Two-qubit mixed states more entangled than pure states: Comparison of the relative entropy of entanglement for a given nonlocality

Amplitude damping changes entangled pure states into usually less-entangled mixed states. We show, however, that even local amplitude damping of one or two qubits can result in mixed states more entangled than pure states if one compares the relative entropy of entanglement (REE) for a given degree of the Bell–Clauser-Horne-Shimony-Holt inequality violation (referred to as nonlocality). By applying Monte Carlo simulations, we find the maximally entangled mixed states and show that they are likely to be optimal by checking the Karush-Kuhn-Tucker conditions, which generalize the method of Lagrange multipliers for this nonlinear optimization problem. We show that the REE for mixed states can exceed that of pure states if the nonlocality is in the range (0,0.82) and the maximal difference between these REEs is 0.4. A former comparison [Phys. Rev. A 78, 052308 (2008)] of the REE for a given negativity showed analogous property but the corresponding maximal difference in the REEs is one order smaller (i.e., 0.039) and the negativity range is (0,0.53) only. For appropriate comparison, we normalized the nonlocality measure to be equal to the standard entanglement measures, including the negativity, for arbitrary two-qubit pure states. We also analyze the influence of the phase-damping channel on the entanglement of the initially pure states. We show that the minimum of the REE for a given nonlocality can be achieved by this channel, contrary to the amplitude-damping channel.

Figure 1

Monte Carlo simulations of about ${10}^6$ two-qubit states $\rho$ for the relative entropy of entanglement $E_R\left(\rho \right)$ versus (a) concurrence $C\left(\rho \right)$, (b) negativity $N\left(\rho \right)$, and (c) nonlocality $B\left(\rho \right)$ corresponding to the Bell-CHSH inequality violation. Red regions correspond to mixed states more entangled than pure states in terms of the $E_R\left(\rho \right)$. Key: $\text{P}$, pure states; $\text{D}$, the Bell-diagonal states ($\alpha =0.5$); ${\text{A}}^{\prime }$ (${\text{A}}^{\prime \prime }$), the MEMS, which are the amplitude-damped states optimized for the case (b) [(c)], $\text{H}$, the Horodecki states, and the coordinates of points ${\rho }_n$ (where $n=1,...,6$) are given in Table .

Figure 2

Ranges of the relative entropy of entanglement for a given nonlocality for the two-qubit mixed states generated from pure states by the amplitude-damping channel, given by Eq. (25) (yellow and green areas) and the phase-damping channel, given by Eq. (32) (blue and green areas).

Figure 3

Ranges of the relative entropy of entanglement for given values of (a) negativity and (b) nonlocality. It is seen that the maximal difference in the REE between the optimal states and pure states is equal to (a) $\Delta =\Delta E_R\left(N^{\prime }\right)=0.0391$ at the negativity $N^{\prime }=0.154$ and (b) $\Delta =\Delta E_R\left(B^{\prime }\right)=0.404$ at the nonlocality $B^{\prime }=0$. Pure states become more entangled than the Horodecki states ${\rho }^{\left(\text{H}\right)}\left(p\right)={\rho }^{\left(\text{A}\right)}(\frac{1}{2},p)$ for (a) $N>N\left({\rho }_1\right)=0.377$ and (b) $B>B\left({\rho }_6\right)=0.586$, while pure states become optimal for (a) $N>N_0\equiv N\left({\rho }_2\right)=0.527$ and (b) $B>B_0\equiv B\left({\rho }_4\right)=B\left({\rho }_5\right)=0.817.$ The states ${\rho }_n$ (for $n=1,...,6)$ are specified in Table .