Schmidt decomposition
E384561
The Schmidt decomposition is a mathematical technique in functional analysis and quantum information theory that expresses a bipartite vector (such as a quantum state) as a sum of orthogonal product states with nonnegative coefficients, revealing its entanglement structure.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Schmidt basis | 1 |
| Schmidt decomposition canonical | 1 |
| Schmidt expansion | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3748398 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Schmidt decomposition Context triple: [Erhard Schmidt, notableWork, Schmidt decomposition]
-
A.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
-
D.
SVD
SVD is the abbreviation for the Special Victims Division, a specialized police unit that investigates sensitive crimes such as sexual offenses and crimes against vulnerable victims.
-
E.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Schmidt decomposition Target entity description: The Schmidt decomposition is a mathematical technique in functional analysis and quantum information theory that expresses a bipartite vector (such as a quantum state) as a sum of orthogonal product states with nonnegative coefficients, revealing its entanglement structure.
-
A.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
-
D.
SVD
SVD is the abbreviation for the Special Victims Division, a specialized police unit that investigates sensitive crimes such as sexual offenses and crimes against vulnerable victims.
-
E.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
decomposition theorem
ⓘ
mathematical concept ⓘ tool in quantum information theory ⓘ |
| alsoKnownAs |
Schmidt expansion
ⓘ
linked to:
Schmidt decomposition
|
| appliesTo |
bipartite quantum state
ⓘ
bipartite vector ⓘ |
| category |
linear algebra
ⓘ
operator theory ⓘ |
| characterizes | bipartite pure state entanglement ⓘ |
| coincidesWith | singular value decomposition of coefficient matrix ⓘ |
| constraintOn | Schmidt coefficients sum of squares equals 1 for normalized states ⓘ |
| criterionFor | separability of bipartite pure states ⓘ |
| defines |
Schmidt number for pure states
ⓘ
Schmidt rank ⓘ |
| expressesAs | sum of orthogonal product states ⓘ |
| extendsTo | certain infinite-dimensional Hilbert spaces ⓘ |
| field |
functional analysis
ⓘ
quantum information theory ⓘ |
| hasComponent |
Schmidt basis
ⓘ
linked to:
Schmidt decomposition
Schmidt coefficients ⓘ Schmidt rank ⓘ |
| hasHistoricalOrigin | Erhard Schmidt ⓘ |
| hasProperty | uses nonnegative coefficients ⓘ |
| holdsIn | finite-dimensional Hilbert spaces ⓘ |
| implies |
Schmidt basis is unique up to phases and degeneracies
ⓘ
Schmidt coefficients are unique up to degeneracies ⓘ existence of orthonormal bases for each subsystem ⓘ state is separable iff Schmidt rank equals 1 ⓘ |
| mathematicalForm | |ψ⟩ = Σ_i λ_i |i_A⟩⊗|i_B⟩ with λ_i ≥ 0 ⓘ |
| relatedTo |
singular value decomposition
ⓘ
spectral theorem ⓘ tensor product structure of Hilbert spaces ⓘ |
| requires | separable Hilbert spaces ⓘ |
| reveals | entanglement structure ⓘ |
| usedIn |
bipartite pure state classification
ⓘ
bipartite state tomography analysis ⓘ entanglement distillation protocols ⓘ entanglement quantification ⓘ quantum channel capacity proofs ⓘ quantum communication ⓘ quantum entanglement theory ⓘ quantum teleportation analysis ⓘ |
| usedTo |
identify maximally entangled states
ⓘ
simplify analysis of bipartite pure states ⓘ |
| usedToCompute |
Rényi entropies of entanglement
ⓘ
entanglement entropy ⓘ von Neumann entropy of reduced state ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Schmidt decomposition Description of subject: The Schmidt decomposition is a mathematical technique in functional analysis and quantum information theory that expresses a bipartite vector (such as a quantum state) as a sum of orthogonal product states with nonnegative coefficients, revealing its entanglement structure.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Schmidt decomposition
linked to: Schmidt decomposition