linear algebra, higher linear algebra
(…)
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 denoted , element are turned into elements of the linear dual space by , whose evaluation on another is the given Hermitian form . Hence if one declares notation such that
then the evaluation-pairing in the Hermitian form is essentially juxtaposition
With the inner product referred to as a bracket this suggest to refer to “” as a “bra” and “” 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 is a finite-dimensional Hilbert space with orthonormal basis , then the compact closure is witnessed by the following isomorphism between the vector space of linear maps out of and a vector space of matrices:
The bra-ket notation is due to:
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]
See also:
Discussion in the (broader) context of string diagram-calculus for dagger compact categories (cf. quantum information theory via dagger-compact categories):
Samson Abramsky, Bob Coecke, §7.2 in: 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]
Samson Abramsky, Bob Coecke, pp. 6 in: Physics from Computer Science: a Position Statement, International Journal of Unconventional Computing 3 3 (2007) pdf, ijuc-3-3-p-179-197
Samson Abramsky, Bob Coecke, §4.4 in: Categorical quantum mechanics, in Handbook of Quantum Logic and Quantum Structures, Elsevier (2008) [arXiv:0808.1023, ISBN:9780080931661, doi:10.1109/LICS.2004.1319636]
Bob Coecke, §3.e in: Kindergarten quantum mechanics [arXiv:quant-ph/0510032]
Bob Coecke, §3.3 in: Quantum Picturalism, Contemporary Physics 51 1 (2010) [arXiv:0908.1787, doi:10.1080/00107510903257624]
Last revised on June 20, 2024 at 14:30:08. See the history of this page for a list of all contributions to it.