This article deals with quantum computations in the Pauli basis whose elements are usually identified with the Pauli strings. This approach allows us to represent quantum states, observables, and unitary operators in the unified form of linear combination of Pauli strings, so that all operations can be reduced to the string compositions. Nevertheless a formal justification of the Pauli basis for quantum computations should be based on the strong results of complex linear algebra and the theory of Hilbert spaces. We briefly review the main features of Pauli strings for quantum states and unitary operators, and also the key operations with them, including an algorithm for string compositions and a transformation algorithm from the standard basis to the Pauli basis.
Using normal operators as a common footing, the basic properties of Hermitian operators, unitary operators, and anti-Hermitian operators are treated from a unified viewpoint. The interlinks among the eigenvalues of these operators are elucidated with the help of suitable prescriptions for mapping in the plane of complex eigenvalues. Further, we show that some of the characteristics of Hermitian operators hold good under a perspective which is broader than the defining criterion for Hermitian operators.
In this thesis, real quantum mechanics (QM) is constructed over the field
calFsbPsb0oplusPsb3 in the Pauli algebra, where the trivector
i=sigmasb1sigmasb2sigmasb3 arises naturally as a product of the unit basis vectors in 3-dimensional Euclidean space. In this scheme, i is the volume element of the algebra with
isp2=−1 and has geometrical meaning. Components of the state vector in Hilbert space take the form of a real element in the field
calFsbPsb0oplusPsb3 and the hermitean conjugate of the state vector is just its transpose, so that all the observables, represented by hermitean operators, are symmetric; i itself is an anti-hermitean operator. In this construction, the time reversal operator
calT is a combination of the transpose "+" and the unitary operator which changes the sign of t. Since
calT is an anti-automorphism the interpretation of its action must be carefully analyzed in terms of the state vectors and their inner products. Furthermore, the parity transformation is not represented by the spatial inversion
tildePi (an antilinear, anti-unitary transformation). Instead, the spatial reversal
barPi (an anti-automorphic, linear transformation) is proposed as a parity operator. It is unitary as well as hermitean, and it commutes with the Hamiltonian, H, of a system, thus, it is a constant of motion. This thesis mainly deals with non-relativistic QM for spinless particles.Dept. of Physics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1991 .W444. Source: Masters Abstracts International, Volume: 31-01, page: 0331. Chair: W. E. Baylis. Thesis (M.Sc.)--University of Windsor (Canada), 1991.
This chapter provides a simple general introduction to some basic mathematics necessary for a subsequent discussion of Lie algebras, operator methods, and their applications to theoretical physics and chemistry. It also provides some useful mathematical tools to derive important theoretical results and expressions in a simple and elegant way. In quantum mechanics the state of a system is represented by an element of a particular vector space, in which one defines an appropriate scalar or inner product, known in mathematics as Hilbert space. Hermitian operators with a complete set of eigenvectors belonging to that state space represent physical observables. Moreover, in certain cases it proves useful to work with vector spaces formed by linear operators. Hermitian operators are particular cases of linear operators and physical observables are particular cases of Hermitian operators. In many physical applications of operator methods one needs to reorder products of exponential operators or to rewrite one such operator as a product of simpler ones.
In the frame of a vectorial Pauli algebraic approach it is shown that the product of two exponentials of any two-by-two linear operators can be put in a single-exponential closed form. As a first application, a compact vectorial expression for the characteristics (angle and axis) of the product of two R 3 rotations is established. The same mathematics can be used in a large diversity of problems of the whole class of two-states physical systems. An exemplification in the field of polarization optics is given.
Abstract A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.