nLab bra-ket

Contents

Context

Linear algebra

Idea
Related concepts
References

Idea

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 $H$ denoted $\langle-\vert-\rangle$ , element $\psi \,\in\, H$ are turned into elements of the linear dual space $H^\ast$ by $\langle \psi \vert -\rangle \,\in\, H^\ast$ , whose evaluation on another $\phi \in H$ is the given Hermitian form $\langle \psi \vert \phi\rangle$ . Hence if one declares notation such that

\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

\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 “ $\langle \psi \vert$ ” as a “bra” and “ $\vert \phi \rangle$ ” as a “ket” [Dirac 1939, last line].

The notation may be understood 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 $\big(\left\vert w \right\rangle\big)_{w \colon W}$ , 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:

\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{\,.} }

Related concepts

quantum probability theory – observables and states

References

The bra-ket notation is due to:

Paul A. M. Dirac, A new notation for quantum mechanics, Mathematical Proceedings of the Cambridge Philosophical Society 35 3 (1939) 416-418 [doi:10.1017/S0305004100021162, pdf]

Textbook accounts:

Jun John Sakurai, Jim Napolitano, §1.2 in: Modern Quantum Mechanics, Cambridge University Press (1985, 1994, 2020) [doi:10.1017/9781108587280, Wikipedia]
Michael A. Nielsen, Isaac L. Chuang, Fig. 2.1 in: Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]
Robert B. Griffiths, Linear Algebra in Dirac notation, Section 3 in: Consistent Quantum Theory, Cambridge University Press (2002) [doi:10.1017/CBO9780511606052, webpage]