Suppose a single spin-1/2  is in the state , which is an eigenstate of the z-spin operator , that is . Similarly . Making use of the expansion of a Hermitian operator  in terms of projections onto its eigenkets  with eigenvalues  as , the associated density operator is
where  is the identity operator. Similarly, the density operator for the state  is
Since the spin operators  are all traceless and the expectation value of an operator  for a system with density operator  is , the terms proportional to the unit operator  do not affect the expectations of the spin operators. Additionally those parts do not evolve in time, since they trivially commute with the Hamiltonian. Therefore those terms can be ignored, and the state  corresponds to a density operator , while the state  corresponds to a density operator . In exactly the same manner, polarisation along the positive x-axis, that is a state , corresponds to a density operator . This idea extends naturally to multiple spins, where the states and operators are direct products of single-spin states and operators. Hence operator terms in the density operator have a direct duality with states.
In the case of two spins , the terms in the density operator (ignoring the identity on its own) can be interpreted as representing
*  - longitudinal magnetisation
*  - in-phase transverse magnetisation, which is the observable quantity in NMR.
*  - anti-phase longitudinal magnetisation
*  - longitudinal two-spin order
*  - other coherences, which are more difficult to interpret, but may evolve into other terms
where eg  is a shorthand for the Kronecker product , where  is the identity operator on the  spin, and similarly  is a shorthand for .
The factors of two in the true two-spin operators are to allow for convenient commutation relations in this specific spin-1/2 case - see below. Note also that we could instead choose to expand the density operator in the basis  etc, where the transverse operators have been replaced with raising and lowering operators. With quadrature detection, the observable associated with an individual spin is effectively the non-Hermitian , so this is sometimes more convenient.