Reliable and robust entanglement witness

Entanglement, a critical resource for quantum information processing, needs to be witnessed in many practical scenarios. Theoretically, witnessing entanglement is by measuring a special Hermitian observable, called an entanglement witness (EW), which has non-negative expected outcomes for all separable states but can have negative expectations for certain entangled states. In practice, an EW implementation may suffer from two problems. The first one is reliability. Due to unreliable realization devices, a separable state could be falsely identified as an entangled one. The second problem relates to robustness. A witness may not be optimal for a target state and fail to identify its entanglement. To overcome the reliability problem, we employ a recently proposed measurement-device-independent entanglement witness scheme, in which the correctness of the conclusion is independent of the implemented measurement devices. In order to overcome the robustness problem, we optimize the EW to draw a better conclusion given certain experimental data. With the proposed EW scheme, where only data postprocessing needs to be modified compared to the original measurement-device-independent scheme, one can efficiently take advantage of the measurement results to maximally draw reliable conclusions.

Figure 1

Entanglement witness and the reliability problem.

Figure 2

Optimization of entanglement witnesses. (a) To get the optimal witness of an unknown entangled state $\rho$, one has to run over all possible witnesses. Intuitively, this is done by scanning over all witnesses that are tangent to the set of separable states. (b) The optimization can be efficiently done if a certain failure probability can be tolerated.

Figure 3

Bipartite nonlocal game with classical and quantum inputs. (a) Nonlocal game with classical inputs. Based on the classical inputs $x$ and $y$, Alice and Bob each perform a local measurement on the preshared entangled state ${\rho }_{AB}$ and get classical outputs $a$ and $b$, respectively. A linear combination of the probability distribution $p(a,b|x,y)$ defines a Bell inequality as shown in Eq. (3). (b) Nonlocal game with quantum inputs. The quantum inputs of Alice and Bob are, respectively, ${\omega }_x$ and ${\tau }_y$. It is shown [16] that any entangled quantum states can be witnessed with a certain nonlocal game with quantum inputs. Equivalently, if we consider that Alice and Bob each prepares an ancillary state and a third party Eve performs the measurement, this setup also corresponds to the case of an MDIEW.

Figure 4

Simulation results of the original and optimized MDIEW protocols. The to-be-witnessed state is the two-qubit Werner state defined in Eq. (24). Here, we consider that Alice projects onto $\left|{\mathrm{\Phi }}_{AA}^{+}\right.〉$ and Bob projects onto $\left|{\mathrm{\Phi }}_{BB}^{-}\right.〉$. In this case, the original MDIEW cannot detect entanglement, while the optimized MDIEW protocol detects all entangled Werner states.