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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities. There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions.
The Möbius–Kantor graph, the Cayley graph of the Pauli group with generators X, Y, and Z In physics and mathematics , the Pauli group G 1 {\displaystyle G_{1}} on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I {\displaystyle I} and all of the Pauli matrices
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension . [2] [3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by
Arbitrary Clifford group element can be generated as a circuit with no more than (/ ()) gates. [6] [7] Here, reference [6] reports an 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C-, where H, C, and P stand for computational stages using Hadamard, CNOT, and Phase gates, respectively, and reference [7] shows that the CNOT stage can be implemented using (/ ()) gates (stages -H- and -P ...
The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as [3] =, where denote the Pauli matrices, is the magnetic moment, and B is the magnetic field.
A graphical intuition of purity may be gained by looking at the relation between the density matrix and the Bloch sphere, = (+), where is the vector representing the quantum state (on or inside the sphere), and = (,,) is the vector of the Pauli matrices. Since Pauli matrices are traceless, it still holds that tr(ρ) = 1.