleanly in one short formula. ## Worked Example: Reading Entanglement From Singular Values For $ |\psi\rangle=\sqrt{\frac34}|00\rangle+\sqrt{\frac14}|11\rangle, $ the coefficient matrix is already diagonal: $ M= \begin{bmatrix} \sqrt{3/4}&0\\ 0&\sqrt{1/4} \end{bmatrix}. $ Its singular values are $\sqrt{3/4}$ and $\sqrt{1/4}$. The Schmidt weights are therefore $3/4$ and $1/4$. Because two Schmidt weights are nonzero, the state is entangled, but it is not maximally entangled like a Bell pair. For a product state such as $ \frac{|00\rangle+|10\rangle}{\sqrt2}, $ the coefficient matrix has two proportional columns/rows and only one nonzero singular value. Schmidt rank 1 means separable. ## Source Reading Lens Use `qclec.pdf` Lecture 11 here. Treat Schmidt decomposition as the computational version of the entanglement definition: it turns "does this factor?" into singular values, rank, and reduced-state eigenvalues. ## Why It Matters - It gives a direct separability test for pure bipartite states. - It links statevector structure to reduced-density-matrix structure. - It is the fastest route to understanding entanglement entropy for pure states. ## Study Checks Use these after the explanation, not before it. ### Quick Checks - What is Schmidt rank? **Answer:** The number of nonzero Schmidt coefficients in a pure bipartite state's Schmidt decomposition. - MCQ: A pure bipartite state is separable if and only if its Schmidt rank is: A. 0 B. 1 C. 2 D. maximal **Answer:** B. 1. - What do Schmidt coefficients determine? **Answer:** They determine the nonzero eigenvalues of the reduced density matrices and therefore the pure-state entanglement entropy. ### Oral Exam Anchors > [!question] Oral exam anchor > Explain the Schmidt decomposition and why it is useful for entanglement. A good answer should say: Any pure bipartite state can be written as |psi> = sum_i sqrt(lambda_i) |alpha_i>|beta_i>, where the lambda_i are nonnegative and sum to one, and the local basis vectors are orthonormal. The number of nonzero lambda_i values is the Schmidt rank. This form is useful because it makes pure-state entanglement transparent. A pure bipartite state is separable if and only if its Schmidt rank is 1. The same lambda_i values are also the nonzero eigenvalues of the reduced density matrices. Therefore the Schmidt decomposition directly gives both a separability test and the ingredients for entanglement entropy. ## Practical Lab Treat this as the linear-algebra lab for pure-state entanglement. - Write a two-qubit state as a coefficient matrix and compute its singular value decomposition. - Extract the Schmidt coefficients and Schmidt rank. - Compare those coefficients with the eigenvalues of the reduced density matrix. ## Homework Use the homework to make Schmidt rank and separability feel automatic. - Compute the Schmidt decomposition of at least one two-qubit pure state. - Use Schmidt rank to decide whether the state is separable or entangled. - Explain why the Schmidt coefficients determine the reduced-state spectrum. ## References - Scott Aaronson, [Introduction to Quantum Information Science](https://www.scottaaronson.com/qclec.pdf), Lecture 11. - QuTiP documentation, [main docs](https://qutip.org/). - IBM Quantum Learning, [learning portal](https://quantum.cloud.ibm.com/learning/en).