Search results
Results From The WOW.Com Content Network
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 density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. [2] In practice, the terms density matrix and density operator are often used interchangeably.
Pauli matrices: A set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I 2 identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices. Redheffer matrix: Encodes a Dirichlet convolution. Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius ...
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 ...
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 approach allowed Pauli to develop a proof of his fundamental Pauli exclusion principle, a proof now called the spin-statistics theorem. [7] In retrospect, this insistence and the style of his proof initiated the modern particle-physics era, where abstract quantum properties derived from symmetry properties dominate.
Another possible definition of the Clifford group can be obtained from the above by further factoring out the Pauli group {,,,} on each qubit. The leftover group is isomorphic to the group of 2 n × 2 n {\displaystyle 2n\times 2n} symplectic matrices Sp(2 n ,2) over the field F 2 {\displaystyle \mathbb {F} _{2}} of two elements. [ 4 ]
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 ...