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Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. 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]
When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.
Penrose diagram of an infinite Minkowski universe, horizontal axis u, vertical axis v. In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity.
For periodic boundary conditions,Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. A matrix product state (MPS) is a representation of a quantum many-body state.
Spin network diagram, after Penrose In physics , a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics . From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of matrix groups .
Foundational research on tensor networks began in 1971 with a paper by Roger Penrose. [9] In “Applications of negative dimensional tensors” Penrose developed tensor diagram notation, describing how the diagrammatic language of tensor networks could be used in applications in physics. [10]
The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved. [3]
The ZX-calculus is a rigorous graphical language for reasoning about linear maps between qubits, which are represented as string diagrams called ZX-diagrams. A ZX-diagram consists of a set of generators called spiders that represent specific tensors. These are connected together to form a tensor network similar to Penrose graphical notation.
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