Introduction

Large scale quantum networks require the capability of transferring quantum information between distant nodes1,2. Currently, photons represent the best candidate to carry out such a task: they are compatible with pre-existing optical fiber3 and free-space infrastructure4, they are extremely resilient to environmental decoherence and they feature a variety of easily manipulable degrees of freedom, for which quantum correlations have been demonstrated5,6,7,8. More specifically, the flexibility and noise resilience of the polarization degree of freedom makes it the current optimal solution to distribute quantum states and quantum correlations over long distances9,10. However, photon distribution over the globe will inevitably be affected by losses and noise, stressing the need of quantum relays and quantum repeaters, i.e., “devices" that mitigate the impact of losses by transferring quantum information between distant nodes of a quantum network11. These architectures are based on fundamental quantum information protocols such as quantum teleportation and entanglement swapping12,13,14, whose successful realization depends on the degree of indistinguishability and entanglement of the involved particles. Thus, the all-photonic implementation of such quantum repeater primitives15, necessary to efficiently distribute flying qubits, sets stringent requirements on the photon sources to be employed in a quantum network. These include high production rate, wavelength tunability, the need of high photon indistinguishability, and near-unity degree of entanglement, to mention a few. In this perspective, semiconductor quantum dots (QDs) have been under the spotlight for two decades, mainly due to the lack of trade-off between brightness and single photon purity (or degree of entanglement) that is typically plaguing Poissonian sources of light16. More specifically, QDs have shown on-demand generation of single and entangled photons with high indistinguishability17,18, near-unity degree of entanglement19, ultra-low multi-photon emission20 and, most importantly, high brightness21,22,23. For this reason, and for their compatibility with quantum memories24,25,26, semiconductor QDs have been acknowledged as a promising candidate to become the ideal quantum light source for quantum networking16,27,28. Following this direction, QDs have been employed for successful demonstrations of photonic quantum teleportation and entanglement swapping using a single emitter (including emitters in the telecommunication band)29,30,31,32,33,34,35,36, even using non-ideal devices37. Moreover, two distinct QDs have been interfaced to achieve quantum teleportation from a photonic qubit to a spin qubit38, and distinct QD spin qubits have been made entangled through a heralded protocol39. Nonetheless, the use of independent QDs to implement all-photonic quantum teleportation in a quantum relay scenario demands state-of-the-art sources of both single and entangled photons to be interfaced via two-photon interference. This, in turn, requires the indistinguishability of photons generated by independent QDs, which inevitably feature dissimilar optical properties. Although pioneering steps along this direction have been moved18,38,40,41,42,43, the implementation of quantum relays interfacing distinct QDs has remained elusive because of the complexity of realizing a device that meets all the requirements set by all-photonic teleportation at the same time. Indeed, achieving efficient generation of single and entangled photons from QDs requires the use of sophisticated nanophotonic devices44,45,46 that exploit light-matter interaction to boost the flux of QD photons. Moreover, since QDs are to deliver highly entangled photons, their electronic structure has to be controlled with high accuracy47,48,49.

Here, we overcome these limitations by combining a variety of cutting-edge techniques developed in the context of QD quantum photonics. We fabricate state-of-the-art semiconductor QDs deterministically embedded in Circular Bragg Resonators (CBRs)23,50, designed to enhance the collection efficiency of both single and entangled photon pairs51. Then, we integrate them onto piezoelectric actuators to enable energy tunability and FSS erasing51. We use external perturbations, specifically multi-axial strain and magnetic fields, to reshape the electronic structure of dissimilar QDs so as to make them suitable for quantum teleportation. Finally, we take advantage of ultrafast nanowire single photon detectors to mitigate the effects of residual photon distinghuishability. These tools enable the demonstration of all-photonic quantum state teleportation with separate and originally dissimilar QDs. In addition, we manage to implement this protocol in an urban communication scenario, distributing teleported photons in a hybrid quantum network (composed of both fiber and free-space links) built over the campus of Sapienza University in Rome52,53.

Results

The quantum teleportation scheme and setup is showcased in Fig. 1. Two independent and remote QDs are driven under two-photon resonant excitation (TPE)54 to generate nearly on-demand photon pairs through the biexciton-exciton radiative cascade55. We denote the photon emitted by the biexciton recombination as XX, and the one after the exciton recombination as X (Fig. 1a). The input state \(\left\vert \phi \right\rangle\) is prepared in the polarization degree of freedom of the X1 photons, generated by QD1 in a first laboratory, acting as Node 1 (see Fig. 1b). This photon is sent to a second laboratory, Node 2, where QD2 generates the XX2-X2 entangled photon pair. There, the quantum interference between X1 and X2 photons enables the teleportation of the \(\left\vert \phi \right\rangle\) target state onto the polarization state of XX2, which is entangled to X2. We first benchmark the teleportation between the two separated nodes 1 and 2 in the Marconi building of the Sapienza Physics Department, realizing a demonstration of all-photonic quantum teleportation between photons emitted by distinct QDs. Then, the XX2 photon is sent via a 270 m free-space optical link to a third laboratory in a different building of the campus (named after Fermi), acting as Node 3. The link is equipped with specifically designed synchronization devices, and stabilization systems to compensate for atmospheric turbulence. Overall, we achieve the successful teleportation of a polarization qubit using dissimilar QDs in an urban quantum network comprising both fiber and free-space links. The key to the success of the experiments lies in the use of state-of-the-art photonic cavities in combination with quantum-engineering techniques (as sketched in Fig. 1b) to make QD1 and QD2 suitable for quantum teleportation, as explained in the next section.

Fig. 1: Sketch of the urban quantum teleportation network.
Fig. 1: Sketch of the urban quantum teleportation network.
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a Scheme of the quantum teleportation protocol as implemented with two QDs over a network including a free-space link. b Depiction of the teleportation experimental realization over the Sapienza University campus. In Node 1, highlighted in teal and placed in the Marconi building, the X1 photon from QD1 is selected and prepared in a well-defined polarization state. Node 1 is connected with a 15 m optical fiber link to Node 2, highlighted in red, where QD2 emits a polarization-entangled biexciton-exciton photon pair, XX2 and X2. Spatial displacement between Node 1 and Node 2 is accentuated on the map for clarity. A BSM is performed in Node 2, while XX2 is sent to Node 3, highlighted in purple and placed in the Fermi building, through a 270 m free-space optical link. The strain and the magnetic field harnessed to optimize the X2-XX2 degree of entanglement and the X1 emission wavelength, are featured in orange and red, respectively. The experimental setup accommodates Volume-Bragg Gratings (VBGs), polarizing beam-splitters (PBSs), waveplates (WPs), linear polarizers (LPs), avalanche single photon detectors (APDs), piezoelectric mirrors (PMs), fast steering mirrors (FSMs), photodiodes (PDs) and client-server PCs for the optimization of the free-space optical link. Inset: map depicting the hybrid quantum network laid over the Sapienza campus, where the Marconi and Fermi buildings are framed in orange. Map data from ⓒ Mapbox, ⓒ OpenStreetMap.

Engineering quantum light sources

For a successful all-photonic quantum teleportation with two remote QDs, three fundamental ingredients are required: (i) a bright source of highly entangled photon pairs; (ii) a bright source of single photons with tunable energy; (iii) the capability to perform a successful Bell state measurement (BSM). We accomplish (i) by using state-of-the-art epitaxial GaAs quantum dots fabricated by the droplet etching method56. They can act as bright and reliable sources of single and highly polarization-entangled photon pairs, especially when embedded in suitable photonic structures23. In our case, we employ GaAs QDs coupled to CBRs, also known as bullseye cavities, in turn integrated onto micromachined piezoelectric actuators made of \({[{{{\rm{Pb}}}}({{{{\rm{Mg}}}}}_{\frac{1}{3}}{{{{\rm{Nb}}}}}_{\frac{2}{3}}){{{{\rm{O}}}}}_{3}]}_{0.72}\) -\({[{{{{\rm{PbTiO}}}}}_{3}]}_{0.28}\) (PMN-PT)51, as depicted in Fig. 2a. The CBR cavity engineers light-matter interaction to boost the flux of both X and XX photons generated during the radiative cascade. Single-photon (photon-pair) extraction efficiencies as high as 85% (65%) have been demonstrated with this type of cavity23. The micro-machined PMN-PT actuators are instead used to control the QD electronic structure via the application of independent stress fields applied along different crystal directions. These are needed to cancel out the energy splitting between the two X states—the fine structure splitting (FSS)—that is usually induced by asymmetries in the QD confining potential (see Fig. 2b). Erasing this “which-path" information in the radiative cascade is fundamental to obtain near-unity degree of entanglement without temporal post-selection47,49,51. In our experiment we employ PMN-PT in combination with CBR cavities: by varying the voltages applied to the piezoelectric actuators, we reach a minimum value of FSS = 0.3 ± 0.2 μeV for QD2, as reported in Fig. 2b. By performing quantum state tomography on the generated photon pairs we estimate a fidelity of F = 0.94 ± 0.01 to the \(\left\vert {\phi }^{+}\right\rangle\) Bell state without resorting to any time-filtering procedure (see Fig. 2c), close to the best obtained with QDs in optical microcavities51. The source of the teleported qubit (ii) is another GaAs QD embedded in a CBR cavity, denoted as QD1 (Fig. 2d), driven under TPE and generating X1 and XX1 photons. The latter are not relevant for the teleportation protocol discussed here and they will not be considered further. The X1 photon features a wavelength similar to X2 - but not identical, as we report in detail in the “Methods” section. This is a clear indication that the two independent QDs feature slightly different size, shape or alloy intermixing57. In order to achieve spectral indistinguishability between X1 and X2, we employ a magnetic field that shifts the emission by the combination of Zeeman splitting and diamagnetic shift. More specifically, we select one of the Zeeman-split X1 transitions and use the magnetic field to tune its energy in resonance with the X2 photon, as reported in in Fig. 2e. This step is crucial to achieve (iii). In its most efficient implementation with linear optics, a BSM exploits two photons that interfere on a symmetric beam splitter (BS) coming from two distinct input ports. Then, they undergo a polarization projective measurement at the two output ports, allowing to sample two of the four Bell states, \(\left\vert {\psi }^{+}\right\rangle=\frac{\left\vert HV\right\rangle+\left\vert VH\right\rangle }{\sqrt{2}}\) and \(\left\vert {\psi }^{-}\right\rangle=\frac{\left\vert HV\right\rangle -\left\vert VH\right\rangle }{\sqrt{2}}\). To perform a successful BSM, the impinging photons have to be indistinguishable in all degrees of freedom. While the magnetic field tuning allows us to achieve energetic resonance, X1 and X2 feature also different lifetimes and linewidths, a consequence of a slightly different Purcell acceleration as well as different interaction with the solid-state environment. This can be indirectly observed in the Hong-Ou-Mandel (HOM) interference measurement of X1-X2 photon pairs (realized by performing cross- and co-polarized interference, see the inset of Fig. 2f), exhibiting the typical “volcano shape”58,59. Collecting all the photons coming out from the BS results in a limited HOM visibility, an evidence which can be fully explained by the lifetimes and linewidths of the X1 and X2 photons (see the SI). However, the level of indistinghuishability can be improved by post-selecting two-photon interference events via ultrafast superconductive nanowire single photon detectors (SNSPDs) with a temporal resolution  <15 ps (FWHM) and a time-to-digital converter with 19 ps time resolution (FWHM), resulting in a total setup time resolution of around 24 ps (FWHM) for coincidences. By narrowing down the considered coincidence detection window Δτ, we can achieve HOM visibilities as high as 60% (see Fig. 2f), a value limited by the SNSPDs time resolution. This technique clearly comes at the cost of the number of useful three-fold coincidences exploitable for the teleportation protocol, but not as much as one would intuitively expect. In fact, we point out that the teleportation fidelity that can be achieved in an experiment depend on both the degree of entanglement and photon-indistinghuishability, and overcoming the classical limit is possible even if the available quantum resources are unbalanced, i.e., even with imperfect and dissimilar quantum dots. We have explored this possibility by using a model that computes the maximum theoretically achievable teleportation fidelity FT as a function of the degree of entanglement in the X2-XX2 photon pair (which is influenced by FSS and quantified by the fidelity of the photon pair state to the \(\left\vert {\phi }^{+}\right\rangle\) Bell state) and the indistinguishability of the X1-X2 photons (measured by their HOM visibility). FT is computed considering the fidelity \(F=\left\langle \phi \right\vert \rho {\prime} \left\vert \phi \right\rangle\) of the teleported state \(\rho {\prime}\) to the expected output state \(\left\vert \phi \right\rangle\) for a perfect teleportation, and averaging over the set of possible input states. The details of the model are reported in the SI while the achieved results are illustrated in Fig. 3a. For the characteristics of our emitters and a BSM coincidence time window of 30 ps, the predicted teleportation fidelity is FT = 0.827 ± 0.006, represented by a white star in Fig. 3a. Most importantly, the same figure highlights that the classical limit can be overcome even with a poor HOM visibility (below 20%) in combination with a high level of Bell state fidelity (above 90%). This suggests that successful teleportation with our independent QDs can be achieved by a loose temporal post-selection, at moderate expenses of teleportation rates. This is exactly what motivated our field demonstration of quantum teleportation, as presented in the next section.

Fig. 2: Quantum resources engineering.
Fig. 2: Quantum resources engineering.
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a Schematic representation of the CBR QD2 integrated onto PMN-PT to enable the application of mechanical stress. b The strain-tuning curve along one axis showing our capability to erase the FSS. The solid line is a fit from the model equations reported in ref. 48. c Experimental two-photon density matrix (real part) measured for FSS = 0.3 ± 0.2 μeV, yielding a fidelity \({F}_{\left\vert {\phi }^{+}\right\rangle }=0.94\pm 0.01\) and a concurrence C = 0.89 ± 0.01. d Schematic representation of QD1 inserted in a magneto-cryostat. e Diamagnetic shift and Zeeman splitting of the emission spectrum of QD1: thanks to a fine tuning of the B intensity, we achieve spectral overlap between the two X photons applying a magnetic field of B = 0.9 T. f HOM visibility between photons from remote QDs as a function of the coincidence time window Δτ, setting the magnetic field at B = 0.9 T: post-selection on the HOM events allows us to reach a maximum HOM visibility of VHOM = 0.598 ± 0.025 for a 20 ps time window. It is worth noting that the overall time resolution of the setup adds up to 24 ps. Inset: time histogram of coincidences in the HOM experiment for co-polarized photon pairs (red) and cross-polarized photon pairs (blue) at the optimal B = 0.9 T.

Fig. 3: Quantum state teleportation.
Fig. 3: Quantum state teleportation.
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a Theoretical teleportation fidelity as a function of the quantum resources available for the protocol, computed for the case of a 50% BSM. The white curve highlights the maximum achievable fidelity with classical resources, i.e., for a protocol only relying on classical correlations. The white star points out the expected teleportation fidelity as estimated with our theoretical model. In particular, our Bell state fidelity is 0.94 ± 0.01 and the HOM visibility for 30 ps coincidence window is VHOM = 0.57 ± 0.02, yielding an average teleportation fidelity FT = 0.827 ± 0.006. b Experimental process matrices \({\chi }_{{\psi }^{+/-}}\) for the teleportation protocol, as defined in ref. 62, reported both for the projection on \(\left\vert {\psi }^{-}\right\rangle\) (top) and \(\left\vert {\psi }^{+}\right\rangle\) (bottom), and in comparison with the corresponding ideal process matrix (dashed). The matrices are represented in terms of the quantum operations I (the identity), X = σX, Y = −iσY and Z = σZ, where {σX/Y/Z} are the Pauli matrices.

Teleportation protocol over a quantum network

The protocol we used builds upon the scheme employed in one of the seminal demonstration of quantum teleportation60. We perform the experiment by preparing X1 photons in the target polarization states \(\left\vert \phi \right\rangle\) in Node 1 and sending them through a 15 m single-mode fiber to Node 2, where they interfere with the X2 photons through the BSM. At the same time, the XX2 photons undergo a polarization measurement in Node 2, where we collect the three-fold coincidences among their detection and X1-X2 coincidence events in the chosen projection of the BSM. After the X1-X2 interference, the joint polarization state of X1-X2-XX2 can be written as61:

$$\begin{array}{rcl}&&\left\vert \Psi \right\rangle=\frac{1}{2}\left({\left\vert {\phi }^{+}\right\rangle }_{{X}_{1},{X}_{2}}\hat{I}+{\left\vert {\psi }^{+}\right\rangle }_{{X}_{1},{X}_{2}}{\hat{\sigma }}_{X}+\right.\\ &&-\left.i{\left\vert {\psi }^{-}\right\rangle }_{{X}_{1},{X}_{2}}{\hat{\sigma }}_{Y}+{\left\vert {\phi }^{-}\right\rangle }_{{X}_{1},{X}_{2}}{\hat{\sigma }}_{Z}\right){\left\vert \phi \right\rangle }_{X{X}_{2}}\end{array}$$
(1)

where \(\hat{I}\) is the identity operator and \({\hat{\sigma }}_{X/Y/Z}\) are the Pauli operators. The BSM projection on one of the two Bell states \(\left\vert {\psi }^{-}\right\rangle\) and \(\left\vert {\psi }^{+}\right\rangle\) corresponds to the application of a different operation to the teleported qubit. The entire teleportation protocol from X1 to XX2 can be fully characterized by performing a quantum process tomography (QPT)62, which produces the process matrix. By collecting three-fold coincidences for different input states, we can reconstruct the teleportation process matrix for different values of BSM coincidence time window Δτ. Those are reported for both projections and Δτ = 30 ps in Fig. 3b. From these matrices, we estimate the overall teleportation fidelity which is found to be \({F}_{T}^{-}=0.82\pm 0.01\) for the \(\left\vert {\psi }^{-}\right\rangle\) projection and \({F}_{T}^{+}=0.77\pm 0.01\) for the \(\left\vert {\psi }^{+}\right\rangle\) one.

Both values are well above the classical limit of \(\frac{2}{3}\)63 by more than 10 standard deviations. Yet, it is worth noting that in the \(\left\vert {\psi }^{+}\right\rangle\) projection case we do not reach the theoretical limit of FT = 0.827 ± 0.006 estimated for our QDs. This is due to hardly avoidable setup imperfections and other sources of undesired background signal. In particular, we estimate that a slightly imperfect polarization compensation procedure may have decreased the teleportation fidelity for the \(\left\vert {\psi }^{+}\right\rangle\) projection of around 2%. We provide further discussion of these points in the SI. As mentioned above, our scheme uses temporal post-selection and we investigate how the teleportation fidelity changes as a function of Δτ. The red plot in Fig. 4 displays the teleportation results averaged over the \(\left\vert {\psi }^{-}\right\rangle\) and \(\left\vert {\psi }^{+}\right\rangle\) projections applying different coincidence time windows to the BSM up to Δτ = 30 ps. It can be clearly seen that the classical limit can be overcome by a relatively moderate temporal post-selection, with a Δτ of about 200 ps. This value allows to retain more than 60% of the total three-fold coincidence events, corresponding to  ≈0.5 Hz rate of teleportation events. The best teleportation fidelity is obtained for the Δτ = 30 ps window, which allows to retain about 11% of the total teleportation events, corresponding to a  ≈0.1 Hz teleportation rate. These results stimulated us to explore the possibility to implement our teleportation scheme in an urban communication scenario, specifically using the Sapienza free-space link52,53: a 270 m free-space channel laid over the university campus in the city center of Rome. Rather than being locally analyzed, XX2 photons are sent through the quantum channel and measured in Node 3, in the Fermi building of Sapienza. This operation presents many challenges, including the presence of substantial losses (almost 90% of the signal is lost through the channel), the need to synchronize photon detection, as well as the presence of atmospheric turbulence which makes the coupling of the signal to single-mode fibers unstable. The synchronization and instability issues are alleviated by the employment of GPS disciplined oscillators and a stabilization system based on both slow and fast steering mirrors, respectively52. The results are shown in Fig. 4, where we report the teleportation fidelities we obtain for a given Δτ, in comparison with those achieved in the fiber-only network (see SI for details). Given the losses present in the free-space link, the uncertainty on the fidelity values is considerably larger with respect to the fiber-only network. However, for Δτ = 30 ps, we obtain an average teleportation fidelity of FT = 0.80 ± 0.04, more than 3σ above the classical limit. The slightly lower teleportation fidelities that can be observed in the hybrid network are related to the presence of the free-space channel, which favors the teleportation of some polarization states in comparison with others, due to imperfect polarization compensation procedures. We discuss these points in more detail in the SI.

Fig. 4: Temporal post-selection of quantum teleportation events.
Fig. 4: Temporal post-selection of quantum teleportation events.
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Teleportation fidelity averaged over the \(\left\vert {\psi }^{-}\right\rangle\) and \(\left\vert {\psi }^{+}\right\rangle\) projections as a function of the BSM coincidence time window Δτ. We compare the results obtained for the fiber network case (red points) with the hybrid scenario (blue points), where teleported photons are sent over a 270 m free-space channel. On the top axis, the fraction of retained events over the total (corresponding to a coincidence time window of 400 ps) is reported as a function of Δτ.

Discussion

In this work, we demonstrate all-photonic quantum state teleportation using photons generated by two independent and dissimilar quantum emitters, specifically epitaxial QDs. Although our results are still not comparable to those achievable with single photons from SPDC and weak coherent pulses, in terms of joint fidelity and success rate64,65,66,67, we managed to exploit solid-state deterministic emitters to realize a quantum relay, while also successfully implementing the protocol in an urban quantum network that presents several challenges, including photon losses, atmospheric turbulence, and the need of synchronizing different nodes. We achieve this results by overcoming some of the roadblocks that have long prevented the implementation of a fully photonic quantum teleportation with quantum emitters. The key to this accomplishment was the use of strain- and magnetic-field-tunable QDs embedded in nanophotonic cavities and the exploitation of state-of-the-art techniques for single photon detection and event analysis. This allowed us to surpass the classical threshold for state teleportation while retaining more than 60% of the total events rate. Moreover, we recorded an average fidelity of up to 79 ± 1% for about 11% of the total teleportation events. Our achievement opens an avenue for the development of practical quantum networks and will stimulate additional research endeavors. Furthermore, a contemporary work68 also realized all-photonic quantum teleportation, using frequency conversion to achieve both the necessary photon indistinguishability and efficient fiber communication28,41, although at the cost of comparably high noise levels. These results offer a perspective of scalability over pre-existing infrastructure that further motivate research in the direction of QD-based networking. This considered, some issues still have to be addressed to reach application-ready performances, but the path is already set for most of them. For instance, several approaches can be put in place to reach near-unity teleportation fidelities at the maximum possible rates. First, the implementation of electric fields to suppress charge noise69 will allow blinking suppression70 and the improvement of the indistinguishably of photons generated by remote QDs. This possibility has been demonstrated for single photon sources18, but additional efforts are still needed to improve the brightness of the devices and extend the method to entangled photon sources. Recent results71 also suggest that differential Purcell enhancement can be implemented to mitigate photon distighuishability arising from time-correlation between XX and X photons59. Regarding the free-space link, the integration of adaptive optics for compensation of high-order aberrations due to atmospheric turbulence as well as the use of efficient synchronization strategies may allow reducing photon losses to the minimum possible. We can also envision a combination with a frequency conversion approach to enable deployed fiber communication, although a significant improvement in noise levels and efficiency would be required to grant performances comparable to in-lab results. The realization of a QD-based quantum relay also represents a step towards the demonstration of entanglement swapping with remote QDs, and then, given the well-established compatibility between QD emitters and efficient quantum memories, the realization of a quantum repeater based on QD emitters. The requirements on the light sources are more stringent in this case, but our results provide a strong indication that the realization of the protocol is within sight. More in general, our demonstration of quantum teleportation in an urban communication scenario highlights that the implementation of a QD-based quantum network for quantum information processing is a likely perspective in the foreseeable future.

Methods

Source characterization

In Fig. 5, we report the normalized spectra of both QDs under resonant two-photon excitation. Even without the application of an external magnetic field, we already observe a partial overlap between the exciton emission lines (see inset of Fig. 5). We measure a multiphoton emission probability for the three transitions of \({g}_{{X}_{1}}^{(2)}(0)=0.013\pm 0.001\), \({g}_{{X}_{2}}^{(2)}(0)=0.016\pm 0.001\) and \({g}_{X{X}_{2}}^{(2)}(0)=0.039\pm 0.001\). Lifetimes of 100 ± 10 ps, 67 ± 5 ps, and 259 ± 30 ps are measured for X2, XX2, and X1 respectively. Comparison with an unprocessed sample35 shows a Purcell factor \({F}_{P}^{{X}_{2}}\approx 2.5\) and \({F}_{P}^{X{X}_{2}}\approx 1.8\), while X1 emission rate does not show Purcell enhancement. This unbalance in the excited state lifetimes results in a lower wavepackets overlap between the X photons arriving at the BSM setup, which lowers the degree of indistinguishability72. A detailed computation of the expected overlap is provided in the SI. The low Purcell acceleration we record shows that, for our case and in general for CBR cavities employed for photon pair extraction51, the main advantage coming from coupling the emitter to a nanostructure is represented by the enhanced extraction efficiency. QD2 exhibits the optimal extraction efficiency reported in ref. 51 (\({\eta }_{ext}^{pair}=0.65\) and \({\eta }_{ext}^{single}\approx 0.85\), much higher than the values achievable with unprocessed emitters35), while QD1 shows a single-photon extraction efficiency a factor 6 lower. Therefore, the brightness of QD1 limits the protocol performance, yielding an estimated rate of teleportation three-fold events of  ≈7.8 × 10−6 per prepared photon (at the best time filtering of Δτ = 30 ps on the BSM). On the other hand, for the same Δτ = 30 ps coincidence window, the BSM events rate per prepared photon is  ≈2 × 10−4. In the SI we provide a detailed description of the experimental setup as well as the characterization setup employed to perform the analyses we report above.

Fig. 5: Optical properties of the single-photon and entangled photons sources.
Fig. 5: Optical properties of the single-photon and entangled photons sources.
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QD1 and QD2 emission spectra for TPE excitation. The emission wavelength of the two excitons X1 and X2, even in absence of external tuning, partially overlaps, as shown in the inset.

Bell state measurement

The quantum teleportation protocol relies on the successful projection of a two-photon polarization state on a Bell state. However, the perfect discrimination between the four Bell states \(\{\left\vert {\phi }^{+}\right\rangle,\left\vert {\phi }^{-}\right\rangle,\left\vert {\psi }^{+}\right\rangle,\left\vert {\psi }^{-}\right\rangle \}\) is not allowed with local operations and classical communication73, unless ancillary degrees of freedom or systems are employed74. The limit for the efficiency of a BSM implemented by linear optics and no ancillary qubits has been set at 50%75. The experimental realization of such projective measurements consists of a HOM interference between the two photons, followed by a polarization-sensitive detection (experimental details are reported in the SI). After HOM interference, photon polarization is measured in the \(\{\left\vert H\right\rangle,\left\vert V\right\rangle \}\) basis at both outputs of the interference BS. Thanks to this measurement stage it is possible to perfectly discriminate between \(\left\vert {\psi }^{-}\right\rangle\) and \(\left\vert {\psi }^{+}\right\rangle\) by collecting coincidences between orthogonally polarized outputs36,37, while all the events corresponding to \(\left\vert {\phi }^{+}\right\rangle=\frac{\left\vert HH\right\rangle+\left\vert VV\right\rangle }{\sqrt{2}}\) and \(\left\vert {\phi }^{-}\right\rangle=\frac{\left\vert HH\right\rangle -\left\vert VV\right\rangle }{\sqrt{2}}\) produce no conclusive outcome.

Quantum process tomography

To characterize the performances of our protocol, we have performed a complete Quantum Process Tomography (QPT)76 of our quantum teleportation, depicting it as a quantum channel applied to the input qubit, conditioned on the BSM result. A quantum channel can be represented as a map \({{{\mathcal{E}}}}\) acting on an input state ρ as follows:

$${{{\mathcal{E}}}}(\rho )={\sum}_{i}{E}_{i}\rho {E}_{i}^{{{\dagger}} }$$
(2)

where the operators Ei satisfy the condition \({\sum }_{i}{E}_{i}^{{{\dagger}} }{E}_{i}={\mathbb{I}}\). The Ei operators must be expressed in function of a fixed set of operators \({\tilde{E}}_{k}\), which form a basis for the set of operators on the state space: \({E}_{i}={\sum }_{k}{e}_{ik}{\tilde{E}}_{k}\). Using this basis, the quantum channel can be expressed as:

$${{{\mathcal{E}}}}(\rho )={\sum}_{mn}{\chi }_{mn}{\tilde{E}}_{m}\rho {\tilde{E}}_{n}^{{{\dagger}} }$$
(3)

where \({\chi }_{mn}={\sum }_{i}{e}_{im}{e}_{in}^{*}\) is a positive hermitian matrix defined as the χ-matrix representation of the process \({{{\mathcal{E}}}}\). For the specific case of a single qubit, it is possible to show that choosing the Pauli matrices as the basis operators (\({\tilde{E}}_{0}=I\), \({\tilde{E}}_{1}={\sigma }_{x}\), \({\tilde{E}}_{2}=-i{\sigma }_{y}\), \({\tilde{E}}_{3}={\sigma }_{z}\)), the χ matrix can be completely determined by measuring the density matrices of the four polarization states \(\left\vert H\right\rangle\), \(\left\vert V\right\rangle\), \(\left\vert D\right\rangle=(\left\vert H\right\rangle+\left\vert V\right\rangle)/\sqrt 2\) and \(\left\vert R\right\rangle=(\left\vert H\right\rangle+i\left\vert V\right\rangle )/\sqrt{2}\) after the action of \({{{\mathcal{E}}}}\)62. In the SI we report the single-qubit density matrices employed to reconstruct the process matrices in Fig. 3. In order to ensure that our analysis yields physically meaningful results we adopt a Maximum Likelihood Estimation (MLE) algorithm developed for QPT77. It is now possible to compute the \(\rho^{\prime}={{{\mathcal{E}}}}(\rho )\) for the experimental quantum teleportation process and, comparing \(\rho^{\prime}\) with the expected outcome \(\left\vert \phi \right\rangle\), by computing the fidelity \(F=\left\langle \phi \right\vert \rho^{\prime} \left\vert \phi \right\rangle\), we can estimate the efficiency of the teleportation. In particular, we average over six possible polarization states \(\{\left\vert H\right\rangle,\left\vert V\right\rangle,\left\vert D\right\rangle,\left\vert A\right\rangle=(\left\vert H\right\rangle-\left\vert V\right\rangle)/\sqrt 2,\left\vert R\right\rangle,\left\vert L\right\rangle=(\left\vert H\right\rangle-i\left\vert V\right\rangle)/\sqrt 2 \}\) to compute FT, as we explain in detail in the SI.