Pauli group
E674930
The Pauli group is the set of all products of Pauli matrices (up to phase factors), forming a fundamental discrete group used to describe qubit operations in quantum mechanics and quantum computing.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Pauli group canonical | 1 |
| n-qubit Pauli group | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7600731 — 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: Pauli group Context triple: [Pauli matrices, belongsToGroup, Pauli group]
-
A.
Pauli matrices
Pauli matrices are a set of three 2×2 complex Hermitian and unitary matrices that form a basis for the Lie algebra su(2) and are fundamental in describing spin-½ particles in quantum mechanics.
-
B.
special unitary group SU(n)
The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.
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C.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
-
D.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
-
E.
Gell-Mann matrices
Gell-Mann matrices are a set of eight 3×3 traceless Hermitian matrices that serve as the generators of the SU(3) Lie algebra in quantum chromodynamics and other areas of particle physics.
- 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: Pauli group Target entity description: The Pauli group is the set of all products of Pauli matrices (up to phase factors), forming a fundamental discrete group used to describe qubit operations in quantum mechanics and quantum computing.
-
A.
Pauli matrices
Pauli matrices are a set of three 2×2 complex Hermitian and unitary matrices that form a basis for the Lie algebra su(2) and are fundamental in describing spin-½ particles in quantum mechanics.
-
B.
special unitary group SU(n)
The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.
-
C.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
-
D.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
-
E.
Gell-Mann matrices
Gell-Mann matrices are a set of eight 3×3 traceless Hermitian matrices that serve as the generators of the SU(3) Lie algebra in quantum chromodynamics and other areas of particle physics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Clifford group subgroup
ⓘ
discrete group ⓘ mathematical group ⓘ matrix group ⓘ |
| actsOn | single qubit Hilbert space ⓘ |
| basisRole | forms operator basis for single-qubit observables up to phase ⓘ |
| closureProperty |
closed under matrix multiplication
ⓘ
closed under taking inverses ⓘ |
| commutationStructure | elements either commute or anticommute ⓘ |
| containsElement |
-I
ⓘ
-iI ⓘ Pauli X matrix ⓘ
linked to:
Pauli matrices
Pauli Y matrix ⓘ
linked to:
Pauli matrices
Pauli Z matrix ⓘ
linked to:
Pauli matrices
iI ⓘ identity matrix ⓘ |
| definedOver | 2-dimensional complex Hilbert space for single qubit ⓘ |
| elementType | 2×2 complex matrices for single-qubit case ⓘ |
| field |
quantum computing
ⓘ
quantum information theory ⓘ quantum mechanics ⓘ |
| generalizesTo |
n-qubit Pauli group
ⓘ
linked to:
Pauli group
|
| generatedBy |
Pauli X matrix
ⓘ
Pauli Y matrix ⓘ
linked to:
Pauli matrices
Pauli Z matrix ⓘ
linked to:
Pauli matrices
global phase factors ⓘ |
| hasCenter | set of scalar phase multiples of identity ⓘ |
| hasProperty |
finite
ⓘ
non-abelian ⓘ unitary ⓘ |
| isAbelian | false ⓘ |
| measurementRole | generators correspond to spin-1/2 observables along orthogonal axes ⓘ |
| namedAfter | Wolfgang Pauli ⓘ |
| nQubitVersionOrder | 4^(n+1) for n-qubit Pauli group including phases ±1, ±i ⓘ |
| order | 16 for single-qubit Pauli group including ±1, ±i phases ⓘ |
| relatedTo |
Clifford group
ⓘ
Heisenberg–Weyl group ⓘ
linked to:
Heisenberg Lie algebra
stabilizer group ⓘ |
| representation | projective representation of Z2×Z2 for single-qubit case ⓘ |
| usedFor |
defining stabilizer formalism
ⓘ
describing qubit errors ⓘ fault-tolerant quantum computation ⓘ modeling qubit operations ⓘ quantum error correction ⓘ stabilizer codes ⓘ |
| usedIn |
Pauli error channels
ⓘ
Pauli twirling protocols ⓘ stabilizer measurement circuits ⓘ |
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: Pauli group Description of subject: The Pauli group is the set of all products of Pauli matrices (up to phase factors), forming a fundamental discrete group used to describe qubit operations in quantum mechanics and quantum computing.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pauli group