Penrose graphical notation

In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971.^[1] A diagram in the notation consists of several shapes linked together by lines.

The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, Categorical quantum mechanics which includes ZX-calculus is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams.

The notation has been studied extensively by Predrag Cvitanović, who used it, along with Feynman's diagrams and other related notations in developing "birdtracks", a group-theoretical diagram to classify the classical Lie groups.^[2] Penrose's notation has also been generalized using representation theory to spin networks in physics, and with the presence of matrix groups to trace diagrams in linear algebra.

^ Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). See Vladimir Turaev, Quantum invariants of knots and 3-manifolds (1994), De Gruyter, p. 71 for a brief commentary.
^ Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press.

[1] Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). See Vladimir Turaev, Quantum invariants of knots and 3-manifolds (1994), De Gruyter, p. 71 for a brief commentary.

[2] Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press.

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