Experimental Study of Nonclassical Teleportation Beyond Average Fidelity

Quantum teleportation establishes a correspondence between an entangled state shared by two separate parties that can communicate classically and the presence of a quantum channel connecting the two parties. The standard benchmark for quantum teleportation, based on the average fidelity between the input and output states, indicates that some entangled states do not lead to channels which can be certified to be quantum. It was recently shown that if one considers a finer-grained witness, then all entangled states can be certified to produce a nonclassical teleportation channel. Here we experimentally demonstrate a complete characterization of a new family of such witnesses, of the type proposed in Phys. Rev. Lett. 119, 110501 (2017) under different conditions of noise. We report nonclassical teleportation using quantum states that cannot achieve average fidelity of teleportation above the classical limit. We further use the violation of these witnesses to estimate the negativity of the shared state. Our results have fundamental implications in quantum information protocols and may also lead to new applications and quality certification of quantum technologies.

Figure 1

Teleportation scenario: Alice and Bob use a shared system in a quantum state ${\rho }^{\mathrm{A}\mathrm{B}}$. Alice measures her subsystem together with system $A_0$ that can be prepared in states $|{\psi }_x^{A_0}⟩$. Conditioned on Alice’s measurement outcome $a$ Bob’s system is left in the state ${\rho }_{a|{\psi }_x}^B$.

Figure 2

Experimental apparatus: Two photon pairs (1-2) and (3-4) are generated via parametric down conversion in two separated nonlinear crystals. Remote state preparation of photon 2 is performed by projective measurement on photon 1 via quarter wave plate, half wave plate (HWP), and polarizing beam splitter. Photon 4 is sent through an interferometer composed by a first calcite displacer (CD), an HWP with a tunable angle $\alpha$, a second CD, and an HWP at 45°. When $\alpha =45°$, photon 4 comes out in path 4a without modifications of the joint state ${\rho }^{\mathrm{A}\mathrm{B}}$. Conversely, when $\alpha =0°$, photon 4 comes out in path 4b projected onto the vertical polarization $|V⟩$, and the joint state ${\rho }^{\mathrm{A}\mathrm{B}}$ becomes $|VV⟩$. Intermediate angles between 0° and 45° allow us to tune the two-qubit state ${\rho }^{\mathrm{A}\mathrm{B}}$ between $|{\psi }^{-}⟩$ and $|VV⟩$. The two paths 4a and 4b are incoherently recombined at the polarizing beam splitter (PBS) of Bob’s measurement stage by means of an HWP at 45° on path 1b. To carry out a teleportation protocol, Alice performs a Bell state measurement (and transmits the outcome “a” to Bob). To test the nonclassicality of teleportation, Bob measures the operators $F_{a|{\psi }_x}$ which allows for testing a teleportation witness, and then shares the outcomes with Alice to check a teleportation witness. A motorized delay line ensures temporal indistinguishability in the beam splitter.

Figure 3

Experimental test of the non-classical teleportation witness. Theoretical predictions [straight lines in (a)] and estimated witness function [dashed lines in (b)] as a function of the free parameter $\theta$, obtained, respectively, from our noise parameter estimation and our experimental data points. Shaded areas correspond to one standard deviation of uncertainty upper and lower the dashed lines, due to Poissonian statistics. The bar legend on the right explains the color relationship with $\gamma$, whereas black points on the curves stand for the minimum theoretical (a) and experimental (b) values obtained for $\gamma =0$, $\gamma =0.14$, $\gamma =0.32$, $\gamma =0.51$, $\gamma =0.68$, $\gamma =0.85$.

Figure 4

Experimental application of teleportation witness. (a) Optimized witness as a function of the channel parameter $\gamma$, corresponding to the dots in Fig. 3. Green line represents the theoretical prediction of $W_{\min }(\gamma )$, accordingly to our noise model (for more information, see the Supplemental Material [43]), whereas blue points show the optimal experimental values of the witness. The black dashed line is the classical teleportation threshold (i.e., the maximal value of $W_{\min }(\gamma )$ necessary to certify nonclassical teleportation), whereas the red dashed line represents the minimal value of $\gamma$ needed to have an entangled teleportation channel, given our noise model. Inset: The optimal estimated negativity for each value of $\gamma$. Green line represents the theoretical prediction, whereas blue points show the experimental value. We stress that for each $\gamma$ a different $\theta$ is used to achieve the best witness violation and the best negativity estimate. (b) Average fidelity of teleportation estimation when considering the outcome $|{\psi }^{-}⟩$. The green line shows the theoretical estimation for the mean value curve ${\overline{F}}_{\mathrm{tel}}$ (given our noise model), whereas blue points depict the experimental values. The red dashed line represents the minimum value of $\gamma$ necessary to have entanglement in the teleportation channel, whereas the orange dashed line shows ${\overline{F}}_{\mathrm{cl}}$, the classical fidelity of teleportation bound. Error bars indicate one standard deviation of uncertainty, due to Poissonian statistics.