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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.
The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. [ 7 ] Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Pauli matrices.
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
The Pauli matrices are traceless and orthogonal to one another with respect to the Hilbert–Schmidt inner product, and so the coordinates (,,) of the state are the expectation values of the three von Neumann measurements defined by the Pauli matrices.
Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients, [ 6 ] [ 7 ] which results in skew-Hermitian matrices .
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 matrix exponential of a Pauli matrix ... ), and the real exponents is a generalization of the series circuit. For example, and = / are ...
Gell-Mann matrices — a generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3). Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear-quadratic regulator (LQR) systems.