Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments

We characterize the boundary of the convex compact set of absolutely separable states, referred as AS, that cannot be transformed to entangled states by global unitary operators, in $2\otimes d$ Hilbert space. However, we show that the absolutely separable states of rank $(2d-1)$ are extreme points of such sets. We then discuss conditions to examine if a given full-rank absolutely separable state is an interior point or a boundary point of AS. Moreover, we construct two-qubit absolutely separable states which are boundary points but not extreme points of AS and prove the existence of full-rank extreme points of AS. Properties of certain interior points are also explored. We further show that by examining the boundary of the above set, it is possible to develop an algorithm to generate the absolutely separable states which stay outside the maximal ball. By considering paradigmatic noise models, we find the amount of local noise which the input entangled states can sustain, so that the output states do not become absolutely separable. Interestingly, we report that with the decrease of entanglement of the pure input state, critical depolarizing noise value, transferring an entangled state to an absolutely separable one, increases, thereby showing advantages of sharing nonmaximally entangled states. Furthermore, when the input two-qubit states are Haar uniformly generated, we report a hierarchy among quantum channels according to the generation of absolutely separable states.

Figure 1

MB represents the maximal ball. It is fully contained in AS, the set of absolutely separable states, while the set of separable states is marked as SEP. Similarly, SEP is the subset of D denoting the entire state space. In $2\otimes 2$, the portion where the boundaries of MB and AS are touching with each other contains the rank-3 extreme points of AS, while the region in the boundary of AS which is not touching that of MB contains rank-4 extreme points of AS. Furthermore, the line segments at the boundary of AS represent the boundary points which are not extreme points.

Figure 2

Creation of absolutely separable states from a pure state $|\psi \rangle =cos(x/2)|00\rangle +e^{-i\varphi }sin(x/2)|11\rangle$. Note that $|\psi \rangle$ is maximally entangled with $x=\pi /2$. It is sent through local depolarizing channels which produce separable as well as absolutely separable states with the variation of parameter $p$ (abscissa). Red, blue, and green lines correspond to $x=\pi /2, x=\pi /6$, and $x=\pi /12$ respectively. Dashed lines correspond to the minimum eigenvalues of the partial transposed output states for different values of $x$ (ordinate), thereby quantifying the entanglement contents of the output states while solid lines represent the quantity ${\lambda }_1-{\lambda }_3-2\sqrt{{\lambda }_2{\lambda }_4}$ (ordinate) for examining the absolute separability in $2\otimes 2$, given in (1). It is clear that there exists a range of $p$ where the output state is separable but not absolutely separable when the input state is a pure nonmaximally entangled state, while for maximally entangled state, the critical value of noise above which state is separable as well as absolutely separable coincide (see red solid and dashed lines). Both axes are dimensionless.

Figure 3

Plot of $p_c$ (critical noise value below which the state is absolutely separable) (vertical axis) against initial entanglement, $E$ (horizontal axis). We generate ${10}^4$ pure states Haar uniformly. Both qubits are sent through the local depolarizing channels. Entanglement of a pure state is characterized by the von Neumann entropy of the local density matrices. The vertical axis is dimensionless while the horizontal one is in ebits.

Figure 4

Regions in $(x,p)$-plane indicates separable and absolutely separable states when $|\psi \rangle$ is sent through local amplitude damping channels. The parameter $x$ corresponds to the state $|\psi \rangle$, and the parameter $p$ corresponds to the channels. The blue (bigger) region is for separability, while the yellowish (smaller) region represents states that are absolutely separable. The parameter $p$ in the vertical axis is a dimensionless quantity, while the parameter $x$ in the horizontal axis is in radian.

Figure 5

Map of separable (blue) and absolutely separable (yellow) states for local PDC with the Werner state as input. The abscissa and ordinate, respectively, represent the noise parameter, $p$, and the mixing parameter, $q$, of the Werner state. Both axes are dimensionless.