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These points may be joined forming a shape like a spider. Joined points represent an "or" condition, also known as a logical disjunction. A spider diagram is a boolean expression involving unitary spider diagrams and the logical symbols ,,. For example, it may consist of the conjunction of two spider diagrams, the disjunction of two spider ...
An example ZX-diagram. This one has two inputs (wires coming from the left), and three outputs (wires exiting to the right), and hence it represents a linear map from to . ZX-diagrams consist of green and red nodes called spiders, which are connected by wires. Wires may curve and cross, arbitrarily many wires may connect to the same spider, and ...
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that = + + where non-negative integers, then is the character of the direct sum of the representations corresponding to .
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication.The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact ( Hausdorff ) topological group and the representations are strongly continuous .
Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example. Let G = R {\displaystyle G=\mathbb {R} } , and let the complex Hilbert space V be L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} .
The unitary group arises as follows: the general linear group over the complex numbers has a diagram automorphism given by reversing the Dynkin diagram A n (which corresponds to taking the transpose inverse), and a field automorphism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the ...