nLab
finite-dimensional Hilbert space
Contents
Context
Linear algebra
linear algebra , higher linear algebra

Ingredients
Basic concepts
ring , A-∞ ring

commutative ring , E-∞ ring

module , ∞-module , (∞,n)-module

field , ∞-field

vector space , 2-vector space

rational vector space

real vector space

complex vector space

topological vector space

linear basis ,

orthogonal basis , orthonormal basis

linear map , antilinear map

matrix (square , invertible , diagonal , hermitian , symmetric , …)

general linear group , matrix group

eigenspace , eigenvalue

inner product , Hermitian form

Gram-Schmidt process

Hilbert space

Theorems
(…)

Contents
Idea
A finite dimensional Hilbert space is a Hilbert space whose underlying vector space is a finite-dimensional vector space .

Since the completeness-condition on a Hilbert space is automatically satisfied in finite dimensions, finite-dimensional Hilbert spaces are equivalently finite dimensional vector spaces (usually complex vector spaces , sometimes real vector spaces ) equipped with a positive-definite Hermitian inner product , aka Hermtian inner product spaces .

Finite-dimensional Hilbert spaces are the spaces of quantum states of principal interest in quantum information theory and quantum computation . In quantum physics more broadly they appear in discussion of internal degrees of freedom (such as spin ) or generally as the quantization of compact phase spaces (cf. eg. the geometric quantization of the 2-sphere or generally the orbit method ).

The fact that they form a dagger-compact category [cf. Abramsky & Coecke 2004 p 10 ; Selinger 2012] has led to the string diagram -formulation of quantum information theory via dagger-compact categories .

References
Paul R. Halmos , Finite-Dimensional Hilbert Spaces , The American Mathematical Monthly, 77 5 (1970) 457-464 [doi:10.2307/2317378 , jstor:2317378 ]

Samson Abramsky , Bob Coecke , p. 10 of A categorical semantics of quantum protocols , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04), IEEE Computer Science Press (2004) [arXiv:quant-ph/0402130 ]

Peter Selinger , Finite dimensional Hilbert spaces are complete for dagger compact closed categories , Logical Methods in Computer Science, 8 3 (2012) lmcs:1086 [arXiv:1207.6972 , doi:10.2168/LMCS-8(3:6)2012 ]

Klaas Landsman , §A.1 in: Foundations of quantum theory – From classical concepts to Operator algebras , Springer Open (2017) [doi:10.1007/978-3-319-51777-3 , pdf ]

Lecture notes:

Last revised on October 26, 2023 at 08:18:19.
See the history of this page for a list of all contributions to it.