nLab finite-dimensional Hilbert space




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.


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