Abstract
Quantum communication employs the counterintuitive features of quantum physics for tasks that are impossible in the classical world. It is crucial for testing the foundations of quantum theory and promises to revolutionize information and communication technologies. However, to execute even the simplest quantum transmission, one must establish, and maintain, a shared reference frame. This introduces a considerable overhead in resources, particularly if the parties are in motion or rotating relative to each other. Here we experimentally show how to circumvent this problem with the transmission of quantum information encoded in rotationally invariant states of single photons. By developing a complete toolbox for the efficient encoding and decoding of quantum information in such photonic qubits, we demonstrate the feasibility of alignmentfree quantum keydistribution, and perform proofofprinciple demonstrations of alignmentfree entanglement distribution and Bellinequality violation. The scheme should find applications in fundamental tests of quantum mechanics and satellitebased quantum communication.
Introduction
Current implementations of quantum communication (QC) use photons as the carriers of qubits (quantum bits), the basic units of quantum information. This is due to the fact that photons, known as socalled 'flying qubits', are easy to transport from one location to another^{1}. Photonic freespace QC has been demonstrated for distances of hundreds of kilometers^{2}, a progress that could soon lead to satellitebased longdistance QC^{3,4,5,6}. However, standard approaches to QC, for example, the one based on encoding qubits into the polarization of photons, require that all users involved have knowledge of a shared reference frame. For instance, in the bipartite scenario, the emitter and receiver, conventionally called Alice and Bob, must initially align their local horizontal (H) and vertical (V) transverse axes, and then keep them aligned throughout the transmission (Fig. 1a). This, in turn, requires the exchange of a large (strictly speaking, infinite) amount of classical information. This represents, in general, an extra technical overhead, which can impose serious obstacles in the particular situations where the users are very far apart from each other, the misalignment between their frames varies in time, or the number of users is large, for example^{7,8}. In general, the lack of a shared reference frame inhibits faithful QC, because it is equivalent to an unknown relative rotation, therefore introducing noise into the quantum channel^{8}.
A possible solution to this problem is to exploit multiqubit entangled states that are invariant under singlequbit rotations acting collectively on all the qubits (see refs 9,10,11,12,13 and references therein). These constitute particular instances of decoherencefree subspaces, originally introduced in the context of faulttolerant quantum computing^{14,15,16,17}. The idea is thus to encode logical qubits into rotationally invariant states of multiple physical qubits. These can, in principle, be realized with multiple photons^{10,11,12}. However, the efficient production and detection of multiphoton states is a technological challenge, they are more susceptible to losses, and the requirement that multiple photons are subject to exactly the same rotation is very seldom perfectly satisfied.
A more efficient way to circumvent misalignments is provided by exploiting multiple degrees of freedom of single photons^{18}. In particular, the polarization and transverse spatial modes stand out for this purpose (Fig. 1b). Just as the circular polarization states are eigenstates of the spin angular momentum (SAM) of light, the helicalwavefront LaguerreGaussian modes are eigenmodes of its orbital angular momentum (OAM). The OAM degree of freedom is attracting a growing interest for applications in both classical and quantum photonics^{19,20,21,22}. The peculiarity of the SAM and (first order) OAM eigenstates together is that, as they are defined with respect to the same reference frame, they suffer exactly the same transformation under coordinate rotation. Therefore, they satisfy the collective rotation requirement exactly, constituting an ideal pair to carry rotationally invariant hybrid qubits (see Fig. 1c).
Here we experimentally demonstrate a complete toolbox for the efficient encoding and decoding of quantum information in such photonic qubits, suitable for alignmentfree QC. The core of our toolbox is a liquid crystal device, named 'qplate'^{22,23}, that maps polarizationencoded qubits into qubits encoded in hybrid polarizationOAM states of the same photon that are invariant under arbitrary rotations around the propagation direction, and vice versa. In other words, the qplate acts as a universal encoder/decoder, where 'universal' refers to the fact that it works for any qubit state. The qplate used in the present work is the result of a very recent technological advance allowing for the manufacture of electrically tunable devices with topological charge q=1/2 (ref. 24). This is the first time such devices are exploited in the quantum regime. In addition, the toolbox requires no interferometric stability as in previous proposals^{18,25}, and can be set entirely in a robust and compact unit that could easily be mounted in a small satellite, for instance. Furthermore, our universaldecoder setup features a builtin filtering mechanism that maps a wide class of physical errors into losses instead of logical errors. We show that, owing to this mechanism, the scheme is robust also against misalignments around axes other than the propagation direction, as well as against other spatial perturbations. We demonstrate the potential of our method by performing a proofofprinciple misalignmentimmune implementation of the singlephoton Bennett–Brassard (BB84) quantum keydistribution (QKD) protocol^{26}, entanglement distribution, and the violation of the ClauserHorneShimonyHolt (CHSH) Bell inequality^{27}.
Results
Hybrid logical qubit encoding
Our logical qubit basis is defined by the hybrid polarizationOAM singlephoton states
Subscript 'p' denotes the polarization Hilbert space, spanned by the left and righthanded circular polarization states L〉_{p} and R〉_{p}, respectively, which are eigenstates of the SAM operator along the propagation direction (z axis) of respective eigenvalues and . In turn, subscript 'o' stands for the OAM bidimensional subspace spanned by the left and righthanded firstorder LaguerreGauss modes l〉_{o} and r〉_{o}, eigenstates of the OAM operator of respective eigenvalues and . Finally, subscript L refers to the logical subspace of zero total angular momentum along z. The latter is spanned by the two hybrid states (1), nulleigenvalue eigenstates of the total SAM+OAM operator . Because the total angular momentum operator is the generator of state rotations, states (1) are both invariant under arbitrary rotations around the z axis (on this issue, see also ref. 28). More specifically, in a physical rotation about the z axis by any angle θ, the circular polarization states and OAM eigenmodes acquire equivalent phase factors on their own: . However, for tensorproduct combinations with opposite handedness as (1), the individual phases cancel each other and the composite states remain intact. As a consequence, because of linearity, any coherent superposition (or incoherent mixture) of the two logical states, that is, the entire logical subspace, is also immune to all possible referenceframe misalignments during the entire QC session.
Universal encoder/decoder
The experimental setup used to encode and decode arbitrary hybrid qubit states in the logical basis (1) is shown in Fig. 2. The qplates are liquid crystal devices that produce a spinorbit coupling of the polarization and OAM contributions to the total angular momentum of photons^{23,22}. The qplate is a birefringent slab having a uniform optical retardation δ and a suitably patterned transverse optical axis, with a topological singularity of charge q at its centre. A 'tuned' qplate with δ=π transfers quanta of angular momentum between the SAM and the OAM. Specifically, each photon suffers a variation in its OAM by an amount determined by the charge q and the SAM of the input polarization. qplates with q=1 have been recently used to demonstrate interesting spinOAM quantum information manipulations^{29,30,31,32,33}.
A tuned qplate with topological charge q=1/2 gives rise to the following transformations:
where 0〉_{o} denotes a zeroOAM state, such as the fundamental Gaussian mode (TEM_{00}). We note that the radial profile of the l〉_{o} and r〉_{o} states generated by the qplate is not exactly LaguerreGauss (see Methods for more details), but this does not affect their pure OAMeigenstate rotational behaviour^{22}. Indeed, the qplate is ideally a unitary device (our qplate was uncoated and had a transmission efficiency of about 85%), but the induced radialmode effects (see Methods) may introduce about 40% of total additional losses in the final recoupling to the singlemode fibre before detection. Consider then a generic polarizationencoded qubit prepared in the TEM_{00} spatial mode. From transformations (2), sending the qubit through the qplate yields
That is, the qubit is now encoded into the desired rotationally invariant space spanned by logical basis (1). Remarkably, the same qplate device works also as a universal decoder, transferring generic rotationally invariant qubits to their polarizationencoded counterparts. Explicitly, by injecting ψ〉_{L} into the qplate, one obtains
that can then be coupled into a singlemode fibre and analysed in polarization using standard methods. The measurement device is sketched in Fig. 2d. Again from the linearity of quantum mechanics, the encoding and decoding transformations (3) and (4) hold, even if the polarization state is part of some larger entangled state. In addition, an outstanding feature of the qplate is that it realizes the polarizationtransversemode coupling in a single compact device that requires no interferometric stability, therefore providing the scheme with a builtin robustness.
Our first step was to experimentally verify that the encoding/decoding apparatus works properly in the case of stationary aligned reference frames. We prepared the input photon in one of the polarization states . The qubit was then mapped by a first qplate into the rotationally invariant encoding, transmitted through free space to the measurement stage, then decoded back to polarization by a second qplate, and finally analysed in polarization using a set of wave plates and a polarizing beam splitter (PBS). The full experimental apparatus is described in the Methods. The average measured fidelity with the input states was F=(98±1)%, indicating that the devices work nearly perfectly.
Alignmentfree quantum key distribution
To experimentally demonstrate that the present QC setup works well for arbitrary relative alignment of Alice and Bob's transverse reference frames, we mounted the qplate, waveplates, PBS and optical fibre couplers in a compact and robust detection stage that can be freely rotated by any angle θ around the light propagation axis, as shown in Fig. 2d,e. Then, using heralded single photons, and for different angles θ, we encoded, transmitted and decoded, the four hybridqubit states required for the BB84 QKD protocol^{26}: 0〉_{L},1〉_{L}, and . We quantified the potential of our setup for QKD by measuring the fidelities of the states prepared and measured with the ideal ones, as well as the qubiterror rates^{34} and for the logical bases and , respectively. The experimental results are reported in Fig. 3a,b.
Figure 3a shows the average fidelity F_{QKD} over the four states, as a function of θ. F_{QKD} is constantly above the value F_{T}=89%, which corresponds to the wellknown ShorPreskill security proof threshold^{35}. Above this, under the usual assumptions that Alice's source emits (logical) qubits, Bob's detectors perform (logical) qubit measurements, and there is no basisdependent flaw in Alice's and Bob's systems^{36}, unconditional security can be guaranteed. In contrast, the fidelity attained using polarizationencoded qubits falls below the security bound for angles θ>20°, even in the ideal noiseless case (blue dashed line). Figure 3b, in turn, shows the fidelity for each state, obtained by uniformly mixing the data over all measured angles θ. Again, all the individualstate fidelities are consistently larger than the security threshold. Indeed, the measured qubiterror rates, estimated as 1−F, were and , from which we expect a high secretkey fraction r=(70±1)% (ref. 34).
Alignmentfree entanglement distribution
To test entanglement distribution, the photon pair is prepared in the polarization entangled state , where the superscripts A and B refer to Alice's and Bob's photons, respectively. The photons are coupled into singlemode fibres that select only states with zero OAM (TEM_{00} mode). A qplate at the output of each fibre transforms the polarizationentangled state to the rotationally invariant entangled state:
To verify the generation of hybrid entanglement, we performed quantum state tomography of the experimental density matrix measured without misalignment (θ=0). The tomographically reconstructed matrix, in the basis , is shown in Fig. 3c. The fidelity with the experimental polarization entangled state input to the encoder is , while the entanglement of , as given by the concurrence, is C=(0.85±0.03). As a first test on the rotational invariance of the state produced, we repeated the tomographic reconstruction with Alice's measurement stage rotated by θ=45°. The corresponding reconstruction is shown in Fig. 3d. The fidelity with is , and the concurrence is C=(0.84±0.03), consistent with the case θ=0. This indicates that our entanglement distribution scheme is immune to relative misalignments of Alice and Bob.
Alignmentfree quantum nonlocality
With the hybrid entangled state (5), we performed a violation of the CHSH inequality > in an alignmentfree setting. In the inequality, a_{x} and b_{y}, with possible values 0 or 1, are the outcomes of Alice's and Bob's measurement settings x and y, respectively, with x and y equal to 0 or 1. Correlators , with 〈〉 the statistical average, quantify the fraction of events where Alice's and Bob's outcomes are observed to coincide. Any localhiddenvariable model satisfies the inequality^{27}. For the rotationally invariant quantum violation of the inequality we chose the following hybrid measurement bases: {0〉_{L},1_{L}〉} and {+〉_{L},−〉_{L}}, corresponding to Alice's settings x=0 and x=1, respectively, and and , corresponding to Bob's settings y=0 and y=1, respectively. Figure 3e reports the measured CHSH parameter S versus the rotation angle θ of Alice's measurement frame. The figure shows that the localhiddenvariable bound is violated for all angles, in striking contrast with the experimental polarization state (blue circles), or even with the ideal maximally entangled polarization state (blue dashed line). For the logically encoded states, we mixed the data of all different values of θ to test the violation's immunity to arbitrarily varying frame orientation, obtaining a value of S=(2.47±0.01)>2. This alignmentfree extraction of nonlocal correlations reconfirms the rotational invariance of the quantum resources created here.
Robustness of rotationalinvariant hybrid qubits
A remarkable feature of our polarizationOAM hybridencoding QC scheme is that it turns out to be robust against the spatialmode perturbations arising in beam misalignments around axes other than the optical one and atmospheric turbulence effects. Such robustness appears at first glance counterintuitive, because the encoding involves the use of OAM, which is quite sensitive to all the abovementioned spatial perturbations^{37,38} (although significant progresses in pure OAMbased classical and quantum communication through the atmosphere have been recently reported^{39,40,41}). The main reason for such robustness is that the OAM spread induced by spatialmode perturbations is neutralized by the polarization degree of freedom, which is in contrast very robust against those spatialmode perturbations. This allows one to filter out, in the receiving unit, most components of the state that would otherwise decrease the fidelities. That is, the particular decoding setup used intrinsically implements an effective quantum errorcorrection procedure that discards (but does not correct) all states outside the logical subspace.
Indeed, spatialmode perturbations will alter a generic hybrid qubit , transforming it into the following state:
where m〉_{o} denotes a generic mode with OAM eigenvalue and C_{m,m′} are the probability amplitudes for the photon OAM to be shifted from to , owing to the perturbation. However, in the decoding unit, the photon undergoes another qplate transformation
followed by a projection onto an m=0 Gaussian spatial mode (for example, by coupling it into a singlemode fibre), which leads to the following final state:
Therefore, if the spatialmode perturbation satisfies the condition
the final polarizationencoded qubit will be identical to the initial one, except for a global phase and amplitude, and the communication fidelity will be preserved (in this simple analysis, we did not consider the radial modes; see Methods for a complete theory).
In particular, every beam transformation that is mirrorsymmetric with respect to a plane containing the initial beam axis will be symmetrical in the sign of OAM and hence will satisfy equation (9). For example, beam parallel displacements, tilts, elliptical deformations, or aperturings with a circular iris (even if off centre) or a halfplane mask (knifeedge), all have this symmetry. An axial misalignment, that is, a misalignment around an axis other than the optical one, between the transmitting and receiving communication units is equivalent to a beam translation and/or tilt, with both contained in the same plane, and can be treated analogously. Only symmetrybreaking combinations of two or more of the above effects may affect the fidelity. For example, a beam tilt combined with a beam displacement along a different plane will break the mirror symmetry and hence might introduce some degree of qubit alteration. (See Supplementary Methods for an explicit analysis of these beammisalignment perturbations.^{42}) Also, the main optical effects arising from atmospheric turbulence, such as beam wandering and spreading are mirrorsymmetric, so that the extent of qubit alteration is expected to be much less significant in our communication scheme than in the case of pure OAM communication.
Another important class of transformations that satisfies equation (9) is that mathematically defined by pure multiplicative factors acting on the optical field, for example, the transformations arising from crossing any arbitrary inhomogeneous medium that is thin as compared with the Rayleigh length. It is easy to verify that these will be described by coefficients C_{m,m′} that depend on the difference m−m′ and on the absolute values m and m′, so that equation (9) is automatically satisfied. Weak turbulence, introducing only pure phase wavefront distortions, falls within this class of transformations and is therefore predicted to leave the qubit fidelity intact^{37}. If we now consider the fact that light propagation in homogeneous media leaves the various OAM components constant, we conclude that equation (9) is satisfied even if the turbulent medium is followed and/or preceded by a longdistance freespace propagation, as in the case of earth to satellite (and vice versa) communication through the atmosphere.
Experimental tests of rotationalinvariant qubit robustness
As a first test, we considered transmission through two types of transverse apertures: a halfplane movable obstruction (knife) covering a variable fraction of the transverse mode, and an iris (or pinhole) with variable radius. We have measured the state transmission fidelity F for different input states, at both aligned and 45°rotated measurement stages, with respect to the transmitting unit, and for an increasing disturbance due to the obstruction. The experimental setup used for this test is illustrated in Fig. 4a. We encoded different polarization qubits using two wave plates and mapped them into the hybrid encoding using a qplate. For the purpose of comparison, we also switched to a pureOAM encoding by inserting a fixed linear polarizer after the qplate, so as to erase the polarization content of the qubit. Then, the photon was sent through the obstruction and to the receiving unit, and the communication fidelity was measured as a function of the obstruction transmittance, by varying the aperture of the pinhole or the transverse position of the knife. Thus, the lowest transmittivity corresponds to a tiny aperture of the pinhole (0.2 times the beam size), or to the almost complete coverage of the beam. All reported experimental fidelities were obtained by averaging over the six eigenstates of three mutually unbiased bases, therefore providing a good representative of the average fidelity over any input qubit state. The experimental results are reported in Fig. 4b,c. It is seen that the average fidelity of hybrid qubits is independent both of the transmittivity of the aperture and of the rotation angle of the measurement kit, with a global average of F=(98±1)% for the case of iris and F=(96±1)% for the knife. Moreover, in the former case the fidelity is not affected by the displacement of the pinhole off the beam axis. For comparison, we tested the resistance of qubits encoded only in the twodimensional OAM subspace o_{1}={+1〉,−1〉}, that is, the same subspace used for the hybrid encoding. In this case, the fidelity remains high (F=(97±1)%) only when the cylindrical symmetry of the modes is not perturbed, as for the centred iris, although for all other cases (noncentred iris or knife), the fidelity drops rapidly with decreasing transmission.
As a second test, we performed a communication run while changing the angle of the measurement kit without reoptimizing the alignment of the singlemode fibre. This corresponds to introducing small uncontrolled tilt and displacements in the beam during the measurement. We found that the system preserves a good QC fidelity (that is, above the security threshold) for rotations up to 30°. Above this angle, it was necessary to slightly readjust the singlemode fibre alignment to restore a high fidelity.
Finally, we tested the communication fidelity dependence on a controlled beam displacement, for two fixed angles of the measurement stage. Fig. 4d shows the behaviour of the average communication fidelity as a function of the beam displacement. The hybridqubit fidelity decreases with the displacement, but much slower than that of pure OAM encoding (see also the Supplementary Discussion).
Discussion
QC has a fundamental role in the modern view of quantum physics and opens the possibility of a variety of technological applications. Uncontrolled referenceframe misalignments limit QC, as they turn the transmitted quantum messages into noisy, classical ones. Here we report the development of a robust and compact toolbox for the efficient encoding and decoding of quantum information into singlephoton states that are invariant under arbitrary rotations around the optical axis. With this, all concerns on relative axisorientation during quantum transmissions reduce simply to the basic requirement of establishing an optical link.
Rotational invariance is achieved by exploiting decoherencefree subspaces spanned by hybrid polarizationOAMentangled states. We experimentally showed the efficacy of these states through the feasibility demonstration of a cryptographickey distribution protocol, distribution of entanglement, and violation of a Bell inequality, all in alignmentfree settings. Importantly, as far as cryptographic security is concerned, our scheme does not introduce loopholes other than those already present in any photonic experiment with conventional encodings. We also emphasize that, even though the states used are themselves invariant only under rotations about the propagation axis, the scheme resists misalignments around other directions too. This is due to a filtering mechanism intrinsic to our universaldecoder setup, which maps errors originating from beam rotations around axes other than the optical link, as well as other spatial perturbations, into signal losses instead of infidelity.
Recently, interesting alignmentfree approaches for QKD^{43} and to extract nonlocal correlations^{44,45,46} have been put forward. These, however, require that the relative axis orientations, though unknown, stay approximately static throughout the quantum data exchange session (see Supplementary Discussion for details). Remarkably, in contrast, our rotationalinvariance protection works even if the relative orientations vary arbitrarily from measurement to measurement. Moreover, another important feature of the present scheme is that it does not restrict to nonlocality and QKD but enables also fully general QC protocols, all misalignmentimmune^{18}. These include, for instance, quantum teleportation, dense coding, and entanglement swapping, the main ingredient of quantum repeaters.
Finally, our scheme should find applications in the forthcoming experiments on longdistance satellitebased QC^{3,4,5,6}. There, apart from misalignments, other issues may impose serious obstacles too, such as precise satellite lasertracking, collection efficiencies, or finitesize effects (for QKD). However, immunity against arbitrarily varying transverse relative orientations not only solves for misalignments but also relaxes requirements on the repetition rates needed to overcome finitesize effects (see Supplementary Discussion).
Methods
Experimental apparatus
The input photon pairs are generated via spontaneous parametric fluorescence in a βbarium borate (BBO) crystal cut for typeII phase matching, pumped by the second harmonic of a Ti:sapphire modelocked laser beam with a 76MHz repetition rate. The generated photons have wavelength λ=795 nm and spectral bandwidth Δλ=3 nm, as determined by two interference filters. The spatial and temporal walkoff is compensated by inserting a λ / 2 wave plate and a 0.75mm thick BBO crystal on each output mode^{47}. The detected coincidence rate of the source is 8 kHz. The photons are delivered to the setup via singlemode fibres, to define their transverse spatial mode to a pure TEM_{00}, corresponding to OAM m=0. After the fibre output, two wave plates compensate the polarization rotation introduced by the fibre. The polarizationencoded photonic states are transformed into rotationally invariant hybrid states by qplates with topological charge q=1/2. The average conversion efficiency of all the qplates employed in the experiment has been optimized to (94±2)% by controlling the electric field applied to the device^{48,49}. For the analysis of experimental data, we referred to the coincidence counts between detectors [D_{A1},D_{B1}] and [D_{A2},D_{B2}] shown in Fig. 2, collected by a coincidence circuit with a gate of 3 ns. Typical coincidence rates were 30 Hz for the CHSH experiment and 300 Hz for the BB84 one.
Theory of qplate encoding/decoding with radial modes
To fully describe the optical action of the qplate, we need to consider both the azimuthal quantum number m, corresponding to the OAM eigenvalue in units of ħ, and a radial quantum number p. These two numbers can be defined, for example, as in the case of LaguerreGauss beams, although this is not the only possible choice and our treatment is general in this respect. We denote with P, m, p〉 the photon quantum state, including the spatial mode defined by m and p and the polarization state P (for example, L, R, H or V). The qplate transformation laws are as follows:
where are coefficients that do not depend on the sign of m and m′, owing to the mirror symmetry of the qplate pattern. These coefficients can be also given explicit analytical expressions in a given radial basis (for example, the LaguerreGauss one), but these expressions are not needed for our purposes here. Let us now consider a generic input polarizationencoded qubit photon in a Gaussian mode TEM_{00} (m=0, p=0):
After the qplate, this photon is converted into the following rotationinvariant hybrid state (corresponding to the logical qubit):
This generalizes the transformation (3), by including also the radial modes. In freespace propagation, the different spatial modes will acquire Gouy phase factors G_{m,p} that depend only on the absolute value of m (see, for example, ref. 30). The final decoding procedure is based on a second qplate followed by a spatial filtering into an output TEM_{00} mode (that is, m=0, p=0; but the output radial basis need not be the same as the input one, as a different beam radius can be defined). This spatial filtering can, for example, be implemented by coupling into a singlemode fibre. This leads to the output polarizationencoded state , where
Therefore η^{2} gives the quantum efficiency of the encoding/decoding process. After decoding, this output state is ready for detection or for further quantum processing.
Effect of spatialmode perturbations during propagation
Let us now consider the action of generic spatialmode perturbation acting on the photons during propagation, as defined by the following transformation laws:
where are suitable complexvalued coefficients (when ignoring the radial modes, they correspond to the C_{m,m′} coefficients used in equation (6)). We assume here that the perturbation does not affect the polarization state. The Gouy phase factors describing freespace propagation can be included in (14) without loss of generality. Let us now consider the effect of the perturbation (14) on the encoded qubit (12), which is transformed into the following state:
Next, we apply again to this perturbed state, the transformations used in the decoding unit (qplate and TEM_{00} filtering), obtaining the following final state
The latter equation shows that the qubit state will be unaffected, except for a global amplitude and phase factor, if and only if the following equality holds true:
for all values of the radial indices p, p′, thus generalizing equation (9). In other words, any spatialmode perturbation that satisfies equation (17) will not alter the qubit transmission fidelity, although it may affect the transmission efficiency by increasing the photon losses. Explicit examples of spatial transformations arising in typical beam misalignment effects, such as beam translations and/or tilting, are considered in the Supplementary Methods.
Additional information
How to cite this article: D'Ambrosio, V. et al. Complete experimental toolbox for alignmentfree quantum communication. Nat. Commun. 3:961 doi: 10.1038/ncomms1951 (2012).
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Acknowledgements
This work was supported by the FETOpen Program, within the 7th Framework Programme of the European Commission under Grant No. 255914, PHORBITECH, FIRBFuturo in Ricerca (HYTEQ), the Brazilian funding agencies FAPERJ, CNPq and the INCTInformação Quântica, and the Spanish Juan de la Cierva foundation.
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V.D., E.N., L.A., S.W., L.M. and F.S. jointly conceived the misalignmentfree scheme, the underlying theory, the experimental layout and methodology. S.S. and L.M. fabricated the qplates. V.D., E.N. and F.S. performed the experiments. All authors discussed the results and participated in the manuscript preparation.
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D'Ambrosio, V., Nagali, E., Walborn, S. et al. Complete experimental toolbox for alignmentfree quantum communication. Nat Commun 3, 961 (2012). https://doi.org/10.1038/ncomms1951
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DOI: https://doi.org/10.1038/ncomms1951
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