Concurrence versus purity: Influence of local channels on Bell states of two qubits

We analyze how a maximally entangled state of two qubits (e.g., the singlet ${\psi }_s$) is affected by the action of local channels described by completely positive maps $\mathcal{E}$. We analyze the concurrence and the purity of states ${\varrho }_{\mathcal{E}}=\mathcal{E}\otimes \mathcal{I}\left[{\psi }_s\right]$. Using the concurrence-versus-purity phase diagram we characterize local channels $\mathcal{E}$ by their action on the singlet state ${\psi }_s$. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.

Figure 1

(Color online) This figure presents the space of unital channels ${\Phi }_{\mathcal{E}}$ parametrized by $\vec{\lambda }=({\lambda }_1,{\lambda }_2,{\lambda }_3)$ (with $-1⩽{\lambda }_j⩽1$). Up to unitary rotations the set of states ${\Omega }_{\mathcal{E}}$ that are obtained by the action of unital channels $\mathcal{E}$ on the singlet state (1.4) form a tetrahedron with vertices corresponding to maximally entangled states that form the Bell basis. The octahedron inside the tetrahedron represents the set of all separable states. The remaining four tetrahedra in each corner of the original tetrahedron represent sets of entangled states. The closer the point is to the vertex, the more entangled the corresponding state is. The planes parallel to faces of the separable region contain states with the same degree of entanglement (“isoconcurrence” planes). States with the same purity form spheres (“isopurity” spheres) centered at the center of the cube—i.e., at the point $\vec{\lambda }=(0,0,0)$.

Figure 2

(Color online) The concurrence-vs-purity phase diagram of two qubits. The region of states ${\Omega }_{\mathcal{E}}=\mathcal{E}\otimes \mathcal{I}\left[{\psi }_s\right]$ (region colored in dark red in the $C\text{−}P$ phase diagram) that are obtained from the maximally entangled two-qubit states by the action of unital channels $\mathcal{E}$ is bounded by two lines corresponding to $C_W$ and $C_D$ for a given value of the purity. The upper bound of the region of physically realizable states (shaded sector of the $C\text{−}P$ diagram) corresponds to maximally entangled mixed states (MEMS’s). We note that the MEMS’s cannot be achieved from Bell states by applying local channels on a single subsystem only. Both the concurrence and purity are presented in dimensionless units.

Figure 3

(Color online) Parametric plots of the time evolution of the concurrence and purity in the processes of decoherence, homogenization, and depolarization. The lines are parametrized in such a way that the initial moment of the time evolution $t=0$ corresponds to the point $C=P=1$ (upper right corner of the $C\text{−}P$ diagram). The decoherence line (2) represents the lower bound $\left(C_D\right)$ of the unital region; the depolarization line (1) forms the upper bound $\left(C_W\right)$ of the unital region. The homogenization (a nonunital process) is characterized by line (3), which is outside the unital region. In the present case the homogenization describes an exponential decay; i.e., the fixed point of the evolution is the state ∣0⟩. Both the concurrence and purity are presented in dimensionless units.

Figure 4

(Color online) We present evolutions of the concurrence and purity for the homogenization process when the ratio of the characteristic times $T_1$ and $T_2$ is constant and $T_1∕T_2=1∕2$ while the fixed point of the evolution varies from the total mixture (a unital process) to a pure state (nonunital process). In particular, when the fixed point of the homogenization is the total mixture, then the homogenization dynamics in the $C\text{−}P$ diagram is described by line (1). This line lies below the line $C_W$ with the final point $C=0$ having the smallest value of the purity. On the other hand, when the fixed point of the evolution is a pure state, then the corresponding $C\text{−}P$ line (2) ends in the point $C=0$ with the largest value of the purity $P=1∕2$. Both the concurrence and purity are presented in dimensionless units.

Figure 5

(Color online) The evolution of concurrence and purity for the homogenization with the fixed point equal to the total mixture. In this case the homogenization is a unital process. We vary the ratio $T_1∕T_2$ in the interval $[1∕2,\infty ]$. For $T_1=T_2$ the homogenization exactly covers line (2) of Werner states $\left(C_W\right)$. In the limit of $T_1∕T_2\to \infty$ this evolution coincides with the line of decoherence $\left(C_D\right)$. Both the concurrence and purity are presented in dimensionless units.

Figure 6

(Color online) The parametric plot of the concurrence and purity for the homogenization with the fixed point that corresponds to a pure state. That is, we consider an exponential decay into a pure state. We vary the $T_1∕T_2$ in the interval $[1∕2,\infty ]$. In this case, the evolution ends in the point $C=0$ and $P=1∕2$ irrespective of the particular value of the ratio $T_1∕T_2$. When $T_1∕T_2=1∕2$ the evolution is represented by the line which is “deep” in the nonunital region of the $C\text{−}P$ diagram. On the contrary, for $T_1∕T_2\to \infty$ the corresponding line coincides with the bound $C_D$ (the decoherence line). Both the concurrence and purity are presented in dimensionless units.