Density Matrix and Product Operator Formalisms

Vladimir I Man$apos$ko, Giuseppe Marmo, E. C. G. Sudarshan +1 more

An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including a visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matrix and the tensor product of partial traces. Entanglement for arbitrary quantum observables for multipartite systems is discussed. Star-product kernels are used to map the formulation of the addition rule of density operators onto the addition rule of symbols of the operators. Entanglement and nonlocalization of the pure state projector and allied operators are discussed. Tomographic and Weyl symbols (tomograms and Wigner functions) are considered as examples. The squeezed states and some spin states (two qubits) are studied to illustrate the formalism.

Timothy F. Havel

We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g. spin = 1/2) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain.

Ravinder Rupchand Puri

H G Venkatesh, N R Roy

The problem of interaction of radiation with a single atom having an isolated pair of energy levels has been solved analytically using the formalism of a spin- 1 / 2 particle, and the T transformation in the interaction picture. This transformation is equivalent to going over to a rotating coordinate system in spin space. The case of the electric dipole transition Delta m=+or-1 is presented. The spin- 1 / 2 formalism is extended to an assembly of atoms. By taking the product of the T operators for single atoms, the T operator for the gas as a whole is obtained. A general expression for the density operator describing the state of the gas at any time t is obtained from density operators for individual atoms. The correlated emission from the gas can be obtained by the unitary transformation of the initial density matrix by the macroscopic T operator.