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The Pauli matrices ... explicit properties of the Pauli matrices, ... is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, ...
This method of generalizing the Pauli matrices refers to a generalization from a single 2-level system to multiple such systems.In particular, the generalized Pauli matrices for a group of qubits is just the set of matrices generated by all possible products of Pauli matrices on any of the qubits.
In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble [1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule.
The matrix exponential satisfies the following properties. [2] We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the conjugate transpose of X. If Y is invertible then e YXY −1 = Ye ...
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
The Clifford group is defined as the group of unitaries that normalize the Pauli group: = {† =}. Under this definition, C n {\displaystyle \mathbf {C} _{n}} is infinite, since it contains all unitaries of the form e i θ I {\displaystyle e^{i\theta }I} for a real number θ {\displaystyle \theta } and the identity matrix I {\displaystyle I ...
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i, so that the Hermitian matrix is written as a Pauli vector. [2] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as ...