linear algebra, higher linear algebra
(…)
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.
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.