Methods to estimate entanglement in teleportation experiments

Quantum state teleportation is a protocol where a shared entangled state is used as a quantum channel to transmit quantum information between distinct locations. Here we consider the task of estimating entanglement of the underlying shared state in teleportation protocols. We show that the data accessible in a teleportation experiment allows us to put a lower bound on some entanglement measures, such as entanglement negativity and robustness. Furthermore, we show cases in which the lower bounds are tight. The introduced lower bounds can also be interpreted as quantifiers of the nonclassicality of a teleportation experiment. Thus, our findings provide a quantitative relation between teleportation and entanglement.

Figure 1

Teleportation-based lower bound on the negativity, and negativity of the state $p|{\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}|+(1-p)|01\rangle \langle 01|$. Alice performs a full Bell state measurement and uses a tomographically complete set of input states. In this case, the lower bound on the negativity is tight, so the curves for the negativity and the negativity lower bound coincide.

Figure 2

Teleportation robustnesses for the state $\rho =p|{\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}|+(1-p)|01\rangle \langle 01|$. The set of quantum inputs consists of all eigenstates of three Pauli operators. For this particular teleportation assemblage the generalized and classical teleportation robustness coincide. We can see that even when the average teleportation fidelity is smaller than $2/3$ the robustness quantifiers are larger than zero, demonstrating nonclassical teleportation.

Figure 3

Teleportation weight for different scenarios involving the state $p|{\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}|+(1-p)\frac{\mathbb{1}}{4}$: Alice either performs a full or partial Bell state measurement, and uses either a tomographically complete set of inputs (eigenstates of $X, Y$, and $Z$), or a tomographically incomplete set of measurements (eigenstates of $X$ and $Z$). The teleportation weight is insensitive to the choice of measurements for both sets of inputs, indicating that it is only the conclusive events (corresponding to POVM elements that are entangled) that count.

Figure 4

Dependence of the teleportation quantifiers introduced here on parameter $a$ the Horodecki state, using a tomographically complete set of input states (chosen randomly to produce this plot), and a partial Bell state measurement. For all values of $a\ne 0$ or 1, nonclassical teleportation is demonstrated. The curves for the classical and the generalized teleportation robustness coincide. Separability of the channel operators was relaxed to the requirement of having a 2-symmetric PPT extension [12].