Bound Entanglement from Randomized Measurements

If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.

Figure 1

Geometry of the three-qubit state space in terms of the second moments of random measurements or sector lengths. The total polytope is the set of all states, characterized by the inequalities $A_k\ge 0$, $A_1-A_2+A_3\le 1$, $A_2\le 3$, and $A_1+A_2\le 3(1+A_3)$ [36]. The fully separable states are contained in the blue polytope, obeying the additional constraint in Eq. (6) in observation 1. States that are biseparable for some partitions are contained in the union of the green and blue polytopes, characterized by the additional equation [Eq. (7)] from observation 2. In fact, for any point in the green and blue areas, there is a biseparable state with the corresponding second moments. The yellow area corresponds to the states violating the best previously known criterion for biseparable states, $A_3\le 3$ [33, 34, 35, 36]. Thus, the red area marks the improvement of the criterion in observation 2 compared with previous results.

Figure 2

Entanglement criterion based on second and fourth moments of randomized measurements for $3\otimes 3$ systems. Separable states are contained in the light-blue area, according to the discussion in the main text. Several bound entangled states (denoted by colored symbols) are outside, meaning that their entanglement can be detected with the methods developed in this Letter. For comparison, we also indicate a lower bound on the fourth moment for PPT states, obtained by numerical optimization, as well as a bound for general states. Further details, such as the forms of the states, are given in Appendix C. Information on the numerical methods is found in Appendix D [40].