Density Matrices

Robert B. Griffiths

Istok P. Mendaš

Properties and structure of the 15-dimensional parameter space of four-state density matrices are examined using the SU(4) generator expansion. Appropriate classification of one-, two- and three-parameter density matrices is obtained, based on the sameness of the characteristic polynomial of density matrices belonging to a given type. It is found that in the one parameter case of 15 different density matrices only three distinct types exist, while in the two parameter case 105 different density matrices group into 11 distinct types. In the three parameter case appropriate classification of 455 different density matrices into 44 types is determined. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are illustrated by randomly drawing matrices for several types of density matrices, providing some insight into the intricate and complex structure of the space of density matrices for a four-state system. Positions of the representative points corresponding to the pure states are found for all types. Global properties of observables are determined by generating, by the Monte Carlo sampling method, and averaging over nearly all density matrices pertaining to a given type.

John E. Harriman

The relationship between densities and density matrices is explored in the case of a finite-basis-set expansion. The space of one-electron density matrices can be divided into two orthogonal subspaces with elements in one of them in one-to-one correspondence with densities. A component in the other does not contribute to the density. The set of densities is convex but there may be densities which cannot be obtained from a density matrix. The matrix of a local potential has a component only in the first subspace, and any such matrix can be obtained from a local potential. It is possible for Hamiltonian matrices differing by the matrix of a local potential to have a common ground-state eigenvector, so a Hohenberg-Kohn theorem cannot always be established. When it can, the explicit local potential with a given ground-state density can be formally obtained when appropriate conditions are satisfied. The details of the decomposition of the space of matrices and of subsequent developments depend on linear-dependency relationships among basis-set products, and are thus basis-set dependent.

Stephen J. Gustafson, Israel Michael Sigal

Y. Ben-Aryeh

Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.

K. T. Hecht