nLab bra-ket




Traditional notation in physics [Dirac 1939] for writing down pure quantum states (elements of Hilbert spaces), their hermitian adjoints and Hermtian inner products:

With the Hermitian inner product on the underlying vector space of a Hilbert space HH denoted |\langle-\vert-\rangle, element ψH\psi \,\in\, H are turned into elements of the linear dual space H *H^\ast by ψ|H *\langle \psi \vert -\rangle \,\in\, H^\ast, whose evaluation on another ϕH\phi \in H is the given Hermitian form ψ|ϕ\langle \psi \vert \phi\rangle. Hence if one declares notation such that

|ϕϕH ψ|ψ|H * \array{ \vert \phi \rangle \;\equiv\; \phi \,\in\, H \\ \langle \psi \vert \;\equiv\; \langle \psi \vert -\rangle \,\in\, H^\ast }

then the evaluation-pairing in the Hermitian form is essentially juxtaposition

H *H ψ|,|ϕ ψ|ϕ \array{ H^\ast \otimes H &\longrightarrow& \mathbb{C} \\ \langle \psi \vert , \vert \phi \rangle &\mapsto& \langle \psi \vert \phi \rangle }

With the inner product |\langle-\vert-\rangle referred to as a bracket this suggest to refer to “ψ|\langle \psi \vert” as a “bra” and “|ϕ\vert \phi \rangle” as a “ket” [Dirac 1939, last line].

The notation may be udnerstood as a lightweight precursor to the string diagram-calculus in dagger-compact categories [Abramsky & Coecke 2004 §7.2, 2007 pp. 6, 2008 §4.4, Coecke 2010 §3.3].

For instance, if \mathscr{H} is a finite-dimensional Hilbert space with orthonormal basis (|w) w:W\big(\left\vert w \right\rangle\big)_{w \colon W}, then the compact closure is witnessed by the following isomorphism between the vector space of linear maps out of \mathscr{H} and a vector space of matrices:

() * (|ww|wA w,w) w,w|wA w,ww|. \array{ \Big( \mathscr{H} \multimap \mathscr{H}' \Big) &\longrightarrow& \mathscr{H}' \otimes \mathscr{H}^\ast \\ \Big( \left\vert w \right\rangle \,\mapsto\, \underset{w'}{\sum} \left\vert w' \right\rangle A_{w', w} \Big) &\mapsto& \underset{w,w'}{\sum} \left\vert w' \right\rangle A_{w', w} \left\langle w \right\vert \mathrlap{\,.} }

quantum probability theoryobservables and states


The bra-ket notation is due to:

Textbook accounts:

See also:

Discussion in the (broader) context of string diagram-calculus for dagger compact categories (cf. quantum information theory via dagger-compact categories):

Last revised on September 24, 2023 at 05:36:00. See the history of this page for a list of all contributions to it.