Einstein-Podolsky-Rosen steering and coherence in the family of entangled three-qubit states

Considering the system of three interacting qubits, we analyze four families of states from the point of view of bipartite correlations appearing in two-qubit subsystems of a three-qubit model, such as Einstein-Podolsky-Rosen steering, entanglement, and coherence. We reveal mutual relations among the steering parameter, concurrence, and three measures of coherence (degree of coherence, first-, and second-order correlation functions). Analyzing in parallel the steerable and unsteerable states, we derive analytical formulas giving the maximal and minimal values of coherence measures as concurrence varies.

Figure 1

Degree of coherence $D_{ij}^2$ versus concurrence $C_{ij}$ for steerable (a) and unsteerable (b) two-qubit states derived from the density matrix ${\rho }_{ijk}^{\left(I\right)}$ given in Eq. (4). Steerable (unsteerable) states are found in green (blue) areas. Black border curves are drawn according to the corresponding analytical formulas.

Figure 2

Steering parameter $S_{ij}$ versus the degree of coherence $D_{ij}^2$ for steerable states.

Figure 3

Degree of coherence $D_{ij}^2$ versus concurrence $C_{ij}$ for steerable (green) and unsteerable (blue) two-qubit states described by the numerically calculated density matrix ${\rho }_{ij}^{\left(II\right)}$ and observed in three-qubit system with double excitation. Black border curves are drawn following the corresponding analytical formulas.

Figure 4

First-order correlation function $g_{ij}^{\left(1\right)}$ versus concurrence $C_{ij}$ for steerable (green area) and unsteerable states (blue area). Black border curves are drawn according to the corresponding analytical formulas.

Figure 5

Steering parameter $S_{ij}$ versus the first-order correlation function $g_{ij}^{\left(1\right)}$ for steerable states occupying the green area. Black border curves are plotted following the corresponding analytical formulas.

Figure 6

Degree of coherence $D_{ij}^2$ versus the first-order correlation function $g_{ij}^{\left(1\right)}$ for steerable (green area) and unsteerable (blue area) states. Black border curves are plotted using the corresponding analytical formulas.

Figure 7

Second-order correlation function $g_{ij}^{\left(2\right)}$ versus concurrence $C_{ij}$ for steerable (green area) and unsteerable (blue area) states. Black border curves are drawn according to the corresponding analytical formulas.

Figure 8

Steering parameter $S_{ij}$ versus second-order correlation function $g_{ij}^{\left(2\right)}$ for steerable set found in the green area. Black border curves are drawn following the corresponding analytical formulas.