Optimal two-qubit gates in recurrence protocols of entanglement purification

We propose and investigate a method to optimize recurrence entanglement purification protocols. The approach is based on a numerical search in the whole set of $\text{SU}\left(4\right)$ matrices with the aid of a quasi-Newton algorithm. Our method evaluates average concurrences where the probabilistic occurrence of mixed entangled states is also taken into account. We show for certain families of states that optimal protocols are not necessarily achieved by bilaterally applied controlled-not gates. As we discover several optimal solutions, the proposed method offers some flexibility in experimental implementations of entanglement purification protocols and interesting perspectives in quantum information processing.

Figure 1

Schematic representation of bipartite entanglement purification, where the entanglement purification protocol trades two entangled qubit pairs for a qubit pair with a higher degree of entanglement, which is quantified by the concurrence $\mathcal{C}$.

Figure 2

Optimized purification protocol for the family of states in Eq. (11). (a) Average cost function $\overline{f}$ with a uniform PDF as a function of $N$, the number of iterations. (b) Concurrence $\mathcal{C}$ as a function of $x$. Four curves are presented for different values of iterations $N$: 0 or the input concurrence (black dashed line), 1 (orange dotted line), 2 (blue dash-dotted line), and 3 (red solid line). (c) Success probabilities as a function of $x$ for different values of iterations $N$: 1 (orange dotted line), 2 (blue dash-dotted line), and 3 (red solid line). (d) Overall success probability after $N=1$ (orange dotted line), $N=2$ (blue dash-dotted line), and $N=3$ (red solid line) iterations as a function of $x$. The sample size has been set to $M=1000$.

Figure 3

Optimized purification protocol for Werner states [see Eq. (8)]. (a) Average cost function $\overline{f}$ with a uniform PDF as a function of $N$, the number of iterations. (b) Overall success probability after $N=1$ (orange dotted line), $N=2$ (blue dash-dotted line), and $N=3$ (red solid line) iterations as a function of $x$. The sample size has been set to $M=1000$.

Figure 4

Optimized purification protocol for the state in Eq. (17) for 1000 parameter samples. (a) Average cost function $\overline{f}$ with a uniform PDF as a function of $N$, the number of iterations. Crosses are the results of the cnot gate, whereas circles display the numerical optimization. They have been connected by lines to guide the eye. (b) Concurrence ${\mathcal{C}}^{\prime }$ as a function of $x$ and $y$ after one purification round. The cone-type surface is obtained with our approach. A less optimal surface with minima at $x=0$ and along the $y$ axis is the result of the purification protocol with the cnot gate. The sample size has been set to $M=1000$.

Figure 5

Overall success probability of the state in Eq. (17) after $N=3$ iterations as a function of $x$ and $y$. Both surfaces are similar in appearance. The one having maxima at $x=1$ belongs to the cnot-based protocol and the other one having maxima at $y=1$ is obtained via the optimized numerical protocol.

Figure 6

Optimized unitaries (a) and (b) $U_N\left({\mathit{\alpha }}^{*}\right)$, the Euler angle parametrization in Eq. (6), and (c) and (d) ${\tilde{U}}_N\left({\mathit{\alpha }}^{\prime *}\right)$, the cnot circuit given in Eq. (30), for $N=1$. The optimal protocols are obtained for the state given in Eq. (17) with $M=1000$ and the procedure discussed in Algorithm 1. For each unitary, (a) and (c) represent the absolute value of the matrix entries taken in the computational basis, whereas (b) and (d) represent the phase of the same entries divided by $2\pi$. The colormap describes values varying from 0 (white) to 1 (dark orange and dark blue).