Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Tensor-product Hilbert space

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The completed tensor product representing composite quantum systems and independent degrees of freedom.

Record metadata

Canonical name: Tensor-product Hilbert space
Concept ID: tensor_product_hilbert_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Hilbert-space construction

Carrier(s)

Hilbert spaces $H,K$

Data

completion of the algebraic tensor product under the canonical inner product

Axioms / constraints

$\langle x_1\otimes y_1,x_2\otimes y_2\rangle=\langle x_1,x_2\rangle\langle y_1,y_2\rangle$

Canonically induces

product states
entangled states
partial traces on trace-class operators

Concept sources

Wikipedia — Tensor product of Hilbert spaces — Tensor product of Hilbert spaces · encyclopedic reference · source ID wp-tensor_product_hilbert_space
nLab — tensor product of Hilbert spaces — tensor product of Hilbert spaces · mathematical reference · source ID nlab-tensor_product_hilbert_space
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Hilbert space Tensor-product Hilbert space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: tensorhilbert_from_hilbert
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: completed tensor product

Authored explanation

The algebraic tensor product is completed in its natural Hilbert norm.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Hilbert space — Hilbert space · expository · source ID wp-hilbert
nLab — Hilbert space — Hilbert space · reference · source ID nlab-hilbert
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis
Wikipedia — Tensor product of Hilbert spaces — Tensor product of Hilbert spaces · encyclopedic reference · source ID wp-tensor_product_hilbert_space
nLab — tensor product of Hilbert spaces — tensor product of Hilbert spaces · mathematical reference · source ID nlab-tensor_product_hilbert_space
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.