Witnessing arbitrary bipartite entanglement in a measurement-device-independent way

Experimental detection of entanglement of an arbitrary state of a given bipartite system is crucial for exploring many areas of quantum information processing. But such a detection should be made in a device-independent way if the preparation process of the state is considered to be faithful, in order to avoid detection of a separable state as an entangled one. The recently developed scheme of detecting bipartite entanglement in a measurement-device-independent way [Phys. Rev. Lett. 110, 060405 (2013)] does require information about the state. Here, by using Auguisiak et al.'s universal entanglement witness scheme for two-qubit states [Phys. Rev. A 77, 030301 (2008)], we provide a universal entanglement detection scheme for two-qubit states in a measurement-device-independent way. We also provide a set of universal witness operators for detecting NPT-ness (negative under partial transpose) of two-qudit states in a measurement-device-independent way. We conjecture that no such universal entanglement witness scheme exists for PPT (positive under partial transpose) entangled states. We also analyze the robustness of some of the experimental schemes—for detecting entanglement in a measurement-device-independent way—under the influence of noise in the inputs (from the referee) as well as in the measurement operator.

Figure 1

Alice and Bob perform Bell state measurement on the states of the joint systems ${\mathcal{A}}_0\mathcal{A}$ and $\mathcal{B}{\mathcal{B}}_0$, respectively. Blue arrows are lossless quantum channels for sending input states to Alice and Bob. Red arrows correspond to the lossless classical channels for receiving the outputs. Green ellipses indicate joint Bell state measurement performed by the players.

Figure 2

Distribution of density matrix spaces acting on the Hilbert space ${\mathcal{H}}_{\mathcal{A}}\otimes {\mathcal{H}}_{\mathcal{B}}.$ [(I) + (II) + (III) + (IV)] $\equiv {\mathcal{D}}_{\mathcal{AB}},$ the set of all density matrices. (I) $\equiv {\mathcal{D}}_{\mathcal{AB}}-{\mathcal{P}}_{\mathcal{AB}}$ is the set of all NPT states, ${\mathcal{P}}_{\mathcal{AB}}$ is the set of all states having PPT. (II) $\equiv {\tilde{\mathcal{P}}}_{\mathcal{AB}},$ the set of all PPT entangled states, [(III) + (IV)] $\equiv {\mathcal{S}}_{\mathcal{AB}},$ is the set of all separable states, [(II) + (III)] $\equiv {\tilde{\mathcal{P}}}_{\mathcal{AB}}^{\prime }$ is the set of a convex combination of PPT entangled states. Region (III) clearly depicts the non-empty-ness of $({\tilde{\mathcal{P}}}_{\mathcal{AB}}^{\prime }\bigcap {\mathcal{S}}_{\mathcal{AB}}).$ This holds due to the fact that ${\tilde{\mathcal{P}}}_{\mathcal{AB}}$ is a nonconvex subset of the convex set ${\mathcal{P}}_{\mathcal{AB}}$.

Figure 3

Distribution of density matrix spaces acting on the Hilbert space ${\mathcal{H}}_{\mathcal{A}}^{\otimes n}\otimes {\mathcal{H}}_{\mathcal{B}}^{\otimes n}.$ [(I) $\bigcup$ (II) $\bigcup$ (III) $\bigcup$ (IV) $\bigcup$ (V)] $\equiv {\mathcal{D}}_{{\mathcal{A}}_n{\mathcal{B}}_n},$ the set of all density matrices. (II)$\equiv {\mathcal{S}}_{{\mathcal{A}}_n{\mathcal{B}}_n}$ be the set of all separable states, (III) $\equiv {\mathcal{S}}_{\mathcal{AB}}^{\prime }\left(n\right)$ is the set of convex combinations of all states like ${\sigma }_{\mathcal{AB}}^{\otimes n},$ where ${\sigma }_{\mathcal{AB}}\in {\mathcal{S}}_{\mathcal{AB}}$. (IV) $\equiv {\widetilde{{\mathcal{P}}^{\prime }}}_{\mathcal{AB}}^{\left(n\right)}$ be the convex combination of all the states like ${\rho }_{\mathcal{AB}}^{\otimes n},$ where ${\rho }_{\mathcal{AB}}\in {\tilde{\mathcal{P}}}_{\mathcal{AB}}.$ (V) contains convex combination of all the states like ${\rho }_{\mathcal{AB}}^{\otimes n},$ where ${\rho }_{\mathcal{AB}}\in {\mathcal{D}}_{\mathcal{AB}}-{\tilde{\mathcal{P}}}_{\mathcal{AB}},$ the set of all NPT states at the single copy level. We conjecture about the non-empty-ness of the intersection (III) $\bigcap$ (IV), and the empty-ness of (III) $\bigcap$ (V).

Figure 4

Bell-state projection $\frac{1}{2}(|HH\rangle +VV)(\langle HH|+\langle VV|)$ measurement via coincidence counts. Photon propagations are denoted by red arrows. PBS, polarizing beam splitter; HWP, half wave plate; D, single-photon detector.

Figure 5

Light ray passing through the half wave plate. The polarization axis, rotates by angle $2\theta$ with respect to the direction of propagation. $\theta$ is the angle between the polarization axis of the incident ray and the fast axis of the HWP. The fast axis of the HWP lies in the $xz$ plane.