Convex Optimization over Classes of Multiparticle Entanglement

A well-known strategy to characterize multiparticle entanglement utilizes the notion of stochastic local operations and classical communication (SLOCC), but characterizing the resulting entanglement classes is difficult. Given a multiparticle quantum state, we first show that Gilbert’s algorithm can be adapted to prove separability or membership in a certain entanglement class. We then present two algorithms for convex optimization over SLOCC classes. The first algorithm uses a simple gradient approach, while the other one employs the accelerated projected-gradient method. For demonstration, the algorithms are applied to the likelihood-ratio test using experimental data on bound entanglement of a noisy four-photon Smolin state [Phys. Rev. Lett. 105, 130501 (2010)].

Figure 1

The normalized log-likelihood ratios $\lambda /N$ at each iterative step by DG and APG, respectively, for the four-qubit $W$ state with white noise: (a) Biseparability; (b) Full separability. The plateaus in the plots imply the process where the algorithms are searching for suitable step sizes for the next update. As shown, the APG algorithm usually returns more accurate solutions than DG does.

Figure 2

Likelihood-ratio test for bound entanglement for the experimental four-photon Smolin state [42]. The top curve labeled “Separable” indicates the $\lambda$ values obtained via maximizing over the fully separable states; while the bottom curve labeled “PPT” gives values obtained by maximizing over the PPT states. Hypothesis testing suggests that the state at noise level $q=0.51$ is both entangled and PPT and, thus, bound entangled.