Activating hidden teleportation power: Theory and experiment

Ideal quantum teleportation transfers an unknown quantum state intact from one party Alice to the other Bob via the use of a maximally entangled state and the communication of classical information. If Alice and Bob do not share entanglement, the maximal average fidelity between the state to be teleported and the state received, according to a classical measure-and-prepare scheme, is upper bounded by a function $f_{\mathrm{c}}$ that is inversely proportional to the Hilbert space dimension. In fact, even if they share entanglement, the so-called teleportation fidelity may still be less than the classical threshold $f_{\mathrm{c}}$. For two-qubit entangled states, conditioned on a successful local filtering, the teleportation fidelity can always be activated, i.e., boosted beyond $f_{\mathrm{c}}$. Here, for all dimensions larger than two, we show that the teleportation power hidden in a subset of entangled two-qudit Werner states can also be activated. In addition, we show that an entire family of two-qudit rank-deficient states violates the reduction criterion of separability, and thus their teleportation power is either above the classical threshold or can be activated. Using hybrid entanglement prepared in photon pairs, we also provide the first proof-of-principle experimental demonstration of the activation of teleportation power hidden in this latter family of qubit states. The connection between the possibility of activating hidden teleportation power with the closely-related problem of entanglement distillation is discussed.

Figure 1

The Werner state $W\left(v\right), v\in [0,1]$ is entangled iff $0\le v<\frac{1}{2}$. We show in this work that for $d>2, W\left(v\right)$ has HTP in the region $0\le v<\frac{d+1}{4d-2}$ (blue and red segment). As $d$ increases from 2 towards $\infty$, the threshold $v_{\mathrm{cr}}=\frac{d+1}{4d-2}$ moves from $\frac{1}{2}$ towards $\frac{1}{4}$, as symbolized by the (shrinking) red segment. The blue segment indicates the region where $W\left(v\right)$ always has HTP whenever $d>2$. For $d=2$, all entangled $W\left(v\right)$ are useful for teleportation.

Figure 2

Theoretical (dashed lines) and experimental (markers) results illustrating the teleportation power before and after filtering for qubit $\rho \left(q\right)$. (a) FEF $F_2$ [evaluated using Eq. (1)] (b) teleportation fidelity $f$. In each plot, the bottom (red) results correspond to the unfiltered states, i.e., $\rho \left(q\right)$ (theory) and ${\rho }_{1^{\prime }2}$ (experiment) in Fig. 3. The middle (blue) results are for the filtered state ${\rho }_{f,\kappa }\left(q\right)$ (theory) and ${\rho }_{1^{\prime \prime }2,\kappa }$ (experiment) in Fig. 3 while the top (turquoise) results are for the filtered state ${\rho }_{f,{\kappa }^{\prime }}\left(q\right)$ (theory) and ${\rho }_{1^{\prime \prime }2,{\kappa }^{\prime }}$ (experiment) in Fig. 3. HTP is shown if a red marker is below the solid line but the corresponding blue or turquoise marker is above the same line, which happens only when $q\in (0,\frac{1}{2}]$.

Figure 3

(a) Experimental scheme used in demonstrating the HTP of $\rho \left(q\right)$. QST may be performed at stage 1, $1^{\prime }$, and $1^{\prime \prime }$ to estimate the density matrix corresponding, respectively, to the initial entangled state ${\rho }_{12}$, the experimentally prepared state ${\rho }_{1^{\prime }2}$ [for $\rho \left(q\right)$], and the locally filtered state ${\rho }_{1^{\prime \prime }2,\kappa }$ or ${\rho }_{1^{\prime \prime }2,{\kappa }^{\prime }}$ [for ${\rho }_{f,\kappa }\left(q\right)$ and ${\rho }_{f,{\kappa }^{\prime }}\left(q\right)$]. (b) Experimental setup. The generated entangled photons are each coupled into a single-mode fiber and sent to Alice and Bob via optical fibers. The fiber-induced polarization drift is corrected by a polarization controller (PC), which is a half-wave plate (HWP) sandwiched by two quarter-wave plates (QWPs). The noisy channel generates $\rho \left(q\right)$ according to ${\theta }_1$, which is then filtered to boost its teleportation power. In our setup, the classical communication was carried out after the experiment, i.e., Bob applies the unitary correction on photon $2^{\prime }$ in a postselected manner to recover the teleported state. See text and Appendix pp3 for details. DM: dichroic mirror. NBF: narrow-band filter. BS: beam splitter.

Figure 4

(Top) FEF of the Werner state before filtering (blue markers) and after filtering (red solid line for the cost function and blue dash-dotted line for the filtered state). (Bottom) Change in FEF as a function of $v$, which depends linearly on the probability of success in filtering $p_{\text{W}}\left(v\right)$.

Figure 5

(Left) Comparison between the FEF of the filtered state ${\rho }_f$ obtained by employing different filtering schemes on $\rho \left(q\right)$ for $d=2$ and $d=3$. Included in the plots are the two-side filtering schemes introduced in Ref. [23] for $n=2,3,5,$ and 10 (see Appendix pp2-s4) as well as the single-side filtering schemes discussed in Appendix pp2-s3. (Center) Comparison of the corresponding success probabilities as a function of the parameter $q$. (Right) Comparison of the corresponding change in FEF as a function of the parameter $q$.

Figure 6

Zoom-in view of the (top) “Photon source” part and the (bottom) “Noisy channel” part of Fig. 2. The top setup aims to generate photon pairs maximally entangled in the polarization degree of freedom (DOF) whereas the bottom setup aims to generate, starting from photon pairs produced using the first setup, two-qubit mixed quantum states $\rho \left(q\right)$ [see Eq. (9)] encoded in the polarization DOF.

Figure 7

Experimental setup [zoom-in view of the “Filter” part of Fig. 2] that performs the filtering operation of Eq. (11) for the $d=2$ case.

Figure 8

Experimental setup [zoom-in view of the “state preparation” part of Fig. 2] that prepares the four pure states to be teleported $|\psi \rangle =\alpha |H_{1^{\prime \prime }}\rangle +\beta |V_{1^{\prime \prime }}\rangle$. We set HWP, respectively, at $0^{\circ }, {45}^{\circ }$ and $22.5^{\circ }$ to prepare $|H_{1^{\prime \prime }}\rangle , |V_{1^{\prime \prime }}\rangle$ and $|{+}_{1^{\prime \prime }}\rangle$, and QWP at ${45}^{\circ }$ to prepare $|R_{1^{\prime \prime }}\rangle$.

Figure 9

Experimental setup [zoom-in view of the “BSM” part of Fig. 2] to implement Bell-state measurement between the path and the polarization DOF of photon $1^{\prime \prime }$.

Figure 10

The real parts of ${\chi }_{\text{exp}}$ based on the shared states ${\rho }_{1^{\prime }2}$ and ${\rho }_{1^{\prime \prime }2,{\kappa }^{\prime }}$. The imaginary parts are omitted here as their experimentally determined values are tiny. The wire grids represent the theoretical values of the elements.

Figure 11

Experimentally determined success probability of filtering $p_{\kappa }\left(q\right)$ and $p_{{\kappa }^{\prime }}\left(q\right)$ for, respectively, filter $A_{\kappa }$ and $A_{{\kappa }^{\prime }}$.

Figure 12

The theoretical predictions of (a), the FEF $F_2$ and (b), the teleportation fidelity $f_2$ assuming an SPDC source described by $\varrho \left(\alpha \right)$. Dashed lines represent the results of $\varrho (\alpha =1)$ while solid lines represent that of $\varrho (\alpha =0.954)$.

Figure 13

Differences between theoretical predictions and experimental results assuming an SPDC source described by (a), $\varrho (\alpha =1)$ and (b) $\varrho (\alpha =0.954)$.