Houria Asri

Olivier Dautel

Armelle Ouali
email: armelle.ouali@enscm.fr

Terpyridine-Ru Complexes Non-Covalently Supported on Cobalt Magnetic Nanoparticles for Nitroarene Transfer Hydrogenation

Keywords: Magnetic nanoparticles (MNPs), -stacking interactions, 2-propanol, Pyrene-tagged catalysts, nitroarene transfer hydrogenation

This work reports the preparation and characterization of active and recyclable supported catalysts involving molecular Ru-terpyridine complexes and able to promote the transfer hydrogenation of nitroarenes in 2-propanol. The system proposed is based on the design of original pyrene-tagged terpyridine ligands, either monomeric "1tpy" (one pyrene tag for one terpyridine) or multivalent "5tpy" (one pyrene tag for five terpyridines), able to interact with the graphene surface

of commercially available cobalt magnetic nanoparticles (MNPs) through -stacking interactions.

The existence of non-covalent interactions between the pyrene-tagged terpyridines and the MNPs surface at 20 °C was assessed by fluorescence spectroscopy for the first time in 2-propanol, an organic medium. Interestingly, 1tpy and 5tpy were shown to be released upon temperature increase (80 °C) while the association was restored upon temperature decrease (20 °C). The reversibility of this phenomenon enabled to reach high activities in nitroarenes transfer hydrogenation for the corresponding Ru complexes (catalytic species are free in refluxing 2-propanol) and high recyclabilities (non-covalent interactions are restored upon cooling). Noteworthy, the multivalent ligand 5tpy displaying as expected the higher terpyridine loading (0.15 mmol of tpy / g of MNPs vs 0.03 mmol for 1tpy), also gave rise to the Ru-catalyst with the best recyclability (8 consecutive cycles with no loss of activity against 5 cycles for the catalyst based on monomeric ligand 1tpy). In both cases, the ruthenium leaching was found to be low and the expected aniline met the requirements of the pharmaceutical industry (< 10 ppm Ru) from the third run.

INTRODUCTION

Heterogenization of homogeneous catalysts on supports including for e. g. polymers, silicas or magnetic nanoparticles has the potential to build bridges between homogeneous and heterogeneous catalyses. [START_REF] Pagliaro | Supported Molecular Catalysts[END_REF] Expected advantages include the combination of the high catalytic activity inherent to the well-defined molecular catalyst and its easy separation from the reaction mixture with the possibility of reuse. However, some drawbacks have been highlighted including catalyst deactivation once immobilized and subsequent lower turnover numbers (as compared to corresponding homogeneous catalysts) as well as low recyclabilities. [START_REF] Hübner | Why Does Industry Not Use Immobilized Transition Metal Complexes as Catalysts ?[END_REF] Besides, except in some rare cases, [START_REF] Keller | An efficient and recyclable dendritic catalyst able to dramatically reduce palladium leaching in Suzuki couplings[END_REF] metal leaching cannot be avoided also leading to poor recycling ability and to the contamination of the products by metal traces. The design of supported catalysts combining high activities, high recyclabilities and low metal leaching thus still constitute a challenge. Among other supports, magnetic nanoparticles (MNPs) are particularly attractive for the development of more efficient and green processes. [START_REF] Gun'ko | Recent Advances in the Application of Magnetic Nanoparticles as a support for Homogeneous Catalysts[END_REF] Indeed, their separation can easily be achieved by magnetic decantation with a simple magnet contrary to other supports, such as polymers or silica, which require time-consuming precipitation and filtration steps. Generally, MNPs are stabilized by coating the magnetic core with polymer or silica shells which are used for the covalent immobilization of homogeneous catalysts. [START_REF] Gun'ko | Recent Advances in the Application of Magnetic Nanoparticles as a support for Homogeneous Catalysts[END_REF] First described by Stark and coll. 5 and now commercially available, 6 graphene coated MNPs also enable the covalent immobilization of catalysts. 5,[START_REF] Schaetz | Immobilization on a nanomagnetic Co/C surface using rom polymerization: Generation of a hybrid material as support for a recyclable palladium catalyst[END_REF] In addition, they offer the unique possibility for non-covalent catalyst attachment by - stacking and interestingly, the efficiency of these non-covalent interactions was found to dependent on the temperature. 8a-c Indeed, while the pyrene-tagged catalysts are associated to the MNPs at low temperature (20-25 °C), a dissociation was observed at higher temperatures (> 60 °C). Such behavior is reminiscent of catch-and-release systems for which the catalytic moiety is released in solution over the course of the reaction and is recaptured at the end thus enabling successful recovery. 9 Reported "boomerang" systems involve different modes of recognition including hydrogen binding, ionic, host-guest or - stacking interactions. 9 Concerning the abovementioned Co/C MNPs, O. Reiser and coll. reported the non-covalent attachment of pyrene-tagged Pd-carbene complexes able to catalyze the hydrocarbonylation of aromatic halides in water. 8a Later on, some of us reported that the introduction of well-defined dendritic spacers between the pyrene tag and Pd-phosphine complexes (Figure 1) enabled to increase the loading of active palladium complexes active in Suzuki couplings. 8b In both cases, the catalyst recovery was ensured in aqueous media (pure water 8a or water/THF (5:2) 8b ) that favor efficient non-covalent interactions. 8d

One aim of this work was to design related systems enabling reversible interactions in organic media and in particular in 2-propanol, a cheap, benign and easy-to-handle H source employed in transfer hydrogenations of various substrates. [START_REF] Barath | Hydrogen Transfer Reactions of Carbonyls, Alkynes and Alkenes with Noble Metals in the Presence of Alcohols/Ethers and Amines as Hydrogen Donors[END_REF] Among the latter, nitroarenes give rise to anilines, key compounds finding widespread applications as intermediates and targets in pharmaceutical, agrochemical, pigments or dyes industries and holding a very high market share in organic chemistry industry. 11 In general, the use of H-donors instead of molecular hydrogen for nitroarenes reduction enables to avoid high-pressure equipments. Besides, it is a more sustainable and benign methodology as compared to the historical stoichiometric Béchamp reduction involving metallic iron in acidic media generating a lot of wastes. A wide array of transition-metal based catalysts, mostly heterogeneous, promote nitroarenes transfer hydrogenations in the presence of different Hsources such as formates, sodium borohydride or hydrazine. 11b Although active in mild conditions and generally more tolerant in terms of functional groups, related homogeneous catalysts have been less studied. Whether homogeneous or heterogeneous, only a few catalysts are able to promote transfer hydrogenation using 2-propanol as the H-donor. 11b,12 Along these lines, Beller and coll. reported a well-defined terpyridine-ruthenium complex catalyzing efficiently and selectively the reduction of a wide array of nitroarenes in 2-propanol under mild conditions. 13 The reuse of ruthenium-based catalysts appears as highly desirable for economic and ecological reasons. Therefore, in this work, pyrene-tagged terpyridine-Ru complexes, either monomeric or multivalent, were prepared and associated to MNPs through -stacking interactions for the first time in 2-propanol, an organic solvent. The activity and recyclability of both systems were evaluated in the transfer hydrogenation of nitrobenzene.

EXPERIMENTAL SECTION • Materials and methods

THF was dried using mBRAUN purification system. All other solvents and chemical reagents were purchased from Sigma Aldrich, Acros or Alfa Aesar and used as received without further purification. Catalytic reactions were performed under nitrogen atmosphere using standard Schlenk technique. Nuclear Magnetic Resonance Spectroscopy (NMR) spectra were recorded with Bruker 400 MHz device at room temperature in the indicated deuterated solvents. All 1 H, 13 C, 31 P NMR spectra were evaluated in parts per million parties (ppm) and coupling constants (J) are reported in Hertz (Hz). The signals in the spectra are described as s (singlet), d (doublet), t (triplet), m (multiplet). Gas chromatography -mass spectra (GC-MS) were recorded on a ThermoFisher Scientific Trace GC Ultra instrument with a ThermoFisher Scientific ITQ 900 Ionic Trap and an Agilent DB-5MS 30 m x 0,25 mm capillary apolar column (Stationary phase: 0,25 μm film). Flash column chromatography was performed on silica gel (230-400 mesh) and analytical thin-layer chromatography (TLC) was carried out using 250 μm commercial silica gel plates. MS spectra were obtained on a Synapt G2-S from Waters with a quadrupole cell collision. High resolution mass spectra (HRMS) were recorded on an Agilent 6210 ESI TOF (time of flight) mass spectrometer. UV-vis spectra and Fluorescence spectra were performed with a LS 55 Luminescence Spectrometer from Perkin Elmer. Solution characterizations were realized on diluted 2-propanol solutions. ICP-MS were performed on a Agilent 7900 apparatus.

• Fluorescence studies: Preparation of the solutions with concentrations ~ 10 -5 mol. L -1 : For compound 3: 20 mg of 3 were dissolved in 2-propanol in a graduated flask of 100 mL. 300 μl of the stock solution were diluted in a graduated flask of 10 ml (final concentration: 0.99.10 -5 mol. L - 1 ). For compound 5: 7 mg of 5 were dissolved in 2-propanol in a volumetric flask of 250 mL (final concentration of 1.02.10 -5 mol. L -1 ). 2): Solutions of pyrenes 3 or 5 in 2-propanol were prepared and their respective fluorescence spectra recorded. Aliquots of Co/C MNPs were then successively added to these homogenous solutions (1 mg for each addition). After each addition, the suspension was sonicated for 15 min at 20 °C in a closed tube (it was checked that no evaporation of the solvent occurred), the NMPs recovered by magnetic decantation and the supernatant analyzed by fluorescence spectroscopy.

Studies at 20 °C (Figure

Note: the addition of MNPs to solutions of 3 and 5 also induced a decrease of the absorption intensity by UV-vis spectroscopy (see Supporting Information, Figure S15). This confirms that the decrease of emission intensity mostly resulted from the decrease of the ligand concentration and not from other possible deactivation pathways (quenching, temperature, impurities…).

Influence of the temperature on the -stacking interactions efficiency (Figure 3): Solutions of pyrenes 3 or 5 in 2-propanol were prepared and their respective fluorescence spectra recorded (blue curves). MNPs were added (1 mg in the case of 3; 0.4 mg in the case of 5) and the suspensions were sonicated for 15 minutes at 20 °C in closed tubes (fluorescence of the resulting supernatants: red curves). Then the supernatants and the MNPs were gathered and allowed to sonicate for 2.5 h at 80 °C (still in closed tubes). After "hot magnetic decantation", fluorescence spectra of the supernatants were recorded (green curves). The latter were one more time associated to the previously recovered MNPs and the corresponding suspensions sonicated for 2h at 20 °C.

The supernatants were analyzed by fluorescence spectroscopy (purple curves). Note: After each sonication steps, it was checked that no evaporation of the solvent occurred.

• General procedure for recycling experiments: In a Schlenk tube equipped with magnetic stirring bar, under nitrogen atmosphere, the catalyst was generated in situ from [{RuCl2(pcymene)}2] (12.0 mg, 0.0125 mmol) and ligand (0.025 mmol of terpyridine moieties: 0.025 mmol = 15.2 mg for 3, 0.005 mmol = 13.7 mg for 5) in 2-propanol (5 ml) for 30 min at 60°C. MNPs (500 mg for 3, 100 mg for 5) were introduced and the mixture sonicated for 30 minutes at 20 °C.

The nitroarene (0.5 mmol), KOH (0.075 mmol, 4.2 mg), and 2-propanol (5 ml) were next added.

After 15 hours at reflux (temperature of the oil bath: 100 °C), the reaction mixture was cooled to room temperature, sonicated for 30 minutes at 20 °C and magnetic decantation was performed thanks to the magnet. The MNPs were rinsed twice with 2-propanol (1 mL). The standard 1,3,5trimethoxybenzene was added to the joined solutions and the mixture obtained analyzed by [START_REF] Pagliaro | Supported Molecular Catalysts[END_REF] H NMR (after evaporation of the solvent) and GC-MS. To the hybrid MNPs dried under vacuum, nitroarene (0.5 mmol) chosen for the next run and 2-propanol (10 mL) were added and the same procedure was repeated at each run.

RESULTS AND DISCUSSION

The first objective consisted in preparing terpyridine ligands bearing a pyrene tag able to interact with the graphene surface of the MNPs. Two systems were targeted: the first one involved a pyrene moiety connected to a terpyridine ligand through an organic linker (Figure 1a) and the second one consisted in a pyrene tag linked to five terpyridine ligands through a multivalent spacer (Figure 1b). S6-S7 and S9-S11). In addition, high resolution mass spectroscopy also confirmed the structures (see SI, Fig. S8 andS12). Noteworthy, several compounds associating one pyrene and one terpyridine were reported in the literature: both moieties can be covalently connected (directly 14

or via an organic linker 15 ) or linked in a non-covalent manner (for e. g. via coordination 16 or other weak bonds in supramolecular assemblies [START_REF] Ao | Supramolecular Amphiphile Derived From non-Covalent Alkynylplatinum(II) Terpyridine Molecular Tweezer/Pyrene Complexation[END_REF] ). Other compounds display non-equimolar pyrene-toterpyridine ratios such as structures with 3 pyrenes pods able to interact with graphene and one terpyridine-metal complex bringing optical properties. 18 Conversely, tetra-substituted pyrenes were used as multifunctional cores bearing 2 or 4 terpyridines. 19 The pyrene-tagged mono-and penta-terpyridines reported here clearly differ from the known related compounds. Indeed, compound 3 includes an original covalent flexible linker and, for 5, the multivalency is ensured by the cyclotriphosphazene (N3P3) moiety thus enabling to maintain the pyrene tag unsubstituted for optimized interactions with the graphene surface.

Scheme 1: Synthetic route to pyrene-tagged ligands involving one (3) or five terpyridine (5) moieties.

To assess the existence of interactions between the graphene surface and the pyrene moiety of 3 and 5, fluorescence studies were carried out. [START_REF] Guo | Functionalization of single-walled carbon nanotubes with thermo-responsive poly(N-isopropylacrylamide): effect of the polymer architecture[END_REF] A preliminary UV-vis spectroscopy study allowed the determination of the wavelength of maximum absorbance (max) for 3 and 5 (342 nm and 343 nm, respectively, for solutions at ~ 10 -5 mol L -1 of each compound in 2-propanol (see SI, , blue curve on the top). After each addition of MNPs, the suspension was sonicated for 15 minutes at 20 °C, magnetically decanted and the emission spectra of the supernatant recorded and reported in these graphics (see details in Experimental Section).

Next, the influence of the temperature on the efficiency of the π-stacking interactions was studied. First and as before, the fluorescence emission spectra of 3 or 5 were recorded (Figure 3 The supernatant and the MNPs were next gathered, the mixture heated up to 80 °C for 2.5 h, and the spectrum of the resulting supernatant analyzed (Figure 3 a and b respectively, green curves).

The latter experiment indicated that the concentration of pyrenes 3 or 5 increased in accordance with a partial release of 3 and 5 from the surface of the MNPs at 80 °C. Upon cooling back to 20 °C, the concentration of 3 and 5 in the solution decreased (Figure 3 a and b respectively, purple curves) suggesting the complete reversibility of the π-stacking interactions. The concentration of 3 and 5 in solution are lower for purple curves than for red ones which suggests that the equilibrium was not attained after 15 minutes at 20 °C when recording red curves.

Next the hybrid MNPs were prepared according to the procedure described (see SI, Fig. S1).

Solutions of pyrene-tagged terpyridines 3 and 5 in 2-propanol were allowed to sonicate for 1h at 20 °C in the presence of MNPs. After magnetic decantation, the supernatant containing pyrenes non-specifically bound to the surface was separated and the hybrid MNPs were washed 10 times. MNPs, the loading could be determined. For both 3 and 5 ligands, a loading of 0.03 mmol of pyrene tag per g of MNPs was found which is significantly lower than the one obtained in the case of phosphine ligands previously reported (0.1 and 0.15 for pyrene-tagged ligands bearing one or 5 phosphines respectively 8b ). This lower loading may be rationalized by the absence of water known to favor attractive interactions between aromatic moieties and consequently stronger stacking behaviours. 8d

Next the catalytic activities of the Ru complexes involving 3 or 5 were evaluated in the transfer hydrogenation of nitrobenzene under the conditions previously reported by Beller and coll.

(Figure 4). 13a Therefore, the Ru precursor [{RuCl2(p-cymene)}2] and the ligand were first heated at 60 °C for 30 minutes before MNPs were added and the mixture sonicated for 30 minutes at 20 °C. Next KOH and nitrostyrene were introduced and the reaction allowed to reflux for 15h, time necessary to observe full conversion in the case of both ligands. After cooling down to room temperature, the crude was sonicated for 30 minutes at 20 °C and magnetic decantation was achieved with the magnet. The MNPs were rinced twice with 2-propanol and the joined solutions analyzed. Both Ru complexes involving 3 and 5 yielded the expected aniline selectively in quantitative yield as confirmed by 1 H NMR (using 1,3,5-trimethoxybenzene as the standard). To the dried MNPs, the nitrobenzene and 2-propanol were added to perform the 2 nd run and the same procedure was applied for each run. While a decrease of the activity was observed from the 6 th run in the case of terpyridine 3 (formation of expected aniline and also of diazobenzene, Fig. 4 footnote), the catalytic system involving multivalent ligand 5 could be reused 8 times with no loss of activity. In the latter case, the supported system thus enabled to improve at least by eight the turnover number of the parent molecular catalyst (at least 160 instead of 20).

It is worthy to note that when the reaction was performed with 2,2′:6′,2′′-terpyridine instead of pyrene-tagged terpyridines 3 or 5, the catalytic system could not be recycled since the MNPs recovered by magnetic decantation displayed no catalytic activity. This suggested that the pyrene tag is required for the Ru-terpyridine complexes interact with the MNPs graphene surface. In the absence of Ru(II), a competitive interaction of terpyridines with the graphene surface can however not be excluded. Indeed, the extinction was observed after the addition of 2 mg of MNPs in the case of ligand 5 bearing 5 terpyridine moieties while 6 mg were necessary in the case of ligand 3

(for a same initial concentration of pyrene tags, Figure 2). This might suggest a non-innocent role of the terpyridine moieties. However, such terpyridine-graphene interactions are unlikely to occur when the terpyridine ligands are complexed to Ru(II) for both electronic and steric reasons, wellknown Ru(II)-terpyridine complexes displaying an octahedral geometry. 13b Indeed, the addition of MNPs on a solution of the Ru(II) complex prepared from dichloro(p-cymene)ruthenium(II) dimer) and 2,2′:6′,2′′-terpyridine in 2-propanol did not induce any decrease of the absorption intensity by UV-vis spectroscopy. Accordingly, previous studies focused on the interactions between pyrenetagged terpyridine-metal complexes and graphene did not report the existence of terpyridinegraphene interaction. 18b

With ligand 3 [a] With ligand 5 [b] Figure 4: Catalytic performances of Ru catalysts for the transfer hydrogenation of nitrobenzene into aniline [a] [RuCl2(p-cymene)]2 (0.0125 mmol), pyrene-tagged 3 (0.025 mmol), MNPs, KOH (0.075 mmol), PhNO2 (0.5 mmol), 2-propanol (10 mL). For the 6 th run: 4% of azobenzene; for the 7 th run: 17% of azobenzene and 15% of azoxybenzene; for the 8 th run: 10% conversion of PhNO2. [b] [RuCl2(p-cymene)]2 (0.0125 mmol), pyrene-tagged 5 (0.005 mmol), MNPs, KOH (0.075 mmol), PhNO2 (0.5 mmol), 2-propanol (10 mL). Yields were determined by 1 H NMR spectroscopy using 1,3,5-trimethoxybenzene as the standard.

For recycling experiments: see general procedure in Experimental Section and SI p. 5.

The Ru amount present in the crude obtained after magnetic decantation was measured by ICP-MS at each of the 6 first runs. In the case of both ligands, the most important Ru leaching was observed for the first two runs (see SI, Table S1, entries 1-2). From the 3 rd run, the amount of Ru leached was found to be low and stable (around 2-7 ppm in the case of 3, less than 4 ppm in the case of 5, see SI, Table S1). Noteworthy, the criteria for the final active pharmaceutical ingredients specify that the residual ruthenium has to be less than ten parts-per-million (< 10 ppm 21 ).

Therefore, the amines prepared met the requirements of the pharmaceutical industry from the third run whatever the ligand. The Ru leaching detectable on the first runs may correspond to Ru not specifically bound to the MNPs. HRTEM, scanning transmission electron microscopy (STEM) and energy dispersive X-ray (EDX) spectra were recorded to get insight into the morphology of MNP functionalized by Ru complexes involving ligand 5. Before catalysis (Figure 5a), whatever the size of MNPs (commercial MNPs used are widely polydisperse by nature [5][6][START_REF] Schaetz | Immobilization on a nanomagnetic Co/C surface using rom polymerization: Generation of a hybrid material as support for a recyclable palladium catalyst[END_REF][8] ), the magnetic cores (dark areas) were found to be surrounded by amorphous grey shells containing the characteristic elements of the Ru-5 complex (P, N, Ru, Cl, see SI Table S2b). Elemental mapping confirmed the main localization of atomic phosphorous, nitrogen and ruthenium around the cobalt magnetic cores (SI Table S3).

After 5 catalytic cycles, no obvious change in the morphology of MNPs could be noticed with cobalt cores still being surrounded by organic shells (Figure 5b). EDX spectra still highlighted the presence of N, P and Ru while Cl could not be detected anymore which is consistent with the formation of the active Ru-H complex from KOH and 2-propanol from the first run (SI Table S4). 13a Noteworthy, an analogous morphology was found for hybrid MNPs obtained from 3 before catalysis (SI, Table S2a). However, after 5 catalytic cycles (SI, Table S5), some agglomeration of MNPs could be highlighted which could contribute to explain the loss of activity observed from the 6 th cycle when using the monomeric ligand 3 (Figure 4). Finally, recycling experiments were achieved by changing the nitroarene at each run (Figure 6). As expected, the yields of anilines bearing different substituents (halide, electron-poor or electron donating) formed from 4-nitrotoluene, 4-chloronitrobenzene, 1-nitronaphtalene or 4methoxynitrobenzene were high and comparable to those obtained in the case of the parent molecular Ru-terpyridine complex (GC yields for each aniline: 92-96% 13a ). Noteworthy, the latter is one of the most active and selective homogeneous catalyst known to date for the transfer hydrogenation of nitroarenes using 2-propanol as a cheap, benign and easy-to-handle H-donor. To the best of our knowledge, related reported heterogeneous and homogeneous systems, in particular those based on 3d-non-noble metals, are mainly associated to hazardous hydrazine derivatives or expensive and waste-generating sodium borohydride. 11b The reported methodology thus appears as complementary of existing ones. The anilines formed quantitatively can be easily isolated from the crude after a straightforward work-up including magnetic decantation, washings of the MNPs and evaporation of 2-propanol (see SI Figure S2). [a] [RuCl2(p-cymene)]2 (0.0125 mmol), pyrene-tagged 5 (0.005 mmol), MNPs, KOH (0.075 mmol), ArNO2 (0.5 mmol), 2-propanol (10 mL) for the first run and general procedure for recycling experiments (see Experimental section and SI p. S5). [b] Yields were determined by 1 H NMR spectroscopy using 1,3,5trimethoxybenzene as the standard. For the fifth run, p-toluidine was directly isolated from the crude obtained after magnetic decantation of MNPs and evaporation of 2-propanol (90% isolated yield, see SI, Figure S2).

CONCLUSION

To conclude, two Ru complexes involving pyrene-tagged terpyridine ligands (either monovalent 3 or multivalent 5) active in nitroarene transfer hydrogenation could be successfully supported on MNPs through -stacking interactions. One main advantage of these systems is that the catalyst is free in refluxing 2-propanol where the H-transfer occurs thus allowing to reach the same activity than the parent molecular terpyridine-Ru catalyst. Besides, when back to roomtemperature, the interactions between the pyrene-tagged complexes and the MNPs are restored enabling the efficient recovery of the catalysts by simple magnetic decantation with a magnet. The catalyst involving the multivalent ligand 5 can be reused at least 8 times and the anilines prepared meet the requirements of the pharmaceutical industry in terms of Ru contaminants (< 10 ppm) from the third run. This system is compatible with an organic environmentally-friendly solvent, do not require chromatographic purification and displays a high activity upon recycling which corresponds to the main challenges that catch-and-release catalysts have generally still to face. 9a Besides, the occurrence of interactions between the pyrene-tagged catalyst and the surface of the MNPs are supported by a network of facts including (i) the fluorescence studies bringing qualitative information, (ii) the quantitative microanalyses achieved on hybrid MNPs involving ligands 3 or 5 after several washings, (iii) the HRTEM images showing an homogeneous repartition of the organic ligands around the MNPs, (iv) the fact that the complex formed by same Ru precursor with 2,2′:6′,2′′-terpyridine (without pyrene tag) could not be recovered with a magnet.

In addition, the existence of reversible interactions between the pyrene tag and the surface of Co/C

MNPs was assessed for the first time in non-aqueous medium which opens new prospects for the use of these MNPs as supports in various organic reactions. In addition, terpyridines being very widely used ligands, [START_REF] Wei | Terpyridine-metal complexes: Applications in catalysis and supramolecular chemistry[END_REF] the systems reported here may find applications in many other catalytic reactions or even in other areas. Along these lines, further studies are currently underway in our laboratories.

ASSOCIATED CONTENT Supporting Information: Procedures to prepare the ligands 2, 3 and 5 and their characterization (NMR and mass spectra); grafting measurements and details about catalytic reactions and determination of Ru amounts present in the crude after each run by ICP-MS ; HRTEM images and related EDX spectra. The Supporting Information is available free of charge on the ACS Publications website.

Figure 1 : 2 .

 12 Figure 1: Targeted pyrene-tagged ligands involving either 1 (a) or 5 (b) terpyridine ligands and their corresponding expected hybrid MNPs

Figure 2 :

 2 Figure 2: Fluorescence emission spectra of pyrenes 3 (a) and 5 (b) in 2-propanol at 20 °C in the absence of Co/C-MNPs (10-5 mol L -1 , blue curve on the top). After each addition of MNPs, the suspension was sonicated for 15 minutes at 20 °C, magnetically decanted and the emission spectra of the supernatant recorded and reported in these graphics (see details in Experimental Section).

Figure 3 ,Figure 3 :

 33 Figure 3: Reversibility experiments: a solution of 3 (a) or 5 (b)(10 -5 mol L -1 , blue curve for reference emission spectra) and MNPs was sonicated for 15 minutes at 20 °C, magnetically decanted and the fluorescence spectrum of the supernatant recorded (red curve). The mixture (supernatant and MNPs) was then heated to 80 °C for 2.5 h, and the spectrum of the resulting supernatant analyzed (green curve). Both the supernatant and the MNPs were next gathered and the mixture sonicated at 20 °C for 2h, the concentration of 3 (a) or 5(b) in the solution decreased (purple curve) (see details in Experimental Section).

  were next analyzed by microanalysis and by comparison to the naked Co/C

Figure 5 :

 5 Figure 5: Hybrid MNPs obtained from 5 before catalysis (a) and after 5 cycles (b).

Figure 6 :

 6 Figure6: Recycling tests by changing the nitroarene at each run (case of ligand 5)[a]

ACKNOWLEDGMENT

The authors gratefully thank CNRS for funding, Labex CheMISyst -ANR-10-LABX-05-01 for HA grant and Erwan Oliveiro, director of the Microscopy Platform, University of Montpellier.

AUTHOR INFORMATION

Corresponding Author Armelle Ouali. Institut Charles Gerhardt Montpellier ICGM, Univ. Montpellier, CNRS, ENSCM, Montpellier 34090, France; orcid.org/0000-0001-7436-776X. Email: armelle.ouali@enscm.fr.

Authors

Houria Asri. Institut Charles Gerhardt Montpellier ICGM, Univ. Montpellier, CNRS, ENSCM, Montpellier 34090, France. Olivier Dautel. Institut Charles Gerhardt Montpellier ICGM, Univ. Montpellier, CNRS, ENSCM, Montpellier 34090, France. orcid.org/0000-0003-2327-5376.

Author Contributions

The manuscript was written through contributions of all authors and all authors have given approval to the final version of the manuscript. H.A. and A.O. participated in the preparation of ligands and catalysis experiments and D.O. carried out the fluorescence spectroscopic studies. A.O. designed the investigations and coordinated the project.

Notes

The authors declare no competing financial interest.


Élie Gouzien

Floriane Brunel

Sébastien Tanzilli

Virginia D'auria

Scheme for the generation of hybrid entanglement between time-bin and wavelike encodings

We propose a scheme for the generation of hybrid states entangling a single-photon time-bin qubit with a coherent-state qubit encoded on phases. Compared to other reported solutions, timebin encoding makes hybrid entanglement particularly well adapted to applications involving longdistance propagation in optical fibers. This makes our proposal a promising resource for future outof-the-laboratory quantum communication. In this perspective, we analyze our scheme by taking into account realistic experimental resources and discuss the impact of their imperfections on the quality of the obtained hybrid state.

1. Losses on hybrid states with time-bin encoding

I. INTRODUCTION

Over the past years, quantum optics information has traditionally followed two distinct approaches, naturally stemming from light wave-particle complementarity [START_REF] Andersen | Hybrid discrete-and continuousvariable quantum information[END_REF]. Discrete variable regime (DV) usually refers to weakly excited optical states, down to single photons, for which information is encoded on discrete spectrum observables such as the polarization or the number of photons [START_REF] Northup | Quantum information transfer using photons[END_REF]. Conversely, continuous variables regime (CV) relies on multiphoton optical states and to encodings on continuous spectrum observables such as amplitude and phase of a light field [START_REF] Adesso | Continuous variable quantum information: Gaussian states and beyond[END_REF]. DV encoding is tolerant to losses and allows high-fidelity teleportation, while CV encoding permits deterministic state generation and unambiguous state discrimination [START_REF] Lie | Limitations of teleporting a qubit via a two-mode squeezed state[END_REF].

Recently, hybrid states entangling DV and CV encoding have been identified as a key tool to switch from one approach to the other and gather the benefits of both [START_REF] Andersen | Hybrid discrete-and continuousvariable quantum information[END_REF][START_REF] Brask | Hybrid long-distance entanglement distribution protocol[END_REF][START_REF] Lee | Near-deterministic quantum teleportation and resource-efficient quantum computation using linear optics and hybrid qubits[END_REF][START_REF] Parker | Hybrid photonic loss resilient entanglement swapping[END_REF][START_REF] Lim | Loss-resilient photonic entanglement swapping using optical hybrid states[END_REF][START_REF] Van Loock | Hy-brid quantum computation in quantum optics[END_REF]. This perspective has motivated an increasing number of theoretical works [START_REF] Kwon | Generation of hybrid entanglement between a single-photon polarization qubit and a coherent state[END_REF][START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF][START_REF] Bergmann | Hybrid quantum repeater for qudits[END_REF][START_REF] Omkar | Resource-efficient topological fault-tolerant quantum computation with hybrid entanglement of light[END_REF][START_REF] Park | Quantum teleportation between particlelike and fieldlike qubits using hybrid entanglement under decoherence effects[END_REF] as well as experiments on hybrid state generation [START_REF] Jeong | Generation of hybrid entanglement of light[END_REF][START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF] or use in proof-of-principle quantum information protocols with single-rail or polarization encodings for the DV part [START_REF] Ulanov | Quantum teleportation between dis-crete and continuous encodings of an optical qubit[END_REF][START_REF] Cavaillès | Demonstration of einsteinpodolsky-rosen steering using hybrid continuous-and discrete-variable entanglement of light[END_REF][START_REF] Jeannic | Remote preparation of continuous-variable qubits using loss-tolerant hybrid entanglement of light[END_REF][START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF]. At the same time, practical quantum communication and networks will require the distribution of hybrid entanglement over long distances, where high losses or polarization instability can play a significant role.

This work addresses future applications of hybrid entanglement for long-distance operation over optical fiber links, by proposing a scheme for the generation of hybrid entangled states with time-bin encoding on their discrete variable part. In the time-bin scheme, information is encoded on two generation or detection times for photons, usually labeled as "early" (e) and "late" (l) and generally obtained by exploiting a Franson interferometer [START_REF] Franson | Bell inequality for position and time[END_REF]. Compared to photon-number or polarization encodings, time bin allows one to comply in a better way with losses, is immune to polarization dispersion, and is particularly well adapted to quantum communication * virginia.dauria@inphyni.cnrs.fr over long optical fibers [START_REF] Gisin | Quantum cryptography[END_REF]. As a consequence, the realization of hybrid entanglement with time-bin encoding is extremely important in view of out-of-laboratory applications. Our scheme permits one to generate the desired state in a heralded fashion without any postselection operation and relies, as enabling resources, on the interference between experimentally achievable states, i.e. an optical Schrödinger cat state [START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF][START_REF] Sychev | Enlargement of optical schrödinger's cat states[END_REF], and a time-bin entangled photon pair [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF]. In order to comply with future practical realizations, it has been conceived so as to be experimentally feasible and fully compatible with existing fiber architectures and with realistic experimental resources, including nonideal heralding detectors [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF][START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF] and input states [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF].

In the following, we illustrate our scheme in detail. For pedagogical reasons, in Sec. II we analyze it in the case of perfect input states and ideal heralding detectors. In Sec. III, we investigate more realistic situations. We start by considering for the heralding feature a minimum number of on-off single-photon detectors [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF][START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF], with no photon-number-resolving ability and finite efficiency. We then examine the impact of realistic input states such as photon pairs generated by a nonlinear process along with vacuum and multiple pairs [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF] on the DV part, and a squeezed vacuum as an approximation of a Schrödinger cat on the CV one [START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF]. We show that our scheme is resistant to these experimental limitations by discussing the heralded state fidelity with respect to the targeted one.

II. PROPOSED EXPERIMENT A. Generation scheme of hybrid entanglement with time-bin DV encoding

The aim of our work is to generate a DV-CV hybrid state, entangling a DV time-bin qubit with a bright CV qubit encoded on two coherent states with opposite phases. We define such a target state as where, for the DV part (mode A), e and l stand for the early and late time bins and the |1 A,i indicates a single photon state in the temporal mode i. For the CV part (mode B), |±α f B is a coherent state of amplitude ±α f . To simplify the notation, in Eq. ( 1) we omitted the vacuum states |0 A,l and |0 A,e that multiply the CV terms |+α f B and |-α f B , respectively. The density matrix associated with this state is |ϕ ϕ|.

|ϕ = |1 A,e |+α f B -|1 A,l |-α f B √ 2 , (1)
The experimental scheme at the heart of our proposal is presented in Fig. 1. Conceptually, it requires pure CV and DV states as initial resources and an interferometric scheme able to entangle them upon the result of a suitable measurement operation. In this section we discuss the case of ideal inputs, whereas more realistic states from the experimental point of view will be treated in the following. Note that the entire realization refers to the case of an experiment operated in pulsed regime, as required for the time-bin encoding.

At the CV input, labeled as mode 3 in the figure, we consider an even Schrödinger cat. In terms of displacement operator D3 (±α), this state can be written as

|cat+ 3 = D3 (α) + D3 (-α) N |0 , (2)
where we recall that D3 (±α) |0 = |±α 3 , i.e. a coherent state of amplitude ±α and N = √ 2 1 + e -2|α| 2 [START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF]. Following an approach analogous to previously reported experiments [START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF], the CV input state |cat+ 3 is sent to an unbalanced beam splitter (BS1) with field reflection and transmission coefficients r and t, respectively. We note that unbalanced and variable beam splitters are easily available in both bulk and fiber configurations [START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF][START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF][START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF]. After the BS1, the state reads

|ψ B,5 = DB (tα) D5 (rα) + DB (-tα) D5 (-rα) N |0 . (3)
Mode B represents the CV part of our final hybrid state. Mode 5 is directed towards the interferometric part of the scheme so as to be subsequently mixed with the DV input state. This action actually permits bridging the CV and the DV parts of the state. More precisely, light in mode 5 is sent to an unbalanced Mach-Zehnder interferometer, where it is mixed with a coherent state of amplitude rα at the input of a 50:50 beam splitter (BS2). As for pure DV experiments [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF], the length of the interferometer arms, 6 and 7, define the late and early time bins required for the encoding. The state at the output of BS2 can be easily computed by recalling that given two input coherent states |γ 5 and |γ 4 , the output of a balanced beam splitter can be obtained by using the relation

D5 (γ) D4 (γ ) = D6 ( γ-γ √ 2 ) D7 ( γ+γ √
2 ) [START_REF] Paris | Displacement operator by beam splitter[END_REF]. For the case under examination, γ = ±rα and γ = rα. Accordingly, due to the interference with the coherent state on mode 4, depending on the sign of ±rα on mode 5, light is routed only towards mode 6 (for -rα) or mode 7 (for rα). As a consequence, right before the balanced beam splitter BS3, we obtain the state

|ψ B,6,7 = 1 N ( DB (tα) D6,l ( √ 2rα)+ DB (-tα) D7,e (- √ 2rα)) |0 . ( 4
)
In the latter expression we use a double index notation to explicitly recall the temporal mode associated with spatial modes 6 and 7.

To obtain the desired hybrid state, light coming from the CV part, and prepared in time-bin modes by the Mach-Zehnder interferometer, must be mixed with the discrete variable part. This is provided by a pair of time-bin entangled photons launched in the input spatial modes 1 and 2 [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF]:

|ξ 1,2 = |1 1,e |1 2,e + |1 1,l |1 2,l √ 2 . ( 5
)
One of the photons is directly routed towards output mode A and represents the DV part of the hybrid state. Its twin, on mode 2, is sent to the balanced beam splitter BS4 so as to be spatially mixed with the two outputs of the Mach-Zehnder interferometer thanks to BS5 and BS6 (see Fig. 1). Balanced beam splitters BS5 and BS6 erase the "which path" information, right before the heralding detectors (in spatial modes C, D, E, F ), such that a given click event from one of the four detectors cannot be attributed to light coming from a certain origin (i.e. from the CV or the DV part). Right before the detectors, the state reads

ψ (1) = 1 2 √ 2N |+tα B |1 A,
where |0 l indicates |0 C,l |0 D,l |0 E,l |0 F,l and analogously for |0 e . We note that rα √ 2 corresponds to the amplitude of the light beam coming from the continuous variable part and reaching the detectors. In the notation ψ (1) , label 1 indicates that we consider exactly one photon pair on the discrete variable input.

Light in the four spatial modes C, D, E, F is measured by using single-photon detectors; for each of them two temporal modes (l and e) must be considered, thus leading to eight possible heralding modes. The combination of their detection signals heralds the hybrid state on A and B.

We will consider here the ideal case of photon-numberresolving detectors with perfect detection efficiency [START_REF] Kwon | Generation of hybrid entanglement between a single-photon polarization qubit and a coherent state[END_REF][START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF] and herald the hybrid state by the simultaneous detection of signals corresponding to the measurement of one photon on detector E in the late time bin, one photon on detector F in the early time bin, and no photon in the six remaining heralding modes. The associated positive operator valued measurement (POVM) [START_REF] D'auria | Quantum decoherence of single-photon counters[END_REF] reads

Πid = |1 1| E,l |1 1| F,e i |0 0| i , (7)
where the label i indicates the heralding modes (C, e), (C, l), (D, e), (D, l), (E, e) and (F, l). In the previous equation, identity is implicit on unmeasured channels A and B. As it can be seen, only the first and the fourth terms of Eq. ( 6) simultaneously contain light in the (E, l) and the (F, e) modes, and can lead to a detection event as described by Πid . We note that the detected photons are provided one by the DV and the other by the CV part of the scheme. On the contrary, the second and the third terms contain light only in one of the two time bins and do not contribute to the announced states. Accordingly, as desired, the chosen heralding strategy exactly leads to the target state of Eq. ( 1):

ρid = Tr CDEF Πid ψ (1) ψ (1)
Tr Πid ψ (1)

ψ (1) = |ϕ ϕ| , (8)
where the amplitude of the CV qubit is α f = tα. The target state is heralded with a probability:

P id = Tr Πid ψ (1) ψ (1) = 1 16 |rα| 2 e -2|rα| 2 1 + e -2|α| 2 . ( 9
)
As it can be seen, equality [START_REF] Lim | Loss-resilient photonic entanglement swapping using optical hybrid states[END_REF] holds true for any choice of the fraction of light, rα, that is subtracted from the CV part and mixed with the DV one. However, as shown in Fig. 3 (dashed lines), the heralding probability is strongly affected by rα. For a fixed size of the initial cat state, i.e. for a given α, P id increases with rα up to an optimal value and then decreases for higher rα values. This behavior is easily justified by the heralding choice. For rα → 0, the fraction of the continuous variable part directed towards detection modes is extremely weak and the probability of obtaining a detection signal from both (E, l) or (F, e) is low. On the contrary, for high rα, detection of light on modes C, D, (E, e) and (F, l) has an important probability to be triggered by the continuous variable component of ψ (1) and, as for state preparation we impose the absence of photons on these detectors, the overall heralding probability decreases. We also stress that for increasing rα, the condition of detecting only one photon on heralding modes (E, l) or (F, e) is no longer respected as multiple photon contributions become non-negligible.

B. Hybrid states with time-bin encoding in long-distance applications

Hybrid DV-CV entangled states are essential for any quantum networks connecting disparate quantum devices based on CV or DV encodings [START_REF] Andersen | Hybrid discrete-and continuousvariable quantum information[END_REF] and are at the heart of a new generation of quantum communication protocols [START_REF] Bergmann | Hybrid quantum repeater for qudits[END_REF][START_REF] Omkar | Resource-efficient topological fault-tolerant quantum computation with hybrid entanglement of light[END_REF][START_REF] Park | Quantum teleportation between particlelike and fieldlike qubits using hybrid entanglement under decoherence effects[END_REF]. At the same time, these applications require the distribution to remote nodes of one or both parts of the hybrid states; it is thus crucial to discuss their robustness in the context of long-distance operation. We will consider here the experimentally relevant case of fiber connections submitted to propagation losses and dispersion.

Loss effect can be modeled by inserting unbalanced BSs with reflection coefficients r CV and r DV on the paths of the CV or DV parts of the states (see Appendix A). In practical scenarios, the reflection coefficient goes with the propagation distance, z, as √ 1e -βz where, in optical fibers, β ≈ 0.2 dB/km at its best [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF]. Losses on the CV part of a hybrid entangled state lead to decoherence and result in a degradation of the state purity with off-diagonal terms (coherences) of the state density matrix exponentially scaling as e -2|r CV α f | 2 . In addition, the state coherent amplitude is reduced as ± 1r 2 CV α f . These limitations are independent of the chosen DV encoding and they are common to any hybrid CV-DV state submitted to losses in the CV part. Conversely, striking advantages of time-bin encoding appear when considering loss and propagation effects on the DV part of the state.

Losses on the DV part transform |ϕ ϕ| into

ρ = t 2 DV |ϕ ϕ| + r 2 DV |0 0| A |+α f +α f | B + |-α f -α f | B 2 , (10)
where t 2 DV = 1r 2 DV and we kept the label A for the DV part after the loss-beam splitter and B for the CV part. An expression similar to ρ can be derived starting from ρid . The first term of Eq. ( 10) contains the initial state multiplied by t 2

DV and unaffected by decoherence effects. The second term, proportional to r 2 DV , is a nonentangled hybrid state with the DV part, A, in a vacuum state; accordingly, in practical applications, this term can be traced out by DV detection operation as it happens in pure DV regime. As widely discussed in the literature [START_REF] Gisin | Quantum cryptography[END_REF], dispersion effects on time-bin encoding play no significant role and can be easily compensated by standard modules. No decoherence effect is thus observed on the state |ϕ ϕ| when its DV part is submitted to losses or propagates over long distances. Remarkably, these relevant features are not available to other kinds of reported hybrid entanglement. When their DV part propagates over long optical fibers, hybrid entangled states with single-rail (presence or absence of a single photon) or polarization DV encodings are strongly affected by losses and polarization dispersion. As a consequence, they suffer from a decoherence effect going exponentially with the propagation distance (see Appendix A).

The robustness of hybrid entangled states with timebin encoding makes them excellent resources for protocols where local manipulation and detection of a part of the hybrid state are used to herald at distance a certain quantum operation. As an example, we consider the simple case of remote preparation of a CV quantum state upon the results of the measurement of the DV part. This scheme allows transferring quantum information between distant nodes, in a configuration where, in contrast with quantum teleportation, the sender has complete knowledge of the state to be communicated [START_REF] Jeannic | Remote preparation of continuous-variable qubits using loss-tolerant hybrid entanglement of light[END_REF]. Figure 2 reports the fidelity of the prepared CV state with the target one versus the distance traveled by the DV part. Hybrid states with time-bin, single-rail and polarization DV encoding are compared. Beside losses, pertinent dispersion effects have been considered. As it can be seen, the fidelity obtained in the case of time bin is unaffected by the traveled distance, while the performances of hybrid state with single rail and polarization encoding degrade with it. We stress that considerations similar to the ones that we discuss for this protocol can be easily extended to other heralded protocols such as teleportation or entanglement swapping, thus confirming the advantage of hybrid entangled with time-bin encoding for practical quantum communication. For the polarization dispersion, we considered the value reported in a recent out-of-the-laboratory experiment [START_REF] Wengerowsky | Entanglement distribution over a 96-km-long submarine optical fiber[END_REF]. We assumed a DV mode projected onto

|1 A,l +|1 A,e √ 2
for hybrid entanglement with time-bin DV encoding and similar projections for single-rail and polarization DV encodings (see Appendix A). For the initial amplitude of the CV part of the hybrid states, we set α f = 2. The distance traveled by the CV part is taken as negligible for the three states.

III. ROBUSTNESS OF THE GENERATION SCHEME AGAINST EXPERIMENTAL LIMITATIONS A. Imperfect detection and simplified heralding strategy

The ideal case discussed in the previous section relies on the possibility of detecting, i.e., two temporal modes for each of the four spatial modes C, D, E, F . This can be seen from the shape of POVM ( 7) that explicitly contains projectors over all the eight heralding modes. In experiments, the strategy described in Sec. II would imply that each of the heralding detectors should be able to measure both early and late time bins. This implies a separation between time bins greater than the detector dead time or, in alternative, the use of extra detectors so as to map each time bin in an additional spatial mode [START_REF] D'auria | Quantum decoherence of single-photon counters[END_REF]. These solutions have dramatic consequences for the experiment maximum operation rate and required overhead, respectively. At the same time, we note that photon-number-resolving detectors, although often introduced in the literature on hybrid state [START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF], are hardly available off the shelf and, so far, most of the demonstrations involving some counting ability rely on complex spatial multiplexing strategies based on detector arrays [START_REF] Feito Boirac | Measuring measurement: theory and practice[END_REF][START_REF] Schmidt | Characterization of a photon-number resolving SNSPD using poissonian and sub-poissonian light[END_REF].

Heralding probability

FIG. 3. Fidelity of the heralded hybrid state to the target state |ϕ ϕ| (top) and heralding probability (bottom) as a function of rα, for the two proposed heralding protocols: Πid (dashed line) and Π (plain lines). The ideal case is discussed in Sec. II and refers to perfect photon number resolving detectors with unitary efficiency. All other cases refer to on-off detector, whose quantum efficiencies, η, have been chosen according to typical experimental values [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF][START_REF] Comandar | Gigahertz-gated ingaas/inp single-photon detector with detection efficiency exceeding 55% at 1550 nm[END_REF][START_REF] Marsili | Detecting single infrared photons with 93% system efficiency[END_REF]. For the input CV state, we set α = 2. The fidelity and the product rα are adimensional quantities.

In view of future experimental realizations, in this section, we discuss our scheme in the case of a more realistic detection scenario. A practical simplification of our protocol consists of heralding the target state only upon the detection signals from the late time bin in mode E and from the early time bin in mode F , thus choosing to disregard all other six heralding modes that describe the state before the detector (see Eq. ( 6)). By doing so, the heralded state will no longer correspond to ρid . At the same time, such an approach considerably reduces the number of detected modes from eight to two. By doing so, only two gated detectors are required and, as each of them must detect only one temporal mode, dead times are no longer a limiting factor to the operation rate [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF]. At the same time, we consider, at the heralding modes, on-off single-photon detectors, with nonunit detection efficiency η and no photon-number-resolving ability [START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF]. These systems are able to provide only the two generic responses: on, i.e. "at least one photon has been detected", and off, i.e. "no photon has been detected", and they represent the vast majority of available singlephoton counters. Their action is described by the positive operators [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF][START_REF] D'auria | Quantum decoherence of single-photon counters[END_REF][START_REF] Feito Boirac | Measuring measurement: theory and practice[END_REF]:

Πoff i = +∞ k=0 (1 -η) k |k k| i , (11a)
Πon i = 1 -Πoff , (11b)
where i indicates the heralding mode under investigation and, only for the previous equations, |k i indicate a Fock state of mode i containing k photons.

The measurement positive valued operator associated with the simplified heralding strategy and with on-off gated detectors reads

Π = Πon E,l ⊗ Πon F,e , (12)
identity being implicit on all six unmentioned modes (i.e. C, D, (E, e), (F, l)) as they are not measured. We note, in particular, that, with this measurement strategy, the scheme no longer relies on projection on vacuum states.

Analogous to what is done in Eqs. ( 8) and ( 9), the heralded state can be computed from Π:

ρ(1) = 1 2 |α f α f | B |1 1| A,e - η |rα| 2 2 e -2|rα| 2 1 -e -η |rα| 2 2 |α f -α f | B |1 0| A,e |0 1| A,l + |-α f +α f | B |0 1| A,l |1 0| A,e + |-α f -α f | B |1 1| A,l . (13)
Its associated heralding probability reads

P (1) = η 8 1 -e -η |rα| 2 2 1 + e -2|α| 2 , (14)
where, as before, we used the notation tα = α f . The state given by ( 13) belongs to the qubit subspace generated by |+α f B |1 A,e and |-α f B |1 A,l , as the density matrix |ϕ ϕ|, and correctly tends to it when rα → 0.

This can be seen from the fidelity of the heralded state to the target state:

F (1) = ϕ|ρ (1) |ϕ = 1 2 1 + η |rα| 2 2 e -2|rα| 2 1 -e -η |rα| 2 2 . ( 15
)
The heralding probability, P 1 , and fidelity F (1) are plotted in Fig. 3 as functions of rα for different detector efficiencies η. In this regard we underline that the fidelity is a commonly adopted and pertinent criterium to check the validity of any scheme aiming at the generation of a given target state [START_REF] Kwon | Generation of hybrid entanglement between a single-photon polarization qubit and a coherent state[END_REF][START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF]. The efficiency η has a clear impact on the heralding probability, P 1 , on the order of η 2 in the limit of rα → 0. Compared to the ideal case described in the previous section (dashed lines), the heralding probability is no longer decreasing for high rα, as the heralding strategy is no longer sensitive to spurious events firing one of the disregarded modes C, D and (E, e), (F, l). Nevertheless, the bad effect of multiphoton contributions coming from the CV part can be clearly seen on the state fidelity, F (1) , that decreases for increasing rα. Detection efficiency η has little influence on the fidelity. This weak dependency arises from the shape of the on operator in Eq. [START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF].

In experiments, for a fixed input state |cat+ 3 , i.e. for a given α, and for given detector quantum efficiency η, the reflection coefficient r can be chosen so as to reach a desired value for the fidelity. As an example, we consider α = 2 as in Refs. [START_REF] Jeong | Generation of hybrid entanglement of light[END_REF][START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF][START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF][START_REF] Sychev | Enlargement of optical schrödinger's cat states[END_REF] and realistic single-photon detectors based on superconducting nanowire technology exhibiting η = 0.95 [START_REF] Marsili | Detecting single infrared photons with 93% system efficiency[END_REF] working in a gated operation mode. With these parameters, a high target fidelity F (1) = 0.99 imposes rα √ 2 = 0.075, thus giving r ≈ 0.053. Correspondingly, the heralding probability P (1) ≈ 6.4 • 10 -4 . Working with a 1 GHz repetition rate laser [START_REF] Ngah | Ultra-fast heralded single photon source based on telecom technology[END_REF], and fast single-photon detectors compatible with such a fast operation regime, as those reported in Ref. [START_REF] Marsili | Detecting single infrared photons with 93% system efficiency[END_REF], leads to a heralding rate of 0.64 MHz. Note that, as the fraction of light subtracted from the CV input is very small, the size of the CV part of the final hybrid state is α f ≈ 2.

To conclude, we note that, besides studying the fidelity, in some applied situations, it can be useful to quantify the produced states in terms of specific touchstones [START_REF] Kreis | Classifying, quantifying, and witnessing qudit-qumode hybrid entanglement[END_REF][START_REF] Agudelo | Conditional hybrid nonclassicality[END_REF], that can be chosen according to the specific quantum information protocol. The nonclassicality of the hybrid states as resources for quantum communication is often formalized by their ability to produce a nonclassical continuous variable state when performing a suitable projecting measurement on the discrete variable subsystem. This operational notion has been theoretically investigated and experimentally validated [START_REF] Cavaillès | Demonstration of einsteinpodolsky-rosen steering using hybrid continuous-and discrete-variable entanglement of light[END_REF][START_REF] Agudelo | Conditional hybrid nonclassicality[END_REF]. In Fig. 4, left, we plot the negativity of the Wigner function of the CV state obtained after having projected the DV part of ρ(1) on the state

|1 A,e +|1 A,l √ 2
. Analogous to what is observed for the fidelity, while being weakly dependent on the detection efficiency η of the on-off detectors, the negativity of the CV part clearly decreases with rα; at the same time, the Wigner function correctly shows negative values for low rα that correspond to the best fidelity with the hybrid target state |ϕ ϕ|. In order to focus more on the hybrid entanglement quantification, as discussed in [START_REF] Costanzo | Properties of hybrid entanglement between discrete-and continuous-variable states of light[END_REF], a valuable tool is the negativity of the partial transpose (NPT). NPT is proportional to the sum of the negative eigenvalues of a partially transposed state density matrix and verifies 0 ≤ NPT ≤ 1; separable and Bell states are respectively valued at 0 and 1. The NPT of the state ρ(1) is shown by Fig. 4, right. The state is maximally entangled, NPT is close to 1, when rα tends to zero, and becomes factorizable when rα increases (NPT is then close to 0).

As a final remark, we stress that the behaviors of both the Wigner negativity and the NPT can be understood from the expression of ρ(1) . In the limit of small rα, offdiagonal terms in Eq. ( 13) scale as -1 + O (rα) 2 and ρ(1) ∼ |φ φ|+O (rα) 2 , with no contribution at the first order in rα. This gives the robustness of our state against imperfections and a zero slope to the Wigner negativity and the NPT when rα 1. When rα increases the contribution of off-diagonal terms of the density matrix gently decrease and, accordingly, the purity of announced hybrid state is reduced. This maps to a decrease of the Wigner negativity and NPT as shown in the figure. (left). In the ideal case discussed in Sec. II, the negativity of the Wigner function of the conditional state is constant with rα. The optimal value of 2 π ≈ 0.64 corresponds to the Wigner function definition W (x, p) =

1 π x + u 2 ρ(1) x -u
2 e -2ipu du. Negativity of the partial transpose (NPT) as a function of rα for the state ρ(1) (right). The ideal case refers to a hybrid state of Eq. ( 8) and to a perfect projective measurement. For the nonideal case, quantum efficiencies, η, of on-off detectors have been chosen according to typical experimental values [START_REF] Alibart | Quantum photonics at telecom wavelengths based on lithium niobate waveguides[END_REF][START_REF] Comandar | Gigahertz-gated ingaas/inp single-photon detector with detection efficiency exceeding 55% at 1550 nm[END_REF][START_REF] Marsili | Detecting single infrared photons with 93% system efficiency[END_REF]. Wigner negativity, NPT, and the product rα are adimensional quantities.

B. Vacuum and multiple pairs in the DV input

In this section, we further modify our model so as to take into account possible limitations due a more realistic model for the discrete variable input. The case of a nonideal CV input will be discussed in the last subsection.

So far, we have considered a perfect time-bin entangled state, |ξ 1,2 , at the input modes 1 and 2. In usual experiments, the generation of time-bin entangled photons typically relies on a parametric down conversion (SPDC) nonlinear process, where a pump photon is converted in a pair of photons (signal and idler) [START_REF] Gisin | Quantum cryptography[END_REF]. However, this kind of process suffers from unwanted generation of vacuum and multiple pair components [START_REF] Christ | Limits on the deterministic creation of pure single-photon states using parametric down-conversion[END_REF]. In order to discuss their effect, we explicitly include these contributions to the DV input state. Accordingly, we replace the initial state of Eq. ( 5) with the state:

|ξ 1,2 = √ p 0 |0 1,2 + √ p 1 |1 1,e |1 2,e + |1 1,l |1 2,l √ 2 + √ p ε |ε 1,2 , (16)
with p 0 + p 1 + p ε = 1, and where |ε 1,2 represents all multipair contributions. A more detailed discussion on the shape of |ξ 1,2 is given in Appendix B. In experimental situations, in order to reduce the impact of multipairs, the SPDC working point is chosen so as to satisfy the condition p 0 p 1 p ε , thus making the vacuum the most important contribution to |ξ 1,2 [START_REF] Ngah | Ultra-fast heralded single photon source based on telecom technology[END_REF]. We will make here this same choice.

As in Sec. II, we first write the state right before the detection. This reads

|ψ = √ p 0 ψ (0) + √ p 1 ψ (1) + √ p ε ψ (ε) , ( 17a
)
with ψ (1) as given by ( 6), As previously, the state is heralded on the simultaneous detection signals (E, l) and (F, e). As can be seen from (17b), for each of the terms of ψ (0) , only one of the two temporal modes (e, l) is populated. Accordingly, |ψ 0 has a zero probability to give the heralding signal and it will not contribute to the final heralded state. Conceptually, the density matrix of the heralded state has the following form:

ψ (0) = 1 N |tα B |0 A DC,l rα √ 2 DD,l - rα √ 2 DE,l - rα √ 2 DF,l rα √ 2 |0 C,D,E,F + |-tα B |0 A DC,e - rα √ 2
ρ = Tr CDEF Π |ψ ψ | Tr Π |ψ ψ | = 1 1 + pεP (ε)
p1P (1)   ρ(1) +

p ε P (ε) p 1 P (1) ρ(1,ε) + p ε P (ε) p 1 P (1) ρ(ε)   , (18)
with a corresponding heralding probability:

P = Tr Π |ψ ψ| = p 1 P (1) + p ε P (ε) . ( 19
)
In the previous expressions, P (1) is the same as given in Eq. ( 14) and P (ε) is the probability for ψ (ε) to give a heralding signal. The explicit expression of P (ε) can be analytically computed for a given multipair contribution, represented by |ε 1,2 . Similarly, ρ(1) is the density matrix already given in Eq. ( 13), ρ(ε) is a density operator containing multiple photons on mode A, and ρ(1,ε) is a traceless operator whose coefficients are on the order of 1, as required to preserve the positivity of ρ . For the sake of simplicity, we do not provide the explicit expressions of ρ(ε) and ρ(1,ε) but only discuss their relative weight compared to ρ(1) . We note that, as expected, the vacuum component of the discrete variable input is entirely rejected by the heralding process and does not enter the expression of the announced state ρ .

The impact of multipair contributions to the heralded state can be quantified in terms of the fidelity of ρ with the target density matrix |ϕ ϕ|. This reads as

F = 1 1 + pεP (ε) p1P (1)
F (1) , [START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF] with F (1) as expressed in Eq. [START_REF] Jeong | Generation of hybrid entanglement of light[END_REF]. For both the announced state, ρ , and the fidelity, F , multiphoton contributions increase with pεP (ε) p1P [START_REF] Andersen | Hybrid discrete-and continuousvariable quantum information[END_REF] . In order to evaluate the general form of this ratio, we note that in most of the situations multipair terms in Eq. ( 16) are dominated by double-pair contributions [START_REF] Ngah | Ultra-fast heralded single photon source based on telecom technology[END_REF] and, as a consequence, P (ε) ≈ P (2) , which is the probability of a double pair to give the heralding signal. In the limit of a small fraction of light coming from the CV part, it is reasonable to expect P (2) = rα→0 O η 2 , while, based on Eq. ( 14), for small α, P (1) = rα→0 η 2 |rα| 2 /32. Correspondingly, we obtain

p ε P (ε) p 1 P (1) ≈ rα→0 p ε p 1 1 O |rα| 2 . ( 21
)
In this limit, the fidelity F can be written as

F ≈ rα→0 1 1 + pε p1 1 O(|rα| 2 ) 1 -(1 - η 8 )|rα| 2 , ( 22
)
where the numerator is given by ( 15) for rα → 0. Equation [START_REF] Franson | Bell inequality for position and time[END_REF] shows that unwanted multipair contributions to the final heralded state can be avoided provided p ε p 1 |rα| 2 . In particular, for sources based on second-order nonlinear effects, p ε ∼ p 2 1 , and the criterion reads p 1 |rα| 2 . By adequately choosing the single pair generation rate p 1 and product rα it is thus possible to neglect the components containing multiple photons on the discrete part of the output state.

To conclude with an example, we consider the case of sources based on spontaneous parametric down conversion. In this case, the explicit expression for |ξ 1,2 can be obtained from the general expression of the output of the SPDC process [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF][START_REF] D'auria | Quantum decoherence of single-photon counters[END_REF], as a function of an excitation parameter λ 2 proportional to the pump intensity and to the square of the nonlinear coefficient of the source. The time-bin state generation can be seen as the result of an SPDC process on mode e and an SPDC process on mode l. By combining the coefficients of the two individual processes, we obtain for the overall generation the weights (see Appendix B)

p 0 = 1 -λ 2 2 , p 1 = 2 1 -λ 2 2 λ 2 , p 2 = 3 1 -λ 2 2 λ 4 . (23)
The explicit expression for the P (2) is

P (2) = η 48 12 -η -(12 -2η + η 2 2 |rα| 2 )e -η 2 |rα| 2 + ηe -2|α| 2 1 + e -2|α| 2 . (24)
In the limit of rα → 0, this expression leads to P (2) = rα→0 η 2 /48 and, as expected, correctly behaves as η 2 . More explicitly, based on Eqs. ( 23) and ( 24), we find for the SPDC

p 2 P (2) p 1 P (1) ≈ rα→0 λ 2 |rα| 2 . (25)
The Fig. 5 (dashed line) shows the fidelity F as a function of the parameter λ 2 . We considered, as for the previous section, η = 0.95, rα √ 2 = 0.075, with α = 2 and r = 0.05. As seen, for these values, the fidelity in absence of multipair contributions is F ≈ 0.99. The fidelity F approaches F when λ 2 |rα| 2 (see Eq. ( 25)), i.e. λ 2 0.01, and decreases when the contribution of multipair increases.

C. Squeezed states at the CV input

In this last section, we briefly discuss the case of nonideal states on the CV input. So far, we have considered at the CV input a perfect Schrödinger cat state as described by Eq. ( 2). Nevertheless, these states are difficult to generate experimentally and they are often replaced by Schrödinger kitten states, of small size α, generated in a heralded fashion [START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF][START_REF] Sychev | Enlargement of optical schrödinger's cat states[END_REF]. At the same time, the use of such heralded states as a starting resource for the generation of hybrid-entanglement implies a further heralding signal to be combined with the ones considered so far (i.e. (E, l) and (F, e)). This would imply that the overall protocol success would rely on a threefold coincidence signal with a dramatic effect for the generation rate. In view of realistic realizations, a convenient approximation to the state |cat+ with α < 1 consists in using at the input 3 a squeezed vacuum state [START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF],

Ŝ(ζ) |0 3 , where Ŝ(ζ) = e 1 2 (ζ * â2 -ζâ †2
) is the single mode squeezing operator and ζ is the squeezing parameter. These states are deterministically available at the output of many nonlinear optical systems with a huge simplification of required experimental resources [START_REF] Andersen | 30 years of squeezed light generation[END_REF]. The main drawback of this approach is that the interference of light reflected from BS1 with the coherent input |rα at BS2 is no longer perfect, thus modifying the shape of the state of Eq. ( 4) and in turns of ψ (1) . Under these conditions, the heralding signals (E, l), (F, e) could in principle be triggered from the sole CV part and occurring even in the absence of photons in the discrete variable. This would lead to an announced state containing vacuum contributions in its DV part.

The described events are linked to vacuum contributions in the DV input |ξ 1,2 . Accordingly, in analogy to the formalism of the previous paragraph, we will call P (0) their corresponding probability and write the overall heralding probability as P s = p 0 P (0) + p 1 P (1) + p P ( ) , with P (1) and P ( ) already discussed in the previous sections. We note that in realistic situations, to comply with multipair events, the nonlinear process providing the DV input |ξ 1,2 is weakly pumped. This condition comes at the price of a high vacuum contribution represented by a p 0 close to 1. By taking into account both the effects of vacuum and multipairs due to nonideal DV and CV inputs, we can express the fidelity as

F s = 1 1 + p0P (0) p1P (1) + pεP (ε) p1P (1)
F (1) ,

where we have neglected minor changes on P (1) and F (1) . The possibility of having a heralding signal with vacuum on the DV input further reduces the fidelity with respect to the situation described in Sec. III B. To avoid the effect of vacuum contribution, p0P (0)

p1P (1) 1. As for Sec. III B, it is pertinent to analyze this ratio in the limit rα → 0, where F (1) is close to 1. The analytical expression of P (0) is reported in Appendix C. It depends on the input squeezing level, ζ, on the product |rα|, and on the detection efficiency η. Its asymptotic behavior gives

P (0) = rα→0 O η 2 |rα| 2 r 4
. Accordingly, by considering p 0 = O(1) and by taking the limit of Eq. ( 14) for P (1) , we obtain

p 0 P (0) p 1 P (1) ∝ rα→0 |rα| 6 p 1 r 4 . ( 27
)
By putting together Eqs. ( 21) and [START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF], we obtain that, with realistic input states and detectors, optimal fidelity F s is obtained when

|rα| 6 r 4 p 1 |rα| 2 . ( 28
)
Compared to the case of an ideal input cat state |cat+ , where the fidelity is maximized by taking an arbitrary low value of p 1 below |rα| 2 , when considering at the CV input a squeezed state, an optimal value of p 1 must be chosen so as to comply with both conditions of Eq. [START_REF] Paris | Displacement operator by beam splitter[END_REF]. Better approximation of Schrödinger cat states than squeezed states [START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF][START_REF] Sychev | Enlargement of optical schrödinger's cat states[END_REF] would even reduce the lower bound of the previous inequality, thus allowing one to reach a better fidelity with respect to |ϕ .

To conclude, in Fig. 5 we illustrate the behavior of the fidelity F s in the case of a DV input state |ξ 1,2 produced by SPDC as given by Eqs. [START_REF] Ourjoumtsev | Generation of optical 'schrödinger cats' from photon number states[END_REF]. The fidelity is reported as a function of the SPDC excitation parameter, λ 2 , and it is compared with the F corresponding to an ideal CV input state, |cat+ 3 (dashed line). In the numerical computation, we have considered, as for the previous paragraph, η = 0.95 and rα √ 2 = 0.075. In particular, we have imposed for the CV input state the squeezing level ζ minimizing P (0) under the condition rα √ 2 = 0.075 and by choosing α = 0.25. In these regards, we note that, in experiments, low α values allow one to satisfy the approximation of a Schrödinger kitten state with a squeezed state [START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF]. With this set of values, the minimum P (0) = 1.3 × 10 -8 is obtained for a squeezing parameter ζ = -0.061 and leads to the fidelity given in Fig. 5. The optimal value, F s = 0.92, corresponds to λ 2 = 9.4 × 10 -4 and to a heralding probability of P s ≈ 3 × 10 -7 . We note that this value is consistent with similar estimated [START_REF] Kwon | Generation of hybrid entanglement between a single-photon polarization qubit and a coherent state[END_REF][START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF] and measured [START_REF] Ulanov | Quantum teleportation between dis-crete and continuous encodings of an optical qubit[END_REF] values in schemes where hybrid state generation is conditioned upon a double detection signal. By operating the experiment at 1 GHz repetition rate [START_REF] Ngah | Ultra-fast heralded single photon source based on telecom technology[END_REF], the heralding rate reaches 300 Hz. If needed, a higher heralding probability can be reached, at the cost of lower values of the fidelity. We note that compromises between the heralding rate and the quality of the produced states are also required in the generation of other kinds of hybrid DV-CV entanglement [START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF][START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF] and, more generally, in any heralded state preparation.

IV. CONCLUSION

In this article, we have presented an experimental scheme able to generate a time-bin encoded hybrid entangled state of the form |ϕ =

|1 A,e |+α f B -|1 A,l |-α f B √ 2
. Our protocol is fully compatible with experimental realizations with off-the-shelf fiber components. The required input resources are a coherent state, an optical Schrödinger cat state |cat+ and a time-bin entangled photon pair |ξ 1,2 . We have shown that, with ideal perfect photon-number-resolving detectors, the process exactly generates the desired state |ϕ . In the second part of the paper, we studied the case of nonideal detectors and/or input states. In particular, we have shown that even with realistic detectors, i.e. available from today's commercial technology, an arbitrary close approximation of |ϕ can be obtained by playing on the scheme parameters. A major advantage of the presented scheme lies in its ability to tolerate both predominance of vacuum on its discrete variable input, as well as multiple pairs, when operated with well-chosen parameters. In this context, we have studied the case of a realistic DV input as well as the one of a CV one in a squeezed vacuum state instead of the Schrödinger cat state. Our study shows that vacuum and multipair effects can be neglected as long as the experiment is carried out with a p 1 in the DV input respecting the condition Experimental generation of time-bin coded hybrid states, compatible with standard telecommunication systems, will permit pushing the applications of hybrid states of light out of the laboratory with a high impact in the context of future development of fiber quantum network systems.

Losses on the CV part of |ϕ are modeled by an unbalanced beam splitter of transmission coefficient t CV and reflection coefficient r CV that couple the CV channel B with a vacuum populated channel. We will label as B the BS transmitted output and as "lost" the reflected one, that also corresponds to the lost part. Hence the state after the beam splitter is

|ϕ = 1 √ 2 |1 A,e |+r CV α f lost |+t CV α f B -|1 A,l |-r CV α f lost |-t CV α f B . (A1)
The density matrix describing the state after losses on the mode B is obtained by tracing out on the mode "lost":

ρ = Tr lost [|ϕ ϕ |] = 1 2 |1 1| A,e |+t CV α f +t CV α f | B -e -2|r CV α f | 2 (|1 0| A,e |0 1| A,l |+t CV α f -t CV α f | B + |0 1| A,e |1 0| A,l |-t CV α f +t CV α f | B ) + |1 1| A,l |-t CV α f -t CV α f | B . (A2)
When losses on the CV part increase, the coefficient e -2|r B α f | 2 multiplying the off-diagonal terms of ρ decreases, thus resulting in a decoherence effect. Although we report here the calculation for an initial state of |ϕ ϕ|, this effect is common to any hybrid CV-DV entangled state submitted to losses in the CV part.

Losses on the DV part are modeled by an unbalanced beam splitter of transmission coefficient t DV and reflection coefficient r DV that couple the DV channel A with a vacuum populated channel. We will label as A the BS transmitted output and as "lost" the reflected one, that also corresponds to the lost part. Hence the state after the beam splitter is

| φ = 1 √ 2 t DV |1 A ,e |+α f B + r DV |1 lost,e |+α f B -t DV |1 A ,l |-α f B -r DV |1 lost,l |-α f B . (A3)
By tracing out the modes (lost, e) and (lost, l), we obtain the state after losses, described by the density matrix:

ρ = Tr lost [| φ φ|] = t 2 DV |ϕ ϕ| + r 2 DV |0 0| A |+α f +α f | B + |-α f -α f | B 2 . (A4)
It is clear that, for any projection operation (or measurement) on A that is insensitive to vacuum, the result is the same as the one obtained from |ϕ ϕ|, with a success probability multiplied by t 2 DV . Upon the combined effect of CV and DV losses, it is easy to show that the density matrix of the initial state |ϕ ϕ| becomes

ρloss = t 2 DV ρ + r 2 DV 2 |0 0| A |+t CV α f +t CV α f | B + |-t CV α f -t CV α f | B . (A5)
with ρ given by Eq. (A2).

Propagation effect on hybrid states with polarization encoding

The state considered here is a hybrid state with the DV part encoded on polarization [START_REF] Li | Experimentally feasible generation protocol for polarized hybrid entanglement[END_REF][START_REF] Sychev | Entanglement and teleportation between polarization and wave-like encodings of an optical qubit[END_REF]:

|ϕ pol = |1 A,H |+α B -|1 A,V |-α B √ 2 , (A6)
where H stands for horizontal and V for vertical polarization. For this DV encoding, the effect of losses on the CV and DV part is the same as for time-bin DV encoding. However, polarization encoded photons strongly suffer from polarization variations when they propagate in long optical fibers [START_REF] Wengerowsky | Entanglement distribution over a 96-km-long submarine optical fiber[END_REF]. To describe the loss of information about the polarization during the propagation, we follow the approach of Ref. [START_REF] Wai | Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence[END_REF] and divide the fiber into small sections of length z whose local birefringence rotates light polarization of an angle θ. The state after a rotation is expressed as

|ϕ pol,θ = 1 √ 2 cos(θ) |1 A,H -sin θ |1 A,V |+α B -sin(θ) |1 A,H + cos θ |1 A,V |-α B . (A7)
The density matrix of state at the output of the fiber is then obtained by averaging on the angle θ the |ϕ pol,θ weighted by the angle distribution: where we omitted the spatial mode labels A and B. As seen from the previous expression, polarization dispersion can progressively convert vertical polarization, i.e. DV state, |1 A,V , into the horizontal one |1 A,H and vice versa, thus leading to a deterioration of the state purity with the distance. For very long distances, the state ρdepol (z) is a mixture of |ϕ pol and the state ϕ pol,-π/2 is obtained when exchanging the H and V in |ϕ pol .

Propagation effect on hybrid states with single rail encoding

We consider here a hybrid entangled state on which the DV part is encoded on the presence |1 A and the absence |0 A of a single photon in mode A, i.e. exhibiting singlerail DV encoding [START_REF] Jeong | Generation of hybrid entanglement of light[END_REF][START_REF] Morin | Remote creation of hybrid entanglement between particle-like and wave-like optical qubits[END_REF]:

|ϕ s-r = |1 A |+α f B -|0 A |-α f B √ 2 . (A10)
The state after the loss beam splitter is expressed as

| φs-r = 1 √ 2 (t DV |1 A |0 lost +r DV |0 A |1 lost ) |+α f B -|0 A |0 lost |-α f B . (A11)
where as before we label as A the BS transmitted output and as "lost" the reflected one. By tracing out on the mode "lost", we obtain

ρs-r = 1 2 (t 2 DV |1 1| A + r 2 DV |0 0| A ) |+α +α| B -t DV |1 0| A |+α -α| B -t DV |0 1| A |-α +α| B + |0 0| A |-α -α| B . (A12)
The coefficient t DV is related to the fiber length by the Beer Lambert law: t DV = e -1 2 βz where the coefficient β is the linear absorption coefficient of the fiber at the working wavelength. We observe that, for single-rail encoding, losses change the relative weight of the DV qubit term by reducing the contribution of |1 , in favor of |0 . This effect cannot be eliminated by postselection at the detection stage and degrades the hybrid entanglement.

Remote preparation of CV qubit

We consider a remote CV qubit preparation experiment [START_REF] Jeannic | Remote preparation of continuous-variable qubits using loss-tolerant hybrid entanglement of light[END_REF] and compare the performances of three kinds of hybrid entangled states with different DV encodings and in the case of a DV part traveling over long distances in optical fibers.

We start with the case of a hybrid entangled state with time-bin encoding submitted to loss on its DV part as in Eq. (A3). With no loss of generality, we consider the case of a DV measurement described by the projector on The unnormalized state on the CV part after such a conditioning reads

Tr A [ ΠA ρ ] = t 2 DV 4 |+α f B -|-α f B +α f | B --α f | B ∝ |cat-cat-| B . (A14)
this case, the probability of having an announced signal with no photon from the DV part is not zero and it is a function of ζ, of the detection efficiency η, and of the amplitude rα of the coherent state entering the system via the input labeled as 4. As for the previous cases, r and t are the amplitude reflection and transmission coefficients of BS1:

P 0 = e -|rα| 2 cosh(ζ)           +∞ y=0 y z=0 z q=0 1 -1 - η 4 z-q 1 -1 - η 4 q 0≤k4≤z k4≡y[2]
t y-z r z-k4 yk 4 zk 4

(rα)

k4 √ k 4 ! y -k 4 y-k4 2 -tanh ζ 2 y-k 4 2 C k4,z-k4,q √ 2 z 2           . ( C1
)
In the previous expression

C k,l,x =    k+l-x k l x 2 F 1 (-k, -x, l -x + 1, -1) if 0 ≤ x ≤ l, ( -1)
x-l x l k x-l 2 F 1 (-l, -(k + lx), xl + 1, -1)

if l ≤ x ≤ k + l, (C2)
with 2 F 1 (p, q, t, w) the hypergeometric function.

BS1 t:r 3 |1 1 2 |α 1 FIG. 1 .

 31211 FIG.1. Scheme for hybrid entanglement generation with time-bin encoding on the discrete variable part.

NPTFIG. 4 .

 4 FIG. 4. Negativity of the Wigner function obtained after having projected the DV part of ρ(1) onto the state |1 A,e +|1 A,l √ 2

FIG. 5 .

 5 FIG.[START_REF] Brask | Hybrid long-distance entanglement distribution protocol[END_REF]. Fidelities with respect to |ϕ of the states obtained when considering multiple pairs on the discrete variable input as functions of the excitation parameter λ 2 of a realistic source based on parametric down conversion and providing the DV input state. Cat and squeezed states were considered for the continuous variable input. Detection efficiency η = 0.95. We chose rα √ 2 = 0.075, with a value of α = 0.25. Fidelity and λ 2 are adimensional quantities.

  |rα|

6

 6

1 √ 2 L 2 ( 1 + 2 ( 1

 122121 ρdepol (z) = P(θ, z) |ϕ pol,θ ϕ pol,θ | dθ ,(A8)where P(θ, z) = 2πσ 2 z e -θ 2 2σ 2 z is a Brownian probability density subjected to impulsive changes as the fiber length, z, increases. The parameter σ is defined as σ = C , with L c the length over which the angles θ lose correlation[START_REF] Wai | Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence[END_REF]. The expression (A7) givesρdepol (z) = 1 e -2σ 2 z ) |ϕ ϕ| + 1 e -2σ2z ) ϕ pol,-π/2 ϕ pol,-π/2 , (A9)

  |1 A ,e +|1 A ,l √ 2leading to an odd Schrödinger cat state on the CV part of the state. The associated measurement operator isΠA = 1 2 |1 A ,e + |1 A ,l1|A ,e + 1| A ,l . (A13)

ACKNOWLEDGMENTS

All the authors would like to thank Jean Etesse and Anthony Martin for stimulating discussions. This work has been conducted within the framework of the project HyLight (No. ANR-17-CE30-0006-01) granted by the Agence Nationale de la Recherche (ANR). The authors also acknowledge financial support from the European Union and the Région PACA by means of the Fond Européen de Développement Regional (FEDER) through the project OPTIMAL and the French government through its program "Investments for the Future" under the Université Côte d'Azur UCA-JEDI project (under the label Quantum@UCA) managed by the ANR (Grant agreement No. ANR-15-IDEX-01).

Appendix A: Action of losses

We consider losses on DV and the CV parts of state |ϕ defined by Eq. [START_REF] Andersen | Hybrid discrete-and continuousvariable quantum information[END_REF]. Similar expressions can be found for the states obtained in nonideal conditions and discussed in different sections of the paper. Following a very standard approach, losses are modeled as unbalanced beam splitters inserted along the paths of mode A and mode B of the state. The higher the losses, the higher are the reflection coefficients. The other BS input of each of these fictitious BS is in a vacuum state.

The obtained state is exactly the same as it would be without losses and it has a unit fidelity with the target, whatever the propagation distance.

A similar analysis can be done on remote preparation of an odd Schrödinger cat state starting with hybrid entanglement with polarization encoding after the propagation of the DV part as in Eq. (A9). In this case, the measurement on the DV part of the state ρdepol is described by the projector ΠA = 1 2 (|1 H,A +|1 V,A )( 1| H,A + 1| V,A ). The fidelity of the so obtained CV state and an odd cat state |cat -is expressed as

Eventually, by starting from the state of Eq. (A12) and following the same approach as for remote state preparation with hybrid states with time-bin or polarization DV encoding, the fidelity with the target odd Schrödinger cat state is

We observe that for both polarization and single-rail DV encodings, the fidelity of the obtained state with the target one is degraded when long propagation distances, z, are considered.

Appendix B: Time-bin entangled photons source Generic form Typically, the time-bin entangled photon pair |ξ 1,2 (see Eq. ( 16)) can be seen as the result of two identical entangled photon pair generation processes, one for the early (e) and other for the late (l) mode. In the Fock basis, whose elements are indicated here as {|k }, the generated state can be written as

By explicitly taking into account only terms containing at most two photons per spatial mode, we can write

) where, as before, |1 and |2 are single and two photon Fock states, respectively.

By comparing the previous expression with the general form of |ξ 1,2 cut at the second order (p ε ≈ p 2 ), we obtain the values of coefficients appearing in Eq. ( 16):

with the second-order term being

SPDC case The explicit expression of coefficients p m k (k = 0, 1, 2 . . .) in the previous equations depends on the specific generation process. In the special case of SPDC, [START_REF] Gouzien | Quantum description of timing jitter for single-photon on-off detectors[END_REF][START_REF] D'auria | Quantum decoherence of single-photon counters[END_REF], with λ the SPDC excitation parameter as described in the text. By injecting these expressions in Eq. (B3), we obtain the results of Eq. ( 23):

We also observe that, for SPDC, the two photon component reads