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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.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
Pauli introduced the 2×2 Pauli matrices as a basis of spin operators, thus solving the nonrelativistic theory of spin. This work, including the Pauli equation , is sometimes said to have influenced Paul Dirac in his creation of the Dirac equation for the relativistic electron, though Dirac said that he invented these same matrices himself ...
Multi-qubit Pauli matrices can be written as products of single-qubit Paulis on disjoint qubits. Alternatively, when it is clear from context, the tensor product symbol can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:
Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum X(t), P(t), with the restriction that the time integral over one period T of the momentum times the velocity must be a positive integer multiple of the Planck constant = =.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli. The Pauli group on n {\displaystyle n} qubits, G n {\displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {\displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle ...
The operator = = (), where is the second Pauli matrix and is the complex conjugation operator, is antiunitary. It satisfies U 2 = − 1 {\displaystyle U^{2}=-1} . Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries
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 ...