Geometrical aspects of entanglement

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.

Figure 1

(Color online) A schematic representation of the set of density matrices in the vector space of Hermitian matrices. The positive matrices form a cone about the axis defined by the unit matrix, and the normalization condition restricts the density matrices to a convex subset, here represented by the shaded circle.

Figure 2

Geometric representation of the diagonal density matrices for the three cases of Hilbert space dimension 2, 3, and 4. For general dimension $n$ they define a hyperpyramid of dimension $n-1$.

Figure 3

(Color online) A relativistic view of the density matrices of a two-level system. The positive Hermitian matrices are represented by the forward light cone and a density matrix $\rho$ by a timelike or lightlike ray (red line). The standard normalization $\mathrm{Tr}\rho =x^0=1$ breaks relativistic invariance (green horizontal line), but may be replaced by the relativistically invariant normalization $\det \rho =x^{\mu }{x}_{\mu }∕4=1∕4$ (green hyperbola).

Figure 4

(Color online) Geometric representation of the diagonal matrices spanned by orthogonal Bell states of the $2\times 2$ system. The density matrices form a tetrahedron (green, its edges are diagonals of the faces of a cube) while the matrices obtained from this by a partial transposition define a mirror image (blue, its edges are the opposite diagonals). The separable states are defined by the intersection between the two tetrahedra and form an octahedron (in the center).

Figure 5

(Color online) Schematic illustration of the method of finding the separable state closest to a test matrix $\rho$. The matrix ${\rho }_s$ represents the best approximation to the closest separable state at a given iteration step, while ${\rho }_p$ is the product matrix that maximizes the scalar product with the matrix $\sigma =\rho -{\rho }_s$. This matrix is used to improve ${\rho }_s$ at the next iteration step. The shaded area $\mathcal{S}$ in the figure illustrates a section through the set of separable density matrices.

Figure 6

(Color online) The boundary of the set of separable states in a two-dimensional section, as determined numerically. $\mathcal{S}$ denotes the set of separable states, $\mathcal{B}$ the states with bound entanglement, and $\mathcal{E}$ the entangled states violating the Peres condition. The straight line from lower left to upper right (in blue) is determined by the algebraic equation $\det \rho =0$ and gives the boundary of the full set of density matrices. The two curves from upper left to lower right (in red) are determined by the equation $\det {\rho }^P=0$, in particular, the red straight line gives the boundary of the set of states that satisfy the Peres condition. The black dots are the numerically determined points on the boundary of the set of separable states. The coordinate values in the plot are a factor of $\sqrt{2}$ too large according to the definition of distance in the text.