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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.
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.
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. [1]
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 ...
One way to do this is to factor the matrix that encodes the unitary transformation into a product of tensor products (i.e. series and parallel circuits) of the available primitive gates. The group U(2 q ) is the symmetry group for the gates that act on q {\displaystyle q} qubits. [ 2 ]
The matrices α and β are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of 3B satisfy equation ( 2 ); instead he regenerated the equation using a Lorentz-invariant action , via the principle of least action , and application of Lorentz group theory.
The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators. ^ = [(^)] + Substitution into the Schrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field.
This time there is a 2 × 2 identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix σ 0 which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).