Abstract This chapter provides a brief introduction to the terminology and basic concepts of quantum spin systems, which are widely used in QIS and QC. The quantum spin of an elementary particle is an example of a qubit and can be visually represented as a spinning top. The vector direction of spin in state space, however, is different from the direction in physical space. Quantum spin-1/2 systems have two orthogonal states: spin-up and spin-down. The distinction between state space angles and physical space angles is emphasized. Because spin is a qubit, the quantum gates from Chapter 6 can be applied to spin-1/2 quantum states with predictable results.Abstract This chapter provides a brief introduction to the terminology and basic concepts of quantum spin systems, which are widely used in QIS and QC. The quantum spin of an elementary particle is an example of a qubit and can be visually represented as a spinning top. The vector direction of spin in state space, however, is different from the direction in physical space. Quantum spin-1/2 systems have two orthogonal states: spin-up and spin-down. The distinction between state space angles and physical space angles is emphasized. Because spin is a qubit, the quantum gates from Chapter 6 can be applied to spin-1/2 quantum states with predictable results.

Apr 15, 20220 citations

Quasi-equilibria in reduced Liouville spaces

Meghan E. Halse, Jean‐Nicolas Dumez, Lyndon Emsley

The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.

The Journal of Chemical PhysicsJun 14, 201219 citations

Creation of Spin

Ijaz Durrani

Jan 1, 20260 citations

Time and Space

Springer eBooksJan 5, 20090 citations

Relativistic spin operator and Lorentz covariant reduced spin density matrix in a new representation space

Heon Lee

The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable $\frac{1}{2}\boldsymbol{\tau}$ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable $\frac{1}{2}\boldsymbol{\tau}$ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.

May 21, 20210 citations

\frac{1}{2}\boldsymbol{\tau}

from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.