Geometric Aspects of the Theory of Density Matrices and Densities

John E. Harriman

José J. Gil

Jon Magne Leinaas, Jan Myrheim, Eirik Ovrum

We study geometrical aspects of entanglement, with the Hilbert--Schmidt norm defining the metric on the set of density matrices. We focus first on the simplest case of two two-level systems and show that a ``relativistic'' formulation leads to a complete analysis of the question of separability. Our approach is based on Schmidt decomposition of density matrices for a composite system and nonunitary transformations to a standard form. The positivity of the density matrices is crucial for the method to work. A similar approach works to some extent in higher dimensions, but is a less powerful tool. We further present a numerical method for examining separability and illustrate the method by a numerical study of bound entanglement in a composite system of two three-level systems.

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama

Bei Zeng

Wu-zhong Guo

A bstract In this paper, we investigate the spectral projection of density matrices in quantum field theory. With appropriate regularization, the spectral projectors of density matrices are expected to be well-defined. These projectors can be obtained using the Riesz projection formula, which allows us to compute both the density of eigenvalues and the expectation values of local operators in the projected states. We find that there are universal divergent terms in the expectation value of the stress energy tensor, where the coefficients depend universally on the density of eigenvalues and a function that describes the dependence of eigenvalues on boundary location. Using projection states, we can construct a series of new states in quantum field theories and discuss their general properties, focusing on the holographic aspects. We observe that quantum fluctuations are suppressed in the semiclassical limit. We also demonstrate that the fixed area state, previously constructed using gravitational path integrals, can be constructed by suitably superposition of appromiate amount of projection states. Additionally, we apply spectral projection to non-Hermitian operators, such as transition matrices, to obtain their eigenvalues and densities. Finally, we highlight potential applications of spectral projections, including the construction of new density and transition matrices and the understanding of superpositions of geometric states.