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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.
Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices. In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits.
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 ...
Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid. [34] [6] In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg.
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.
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 ...
A quantum depolarizing channel is a model for quantum noise in quantum systems. The -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map, depending on one parameter , which maps a state onto a linear combination of itself and the maximally mixed state,
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).