Yannick Petit

Benoit Boulanger

Jérôme Debray

Thierry Chanelière

Benoît Boulanger

Demonstration of site-selective angular-resolved absorption spectroscopy of the 4 15∕2 → 4 13∕2 erbium transition in the monoclinic crystal Y 2 SiO 5

Keywords:

come     Demonstration of site-selective angular-resolved absorption spectroscopy of the 4I15/2 -4I13/2 erbium transition in the monoclinic crystal Y2SiO5

Introduction

The interest for high resolution spectroscopy in rare-earth samples have been recently renewed by the active quest for quantum photonics devices [START_REF] Zhong | Emerging rare-earth doped material platforms for quantum nanophotonics[END_REF]. Lower doping levels are generally targeted, 50 ppm in our case, as compared to those for laser applications but the host matrices are the same, benefiting from years of development to improve crystal quality. Among them, the monoclinic yttrium orthosilicate (Y 2 SiO 5 ) is certainly the most studied, with an unprecedented record quantum coherence time for the europium dopant [START_REF] Zhong | Optically addressable nuclear spins in a solid with a six-hour coherence time[END_REF]. The low symmetry of both the crystal matrix and involved substitutional sites lead to a strong anisotropy being then exploited to tune the spin properties by accurately adjusting the magnetic field in the crystal frame [START_REF] Böttger | Effects of magnetic field orientation on optical decoherence in Er 3+ :Y 2 SiO 5[END_REF][START_REF] Zhong | Optically addressable nuclear spins in a solid with a six-hour coherence time[END_REF][START_REF] Rančić | Coherence time of over a second in a telecom-compatible quantum memory storage material[END_REF]. Erbium shows remarkable performances in Y 2 SiO 5 in terms of coherence [START_REF] Böttger | Effects of magnetic field orientation on optical decoherence in Er 3+ :Y 2 SiO 5[END_REF][START_REF] Rančić | Coherence time of over a second in a telecom-compatible quantum memory storage material[END_REF]. Among the other lanthanides, Er 3+ holds a lot of promises because of its compatibility with the fiber telecom range which opens many perspectives in classical and quantum processing. Major effort has thus been made to fully understand the orientation of the magnetic properties. Using advanced spectroscopic studies, the anisotropic tensors describing the spin are accurately known, including the crystal-field [START_REF] Li | Spectroscopic properties and fluorescence dynamics of Er 3+ and Yb 3+ in Y 2 SiO 5[END_REF][START_REF] Horvath | Extending phenomenological crystalfield methods to 1 point-group symmetry: Characterization of the optically excited hyperfine structure of 167 Er 3+ :Y 2 SiO 5[END_REF], the different g-tensors [START_REF] Sun | Magnetic tensors for the 4 15∕2 and 4 13∕2 states of Er 3+ :Y 2 SiO 5[END_REF] (in both substitution sites of yttrium and in the ground and optically excited state of Er 3+ ). The hyperfine tensors for odd-isotopes are also known [START_REF] Guillot-Noël | Hyperfine interaction of Er 3+ ions in Y 2 SiO 5 : An electron paramagnetic resonance spectroscopy study[END_REF][START_REF] Chen | Hyperfine interactions of Er 3+ ions in Y 2 SiO 5 : Electron paramagnetic resonance in a tunable microwave cavity[END_REF].

Even if the optical 4 15∕2 → 4 13∕2 transition is put forward to stimulate the applications, the orientation of the absorption principal directions as well as the associated principal values are essentially unknown. Absorption is indeed a key parameter for the efficiency of the different optical functions currently under investigation, like quantum memories [see 8, 4, and references therein] or the more recent opto-RF quantum transduction [see 13, 14, and references therein]. Despite its importance, the absorption is only known along the so-called extinction axes, which corresponds to the principal axes of the dielectric tensor [START_REF] Liu | Spectroscopic properties of rare earths in optical materials[END_REF]. However, the off-diagonal components of the absorption tensor are not known. The impact on the propagation of the mismatch between the absorption and dielectric tensors has been already addressed in Pr 3+ :Y 2 SiO 5 . This pioneering work also performed at low temperature, where all the absorption lines are resolved, corresponds to a simplified situation because the absorption lies in a principal plane of the dielectric tensor [START_REF] Sabooni | Wave propagation in birefringent materials with off-axis absorption or gain[END_REF][START_REF] Kinos | Light-Matter Interaction and Quantum Computing in Rare-Earth-Ion-Doped Crystals[END_REF]. In any case, a complete characterization requires to explore the different crystalline directions, away from the principal axes, in order to access the maximal absorption by properly cutting and orientating the promising Y 2 SiO 5 crystal.

A complete determination of the optical properties in monoclinic crystals is not obvious and requires an accurate control of the sample orientation and incoming polarization. Despite implications in laser physics and non-linear optics, and even though the fundamental background is well established for electric-dipolar transitions, experimental data are quite rare [START_REF] Kaminskii | Anisotropy of spectroscopic characteristics in the biaxial YAlO3-Nd3+ laser crystals[END_REF][START_REF] Owen | Orientation-dependent fluorescence studies and spectroscopic analysis of doped barium yttrium fluoride upconversion laser crystals (BaY 2--Yb Tm F 8 )[END_REF][START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF].

The spherical crystal shape is ideal to explore all possible orientations with a constant propagation length [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF]. However, a full three dimensional rotation of a sphere is not compatible with measurements in our cryostat and it appears technically challenging at low temperature. As the cylindrical geometry (with its revolution axis along the crystallographic b-axis) may be sufficient for monoclinic crystals [START_REF] Dinndorf | Principal axes for transitions in monoclinic crystals[END_REF], we choose this configuration for our experiment performed at cryogenics temperature. Still, as further detailed in section 4, the cylindrical geometry won't be sufficient to fully reconstruct the absorption tensor of the 4 15∕2 → 4 13∕2 transition. Nevertheless, the low temperature measurements give an unprecedented spectral resolution to characterize unambiguously the different crystal field levels of the two substitutional sites as detailed in section 2. We will first present the experiment and then discuss the characterization of the absorption tensor. As we will see, we cannot fully distinguish between the electric-dipole (so called forced ED) and magnetic dipole (MD) contributions, but we can only bound their relative weight. The MD transition is indeed allowed for the the 4 15∕2 → 4 13∕2 of Er 3+ [START_REF] Li | Spectroscopic properties and fluorescence dynamics of Er 3+ and Yb 3+ in Y 2 SiO 5[END_REF].

Optical setup

Following the method in [START_REF] Ménaert | Bulk cylinders and spheres: from shaping to the use for linear and nonlinear optics[END_REF], we prepare a polished cylinder of Y 2 SiO 5 (thickness 3 mm, diameter 5.8 mm) doped with an Er 3+ atomic concentration of 50 ppm grown by Scientific Materials (Fig. 1,(i)). The b-axis, the cylinder axis, is fixed by monochromatic diffraction (±0.1 • precision) in order to properly shape the cylinder in the (a,c) plane (cylindricity error under 1%). Finally, the exact in-plane orientation is determined with a Laue diffractometer with a sub degree precision. We chose this configuration because the b-axis, also called the special monoclinic axis, defines the axis of the dielectric frame that remains fixed as a function of any dispersive parameter of the dielectric permittivity, like the wavelength or the temperature for example. The two other axes, perpendicular to the b-axis, will be called D 1 and D 2 in the following, and may vary as a function of wavelength and temperature. The other reason is that the special axis is a principal axis for both the real and imaginary part of dielectric permittivity [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF].

The cylinder is placed in a variable-temperature-insert liquid helium cryostat. The rotation is controlled manually from room-temperature though an O-ring sealed rotating pole. The precision is 2 • , which corresponds to the rotating mount scale increment. We collimate the beam in the cylinder by properly focusing the incoming beam using cylindrical lenses with a focal length of 75 mm at the input and output of the cryostat. We place a polarizing cube and half-wave plate to control the incoming polarization to independently address each of the two polarization Eigenmodes, i.e. parallel or orthogonal to the b-axis.

Our objective is to scan the 4 15∕2 → 4 13∕2 transition between the lowest crystal field levels of both the sites 1 (6coordinate) and 2 (7-coordinate) which represent the two possible substitutional sites in the Y 2 SiO 5 matrix (see Fig. 2). The exact ratio between both sites cannot be determined quantitatively because a slightly weaker substitution could be potentially compensated by a larger oscillator strength for example. That being said, Er 3+ is abundantly present in both sites having the same valency and size. The crystal fields are very similar which is confirmed by very similar optical absorption values [START_REF] Kurkin | Epr and spin-lattice relaxation of rare-earth activated centres in Y 2 SiO 5 single crystals[END_REF]. So the sites occupancy seems to be well balanced, especially at low doping level.

Then we use the same fibred DFB laser (slow diode Peltier device temperature tuning) centered at 1536.5 nm and 1538.9 nm, for both sites respectively. We operate the cryostat at 45 K. At this temperature, the measured optical depth corresponding to the natural logarithm of the transmission is of the order of one, the transition linewidths being ∼ 2 GHz. At lower temperature, the linewidth is lower (∼ 0.5 GHz) thus increasing the absorption to a point where the transmission is becoming too weak to be accurately measured. We scan the laser frequency across the resonance to record a complete line spectrum for each absorption measurement as illustrated in Fig. 3. The natural logarithm of the transmission spectrum is fitted by a Lorentzian curve whose height gives the optical depth.

Experimental results

As reference frame for the measurement, we use the dielectric frame (b,D 2 ,D 1 ). This notation has been introduced for the first spectroscopic measurements of Er 3+ :Y 2 SiO 5 [START_REF] Li | Spectroscopic properties and fluorescence dynamics of Er 3+ and Yb 3+ in Y 2 SiO 5[END_REF] and is widely used to reference the spin Hamiltonian components. The (b,D 2 ,D 1 ) frame is the dielectric frame (X,Y,Z), these axes respectively bearing the low, intermediate and large principal indices of refraction [16, 25, 2, p.386]. The relative orientation of (b,D 2 ,D 1 ) and (a,b,c) is summarized in Fig. 1. The dielectric frame slightly rotates around the b-axis as a function of wavelength, as observed in [START_REF] Traum | Direct measurement of the dielectric frame rotation of monoclinic crystals as a function of the wavelength[END_REF]. Moreover, such a rotation of the dielectric frame orientation becomes negligible in the infrared (less than 1 • ), allowing to consider that this orientation offset is within our measurement uncertainty (precision 2 • ). Therefore, we did not try to remeasure the exact positions D 2 and D 1 at our wavelength and we simply fix the position of D 1 at 23.4 • from a, as chosen in Fig. (5.a) of [START_REF] Traum | Direct measurement of the dielectric frame rotation of monoclinic crystals as a function of the wavelength[END_REF]. This recent reference value is in good agreement with the previous measurement of a 23.8 • offset for D 1 [START_REF] Li | Spectroscopic properties and fluorescence dynamics of Er 3+ and Yb 3+ in Y 2 SiO 5[END_REF]. In any case, the slight variations of the optical frame orientation are smaller than our measurement precision. Fixing the D 2 and D 1 with respect to the crystal frame allows the crystal to be oriented using X-ray diffraction without depending on the wavelengths or dopants of interest and to keep a common reference for the g-tensor definitions [START_REF] Sun | Magnetic tensors for the 4 15∕2 and 4 13∕2 states of Er 3+ :Y 2 SiO 5[END_REF]. The monoclinic angle between a and c is = 102.65 • .

As we rotate the sample by an angle from D 1 to D 2 , the polarisation is either constant along or it lies in (D 1 ,D 2 ) plane, respectively corresponding to the ordinary or extraordinary polarization eigenmodes, as defined in Fig. 1. In the latter case, the polarization rotates in the plane as well (along a direction close to +90 • , while neglecting the minor influence of double refraction). The measurement of the optical depth is reproduced for site 1 and site 2, and results are plot in Fig. 4.

The different angular distributions in polarized light exhibit contrasted modulations, expect for site 2 when polarization lies in the (D 1 ,D 2 ) plane. As a signature of the local anisotropy in the Y 2 SiO 5 matrix, well-known for the different components of the spin Hamiltonian, the absorption varies a lot (see for site 2, polarisation along b in Fig. 4). As an illustration, the g-factor varies by a factor of 10 to 20 by comparing the maximum and minimum values at different magnetic field orientations for site 2 [START_REF] Sun | Magnetic tensors for the 4 15∕2 and 4 13∕2 states of Er 3+ :Y 2 SiO 5[END_REF]. We retrieve a comparable anisotropy ratio for the absorption of the same site (blue curve in Fig. 4, site 2).

The absorption eigenaxes (maximum and minimum) that we exhibit by rotating the cylinder around b do not coincidence with the dielectric frame (b,D 2 ,D 1 ) axes (see polar plots in Fig. 4) as expected [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF][START_REF] Dinndorf | Principal axes for transitions in monoclinic crystals[END_REF]. Moreover, it is remarkable to observe that the absorption eigenaxes from site 1 and site 2 are different. This results indeed from the different local environments of the two sites, and primarily the different coordination numbers, with oxygen in that case. Beyond the qualitative analysis of the angular distribution, the data can be discussed further by noting first that Y 2 SiO 5 belong to the monoclinic class where b is a fixed and common eigendirection to any optical tensors by symmetry, allowing the dielectric frame as well as any resonant absorbing or emitting frame to show orientation dispersion around b [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF]; and secondly that the 4 15∕2 → 4 13∕2 is MD allowed.

Analysis

The theoretical analysis of absorption anisotropy relies on symmetry considerations [START_REF] Aminov | Theory of luminescence anisotropy of YAlO3: Nd3+ single crystals[END_REF][START_REF] Stedman | Polarization dependence of natural and field-induced one-photon and multiphoton interactions[END_REF][START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF]. The key point for a monoclinic crystal such as Y 2 SiO 5 is the fact that b is an axis of symmetry.

To start up, as judiciously noted in [START_REF] Dinndorf | Principal axes for transitions in monoclinic crystals[END_REF], the transition dipole follows the general projection rules of a spin-1/2 particle submitted to an anisotropic Hamiltonian. In short, the absorption angular distribution can be decomposed as ( ) = 1 cos 2 -0 + 2 sin 2 -0

(1)

= 1 + 2 2 + 1 -2 2 cos 2 -0 (2

Table 1 Fitting parameters

Experimental set where 1 , 2 and 0 allow the complete tensor to be reconstructed. This corresponds to the cross-section of an ellipsoid in general. The formula can be rigorously justified for the Maxwell equations by considering the first-order electric susceptibility as a rank-2 polar tensor in the relevant monoclinic symmetry, and considering the imaginary part as a perturbative absorption behavior with respect to refractive aspects driven by the real part. [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF]. In (1), if one finds 0 = 0, then the principal axes of the investigated absorption transition are aligned with those of the dielectric frame, showing diagonal values of 1 and 2 . As shown in Fig. 4, this is clearly not the case, highlighting here the monoclinic behavior of Y 2 SiO 5 . The four data set in Fig. 4 are fitted by (1) (dashed-dotted curves) to obtain the results reported the table 1.

The two ideal cases independently with either a purely ED or MD transition are considered below.

(i) If the transition was purely ED (so called forced-ED), our configuration would be sufficient to extract the full transition tensor for both sites. Because b is an axis of symmetry, when the electric field polarization stands the ordinary polarization mode along b, the absorption should be constant to a value as is varied: the blue curves in Fig. 4 would be flat. In the (D 1 ,D 2 ) plane, there is no reason for D 1 nor D 2 to be the absorption eigenaxes, so that the absorption angular distribution should be characterized by three values 1 , 2 and 0 . The latter is to be nonzero due to the existence of off-diagonal components in such dielectric plane orthogonal to the monoclinic b-axis. As absorption is not constant while considering a fixed polarization mode along b, it strongly evidences that the transition is not purely ED. This is expected because the 4 15∕2 → 4 13∕2 transition is MD allowed (with Δ = -1 as selection rule [START_REF] Li | Spectroscopic properties and fluorescence dynamics of Er 3+ and Yb 3+ in Y 2 SiO 5[END_REF]).

(ii) If the transition was purely MD, we can follow the previous reasoning for ED transition replacing the electric field by the magnetic field polarization, these fields being typically perpendicular to the propagation axis ⃗ and to each other. Namely, if the electric field lies in the (D 1 ,D 2 ) plane, then the magnetic field is along b and the MD absorption should be constant : the red curves in Fig. 4 would be flat. By analogy, if the electric field is along b, then the absorption angular distribution should be characterized by the three values 1 , 2 and 0 . From the fact that the red curve for site 2 in Fig. 4 is flat, it is extremely tempting to conclude that the site has a dominant MD character. One cannot nevertheless discard the fact that the ED components are accidentally isotropic in the (D 1 ,D 2 ) plane, leading to an apparent purely MD transition pattern. Further analysis is needed to lift the ambiguity.

If now is considered a mixture of ED and MD transitions, then the ED and MD absorptions have to be simply added. Actually, the measurements do stand in the weak absorption approximation and we measure the linear absorption where ED and MD can be taken independently. Following the previous reasoning, the angular distribution when the electric field polarization is along b can be written as blue

( ) = + 1 + 2 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ 1 + 2 2 + 1 -2 2 ⏟⏞⏞⏟⏞⏞⏟ 1 -2 2 cos(2( -0 ⏟ ⏟ ⏟ 0 )) (3)
And when the electric field lies in the (D 1 ,D 2 ) plane, it comes:

red ( ) = + 1 + 2 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ 1 + 2 2 + 1 -2 2 ⏟⏞⏞ ⏟⏞⏞ ⏟ 1 -2 2 cos(2( - 0 ⏟ ⏟ ⏟ 0 )) (4)
In order to determine unambiguously the components of the ED and MD absorption tensors,it is necessary to have 4 values per curve, i.e. , 1 , 2 , 0 for the blue and , 1 , 2 , 0 for the red one, while the fit gives only 3 parameters: 1 , 2 and 0 reported as underbraces in (3) and ( 4). So we can only take the values in table 1 as phenomenological parameters to describe the absorption in the (D 1 ,D 2 ) measurement plane (with two polarizations) but without allowing for a complete determination of the absorption tensors. This is a consequence of the ED and MD mixture of the 4 15∕2 → 4 13∕2 transition of Er 3+ in Y 2 SiO 5 . A complete determination would require to repeat the measurement in different planes as (b,D 1 ) and (b,D 2 ), thus giving redundantly the ED and MD tensors values.

Conclusion

We perform for the first time to the best of our knowledge a low temperature measurement of the absorption angular distribution in polarized light in the plane perpendicular to the monoclinic special axis of Y 2 SiO 5 , which allows us to spectrally resolve and independently characterize the two yttrium substitution sites. Measurements remarkably demonstrate that this transition of interest, namely the 4 15∕2 → 4 13∕2 transition between the lowest crystal field levels of Er 3+ , actively investigated for quantum information processing, is not only a magnetic-dipole allowed transition but indeed a hybrid electric-magnetic transition. Because of such behavior of ED and MD mixture, the absorption tensor components cannot fully be unambiguously determined yet while only considering the currently investigated dielectric plane. The use of a sphere instead of a cylinder would ideally permit to explore the different independent measurement planes by following the procedure from [START_REF] Petit | Recent advances in monoclinic crystal optics[END_REF]. But it would be very difficult to rotate a sphere inside a cryostat. The use of three oriented cylinders should then be used, but at the price of tedious crystal machining and polishing.

Nevertheless, one should note that recent progress in determining the crystal field parameters in the low symmetry site of Y 2 SiO 5 [START_REF] Horvath | Extending phenomenological crystalfield methods to 1 point-group symmetry: Characterization of the optically excited hyperfine structure of 167 Er 3+ :Y 2 SiO 5[END_REF] make possible a calculation of the allowed MD component, as opposed to the ED component which is forbidden or so-called forced. Complemented by a theoretical prediction of the MD component, our measurements would be sufficient to fully characterize the forced ED and allowed MD tensors.

To conclude with the spin Hamiltonian analogy outlined in the introduction, it would be interesting to repeat absorption measurements under magnetic field. One would obtain the equivalent of the g-tensor that can be extracted from electron spin resonance spectroscopy. It may sounds surprising that the g-tensor has 6 independent components [START_REF] Sun | Magnetic tensors for the 4 15∕2 and 4 13∕2 states of Er 3+ :Y 2 SiO 5[END_REF] while the ED and MD optical transition tensors only have 4 independent components each here. This is precisely because we do not apply a magnetic field that the monoclinic symmetry (and not the local 1 symmetry) strictly applies in the sense that b stays an axis of symmetry. Under magnetic field, the optical transition tensors should have 6 independent components because b may not remain an axis of symmetry, forcing the magnetically-stressed crystal to behave similarly to a triclinic crystal. One could alternatively reinterpret this global symmetry analysis as lifting the inequivalence between the so-called magnetic sub-sites, related by the crystal symmetry, whose degeneracy can be precisely lifted by applying a magnetic field, away from the b axis or the (D 1 ,D 2 ) plane. Such experiment is quite challenging because the crystal has to be rotated with the magnetic field in the cryostat. The latter should be sufficiently strong to lift the degeneracy and resolve the Zeeman transition. This is an interesting perspective.

Funding Information

We have received funding from the Investissements d'Avenir du LabEx PALM ExciMol, ATERSIIQ and OptoRF-Er (ANR-10-LABX-0039-PALM). This work was supported by the ANR MIRESPIN project, grant ANR-19-CE47-0011 of the French Agence Nationale de la Recherche.

Figure 1 :

 1 Figure 1: (i): a polished cylinder is centered on a brass holder and placed on a rotating mount in the cryostat. A small mirror is glued on the side of the brass holder, serving as a common orientation reference (laser auto-collimation) between the Laue diffractometer and the measurement setup. (ii): reference dielectric frame (b,D 2 ,D 1 ). The angle is defined as the rotation angle from D 1 toward D 2 . (iii) Projection and identification of the axes in the monoclinic plane perpendicular to b. In practice, the propagation direction ⃗ is kept constant while rotating the sample around b. The electric field polarization can be either aligned with b (blue) or in the (D 1 ,D 2 ) plane (red).

Figure 2 :

 2 Figure 2: Reduced crystal cell representing the two possible substitutional sites in the Y 2 SiO 5 matrix: site 1 (left) and site 2 (right) with 6-coordinate and 7-coordinate respectively. Colour code: cyan -yttrium, red -oxygen, yellow -silicon and green -erbium substituting an yttrium.

Figure 3 :

 3 Figure 3: Examples of the transmission spectrum for the 4 15∕2 → 4 13∕2 transition of site 1 (1536.5 nm). Theses spectra correspond to the maximum and minimum optical depths of 1.65 and 0.54 respectively for an angular rotation = 298.6 • (purple) and = 28.7 • (orange) respectively when the polarization is aligned with b. The dashed lines are Lorentzian fits of the experimental curves.

Figure 4 :

 4 Figure 4: Angular distributions of optical depths for site 1 (top) and 2 (bottom) versus the cylinder orientation , for the two Eigenmode polarizations either aligned with b (blue) or in the (D 1 ,D 2 ) plane (red). When = 0, then ⃗ ∥ D 1 . When = 90 • , then ⃗ ∥ D 2 . Right panel: same data in polar representation. The blue and red dashed curves are fitted with (1). The two measurements represented in Fig.3 have been marked with purple ( = 298.6 • ) and orange ( = 28.7 • ) circles for site 1.

  Site 1 -polar. along b (blue) 1.60 0.55 115 • Site 1 -polar. in (D 1 ,D 2 ) plane (red) 1.59 1.20 18 • Site 2 -polar. along b (blue) 0.98 0.09 68 • Site 2 -polar. in (D 1 ,D 2 ) plane (red) 1.29 1.27 96 •

Acknowledgments

We thank A. Kinos for enlightening discussions and P. Goldner, M. Reid and C. Thiel for their stimulating interest in our work.


Deriving Born's rule from an Inference to the Best Explanation.

Alexia Auffèves (1) and Philippe Grangier (2) (1): Institut Néel, BP 166, 25 rue des Martyrs, F38042 Grenoble Cedex 9, France.

(2): Laboratoire Charles Fabry, IOGS, CNRS, Université Paris Saclay, F91127 Palaiseau, France.

In previous articles we presented a simple set of axioms named "Contexts, Systems and Modalities" (CSM), where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain (or rather infer) Born's rule within this framework. Our approach is compared with other former and recent derivations, and its strong links with Gleason's theorem are particularly emphasized.

I. INTRODUCTION.

Many recent articles [START_REF]Foundations of quantum mechanics and their impact on contemporary society[END_REF] claim to provide new derivations of Born's rule, that is clearly a major theoretical basis of quantum mechanics (QM). Others claim that deriving Born's rule is a nonsense, and that it must essentially be postulated [START_REF]For an interesting non-derivation of Born's rule and a discussion[END_REF]. In this article we take a medium position, that is: Born's rule cannot be logically proven, but it does not have either to be postulated. Actually, it can be inferred from some simple physical requirements or postulates, based on established (quantum) empirical evidence [START_REF]Our approach can be related to an "Inference to the Best Explanation[END_REF]. After the realization of loophole free Bell tests [START_REF] Aspect | Closing the Door on Einstein's and Bohr's Quantum Debate[END_REF] and in the era of quantum technologies [START_REF] Nielsen | Quantum Computation & Quantum Information[END_REF], these postulates can be simply formulated [START_REF] Auffèves | Contexts, Systems and Modalities : a new ontology for quantum mechanics[END_REF][START_REF] Auffèves | Violation of Bell's inequalities in a quantum realistic framework[END_REF][START_REF] Auffèves | Recovering the quantum formalism from physically realist axioms[END_REF][START_REF] Grangier | What is quantum in quantum randomness?[END_REF][START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF][START_REF] Auffèves | A generic model for quantum measurements[END_REF]. They are presented here in a synthetic form, and various consequences are obtained and discussed.

II. DEFINITIONS AND POSTULATES.

Definition 1 : We consider a quantum system S and a specified ensemble of measurement devices interacting with it; this ensemble is called a context1 . The best physically allowed measurement process provides a set of numbers, corresponding to the values of a welldefined and complete set of jointly measurable quantities; these values will be found again with certainty, as long as the system and context are kept the same 2 . The physical situation occurring after such an ideal and repeatable measurement process is called a modality.

As an example using QM notations (not required yet), for K particles with spin 1/2, the set of observables {S (i) z , i = 1...K} constitutes a context, and the observation of a given set of results {m (i) , i = 1...K}, where m (i) equals either + /2 or -/2, constitutes a modality. The modalities are not defined in the same way as the usual "quantum states of the system", since they are explicitly attached to both the context and the system.

From the above definition, justified by empirical evidence, one measurement provides only one modality. Therefore in any given context the various possible modalities are mutually exclusive, meaning that if one result is true, or verified, all other ones are not true, or not verified. We have then the Basic postulate (contextual quantization) : The number N of mutually exclusive modalities for a given quantum system is the same in any relevant context.

In the above example one has N = 2 K . Definition 2 (incompatible modalities) : Modalities observed in different contexts are generally not mutually exclusive, they are said to be incompatible.

Incompatible means that if a result is true, or verified, one cannot tell whether the other one is true or not. Definition 3 (extravalence3 ) : When S interacts in succession with different contexts, certainty and repeatability can be transferred between their modalities. This is called extracontextuality, and defines an equivalence class between modalities, called extravalence4 .

The equivalence relation is obvious, for more details and examples of extravalence classes see [START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF].

The intuitive idea behind these definitions and postulate is that making more measurements in quantum mechanics (by changing the context) cannot provide "more details" about the system, because this would increase the number of mutually exclusive modalities, contradicting the basic postulate. One might conclude that changing context randomizes all results, but this is not true: some modalities may be related with certainty between different contexts, this is why extravalence is an essential feature of the construction.

III. THEOREMS.

Theorem 1 : Given an initial modality and context, obtaining another modality in another context must (in general) follow a probabilistic law.

First, let us emphasize that modalities in different contexts are always considered different, even if they are extravalent, so some care is required when counting modalities. Let us start from an initial modality for a system in context C u , and perform a measurement in another context C v . Several situations can be considered:

(i) From the basic postulate there are N mutually exclusive modalities in each context, and one of them is realized when doing a measurement. Therefore the situation where all modalities in context C v would have a probability p = 0 to occur is excluded by construction.

(ii) If one modality in the new context C v is obtained with certainty, this means that C v contains a modality extravalent with the initial one; then p = 1 for this modality, and p = 0 for all other (mutually exclusive) ones. If the situation is the same for all modalities in C u , then they are all extravalent with a modality in C v , and the modalities in the new context can be seen as a rearrangement (permutation) of the initial ones. So let's try again with another context C w ; if the situation is the same again in all other contexts, it means that there are only N classes of extravalent modalities, going through all contexts. This means that the context is unique up to a rearrangement (permutation) of the modalities; therefore there are no incompatible modalities, and the situation is essentially classical.

(iii) Since case (i) is excluded, and case (ii) is classical (there are no incompatible modalities), the general case (where incompatible modalities do exist) is that obtaining a modality in the new context is probabilistic (0 < p < 1), hence the theorem is demonstrated.

The core of this proof is that measuring in a new context cannot be a "refinement" of the previous measurement, because this would extend the number N of mutually exclusive modalities. To see that more explicitly, let us consider an initial modality u 0 in C u , connected to at least two modalities v 1 or v 2 , according to (iii) above. Now let us measure again in C u : if u 0 is found again with certainty, then there would be two mutually exclusive situations, u 0 → v 1 → u 0 and u 0 → v 2 → u 0 . This would give at least (N + 1) mutually exclusive modalities, in contradiction with the quantization postulate.

Therefore the randomness is not only from C u to C v , but also back from C v to C u [START_REF] Auffèves | A generic model for quantum measurements[END_REF]. This makes clear that probabilities do follow from the fixed value of N , i.e. from the maximum number of mutually exclusive modalities for a given system, imposed by the basic postulate.

Theorem 2 : Given an initial modality and context, the probability to get another modality in another context keeps the same value as long as the initial and final modalities belong to the same respective extravalence class, independently of the embedding contexts.

Let us start again from an initial modality u i and context C u , and follow the same steps as in Theorem 1 when performing a measurement in another context C v .

(i) The situation where no modality can be obtained in the new context (p = 0) is excluded as said above.

(ii) The situation where obtaining one modality in the new context is certain (p = 1) means that the new context contains a modality extravalent with the initial one. Then p = 1 corresponds to modalities in the same extravalence class, this is the definition of extravalence.

(iii) In the general case one gets another modality v j with a probability 0 < p < 1. Given this new modality v j , changing again the context to another one C w containing a modality w k extravalent to v j will yield w k with certainty. In that case the probability for going from u i to v j will be the same as the one for going from u i to w k (Fig. 1). Moreover, if one starts from a modality x l extravalent to u i , and one goes to u i then to v j , the probability for going from x l to v j will be the same as the one for going from u i to v j .

Therefore the probability to get another modality in another context only depends on the extravalence classes of the initial and final modalities, and the theorem is demonstrated 5 . This theorem shows that the probability to get a new modality starting from an initial one is linked neither to the context, nor to the modalities themselves, but to their extravalence class. In some approaches this property is called "non-contextual assignment of probabilities", and this is a very fundamental feature of quantum mechanics, which appears here as a theorem. It also suggests the major next step, i.e. that the probability law should be obtained by attributing a mathematical object to an extravalence class, in such a way that all the above requirements are fulfilled. As a general feature of such an inductive or inference reasoning [START_REF]Our approach can be related to an "Inference to the Best Explanation[END_REF], it cannot be shown that the proposed solution is unique (ie, necessary), but it can be shown that it fulfills all the requirements (ie, that it is sufficient).

Theorem 3 : Let us associate a N × N rank-1 projector P i to each extravalence class of modalities, and a set of N mutually orthogonal projectors to each context. Then the probability law f (P i ) built from these projectors obeys Born's rule, and different sets of mutually orthogonal projectors are related by (complex) unitary matrices.

Since the N × N projectors are associated to extravalence classes of modalities, the probabilities are a function f (P i ) of these projectors, in agreement with Theorem 2. Since a context (set of mutually exclusive modalities) is associated to a set of N mutually orthogonal projectors, the probabilities for this set of projectors sum to 1. This condition only requires to add probabilities for commuting (orthogonal) projectors, avoiding known objections to other derivations [START_REF] Laloë | Do We Really Understand Quantum Mechanics?[END_REF]. Then all the hypotheses for Gleason's theorem [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF] are fulfilled (see § IV), and thus Born's rule applies [START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF]. By construction orthogonal sets of projectors are connected by complex unitary matrices. Complex numbers are required to connect continuously the identity matrix to all permutations of modalities: this cannot be done by (real) orthogonal matrices, which split into two subsets with determinants ±1; see [START_REF] Auffèves | Recovering the quantum formalism from physically realist axioms[END_REF][START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF].

Here we have considered initial and final modalities, i.e. rank 1 projectors [START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF], but more generally Gleason's theorem provides the probability law for density operators (convex sums of projectors), interpreted as statistical mixtures. This clarifies the link between Born's rule and the mathematical structure of density operators [START_REF] Masanes | The Measurement Postulates of Quantum Mechanics are Redundant[END_REF].

IV. AN OVERVIEW OF GLEASON'S

THEOREM.

Gleason's theorem has the reputation of being unpenetrable by physicists, who usually keep away from this frightening monument (see also Discussion below). Therefore we want to present here a "physicist's demonstration", where most mathematical difficulties are deliberately omitted, in order to reveal the big picture. All the (nice) mathematical details can be found in "An elementary proof of Gleason's theorem", by Roger Cooke, Michael Keanes & William Moran, Math. Proc. Camb. Phil. Soc. 98, 117 (1985) [START_REF] Cooke | An elementary proof of Gleason's theorem[END_REF], which is more recent and reader-friendly than the original work by Gleason [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF].

Let us consider a separable Hilbert space H over R or C, and if dim(H) = N we denote it C N (over C) ou R N (over R). Then we define a real-valued non-negative function f acting on the unit sphere of H, such that for any orthonormal basis

{x i } of H, one has i f (x i ) = 1.
The function f (x i ) can be seen as the probability to get the result x i , in a "state" defined by f . Note that if f (x j ) = 1 for the vector x j , then f (x k =j ) = 0 for all other vectors in the orthonormal basis {x i }: the results x i are mutually exclusive as we required. The non-obvious hypothesis is why f (x i ) depends only on x i and f , and not on the other vectors {x k =j } in the orthonormal basis: this is where the discussion above plays a crucial role, by associating x i to an extravalence class of modalities.

Here our goal is to sketch a demonstration of Gleason's theorem: If N ≥ 3, there exists a density operator6 ρ defined on H such that f (x i ) = x i |ρ|x i for all unit vectors x i . Then f is said to be "regular".

For simplicity we will assume that the extreme value f (x i ) = 1 is reached, and then present the (easier) result that in that case ρ is a projector |x x|, so f

(x i ) = x i |x x|x i = | x i |x | 2 :
this is the usual Born's rule for pure states (or for extravalent modalities).

Step 1: prove the following "reduction lemmas" L1 -In R N , f is regular iff it is the restriction to the unit sphere of a quadratic form (this is clear by writing explicitly ρ as a self-adjoint operator) L2 -If f is regular in R 3 , then it is also regular in any 2-dimensional subspace R 2 of R 3 (this is clear by restricting the quadratic form from

R 3 to R 2 ) L3 -If f is regular in any subspace R 2 of C 2 , then it is regular in C 2 (not obvious, see [14]) L4 -If f is regular in any subspace C 2 of C N , then it is regular in C N (not obvious, see [14]) Crucial lemma: If f is regular in R 3 , then it is regular in C N (use L2, then L3, then L4).
Therefore it is enough to show that f is regular in R 3 . This explains why the theorem requires N ≥ 3: in fact, f is regular in any C 2 considered as a subspace of C N , but not in C 2 considered alone. Said otherwise, it is well known, e.g. from Clauser [START_REF] Clauser | Von Neumann's informal hidden-variable argument[END_REF], that one can build a "classical model of a (unique) qubit". However this classical model fails if this qubit is one among several qubits, which is fine as far as QM is concerned.

Step 2: prove that f is regular in R 3 Now one looks for a probability function f (u), where u is a normalized vector in R 3 , so that 0 ≤ f (u) ≤ 1 and f (u)+f (v)+f (w) = 1 for any orthonormal basis {u, v, w} of R 3 . One does not assume that f is continous, but here we assume that the extreme values 0 and 1 are reached (this is only for simplification, and the general case is treated in the full theorem [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF][START_REF] Cooke | An elementary proof of Gleason's theorem[END_REF]).

Given a normalized vector p and an orthonormal basis {u, v, w}, the quantities cos 2 (u, p), cos 2 (v, p), cos 2 (w, p) are the squares of the components of p in the basis, so they sum to 1. Therefore cos 2 (u, p) for a fixed p is an acceptable function f (u), and actually it is the good one. But why is it the only such function? We will split the answer in two parts.

We look for a probability function f(u), where u is a normalized vecto 0 ≤ f(u) ≤ 1 and f(u) + f(v) + f(w) = 1 for any orthonormal basis {u, v For simplification assume that the extreme values 0 and 1 are reached for simplification, the general case is treated in the full theorem).

Gleason's theorem, why and how (2) ?

p q u v w Then define a normalized vector p such f (p) = 1, and put it at the pole of the 3D consequence, f (q) = 0 for any q on the Given an orthonormal basis {u, v, w}, cos 2 (u,p), cos 2 (v,p), cos 2 (w,p) are the components of p in the basis, so they s Therefore h(u) = cos 2 (u,p) is an acceptable function f(u), and actua good one. But why is it the only one ? This can be shown by (simple geometrical arguments, including a nice lemma due to C. Piron.

FIG. 2:

We look for a probability function f (u), where u is a normalized vector in R 3 , so that 0 ≤ f (u) ≤ 1 and f (u) + f (v) + f (w) = 1 for any orthonormal basis {u, v, w} of R 3 . To be simple (see text) we assume that the extreme values 0 and 1 are reached, we define a normalized vector p such that f (p) = 1, and put it at the pole of the 3D sphere. As a consequence, f (q) = 0 for any q on the equator. Given an orthonormal basis {u, v, w}, the quantities cos 2 (u, p), cos 2 (v, p), cos 2 (w, p) are the squares of the components of p in this basis, so they sum to 1. Therefore h(u) = cos 2 (u, p) is an acceptable function f (u), and Gleason's theorem shows that it is the only one.

Why is there no φ ?

In R 3 a normalized vector u is defined by two polar angles θ and φ, and in cos 2 (u, p) there is only one angle, why? To be specific let us choose p as the vector such that f (p) = 1 (since this value is reached), and position it at the pole of the unit sphere in 3 dimensions (see Fig. 2). As a consequence, f (q) = 0 for all vectors on the equator, and for any vector one can define h(u) = cos 2 (u, p), which depends on the "latitude" of u (the polar angle θ), but not on its "longitude" (the azimuthal angle φ).

One can then use two lemmas to show that for any two vectors u, v in the northern hemisphere such that h(u) > h(v), one has f (u) ≥ f (v); this is done in Annex 1. Then define the smallest and largest values of the possible values of f (u) for a given latitude:

m(x) = Inf{f (u) such that h(u) = x} M (x) = Sup{f (u) such that h(u) = x}.
One has M (1) = m(1) = 1 at the pole, M (0) = m(0) = 0 at the equator, and if x < x then M (x) ≤ m(x ) due to the above lemmas. In addition one has obviously m(x) ≤ M (x) and m(x ) ≤ M (x ), so if x → x one gets a contradiction (M less than m), unless m(x) = M (x), i.e. f (u) depends only on the latitude.

Why only cos 2 (u, p) ?

Given that cos 2 (u, p) is an acceptable f (u), one may think that any other function f (u) = g(cos 2 (u, p)) should be acceptable also. To show this not the case, one uses a Magical lemma: Consider a function g over [0, 1], verifying the hypotheses (i) g(0)=0, (ii) a < b ⇒ g(a) < g(b), (iii) a+b+c = 1 ⇒ g(a) +g(b) +g(c) = 1. Then g(a) = a for any a within [0,1].

The proof (subtle but not difficult) is given in Annex 2. It is easily seen that g(cos 2 (u, p)) fulfills the hypothesis of the lemma for any orthonormal basis {u, v, w}, with a = cos 2 (u, p) etc. So from the magical lemma one gets g(cos 2 (u, p)) = cos 2 (u, p), and the additional function g is useless.

Step 3: conclude that f is regular in C N Therefore f is regular in R 3 , and also in C N from the reduction lemmas. The demonstration can be reconsidered in the more general case where the value f (p) = 1 is not reached, and one finds 7 that ρ is no more a projector, but a density matrix associated with a statistical mixture.

V. DISCUSSION.

An essential feature of the contextual quantization postulate, i.e. the fixed value N of the maximum number of mutually exclusive modality, turns out to be the dimension of the Hilbert space. In the spirit of [START_REF]Our approach can be related to an "Inference to the Best Explanation[END_REF] and as 7 In the general case in R 3 , the maximum (resp. minimum) value of f is 0 ≤ M ≤ 1 (resp. 0 ≤ m ≤ 1), and one shows [START_REF] Cooke | An elementary proof of Gleason's theorem[END_REF] that there exist a basis {p, q, r} such that

f (u) = M cos 2 (u, p) + m cos 2 (u, q) + (1 -M -m) cos 2 (u, r) with M + m ≤ 1.
shown is [START_REF] Auffèves | Recovering the quantum formalism from physically realist axioms[END_REF][START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF], this provides one more heuristic reason for using projectors. Then the projective structure of the probability law warrants that, despite the availability of an infinite number of incompatible modalities, N cannot be "bypassed" by getting more details on any of them.

This would not be the case in the usual probability theory, based on partitions : making a partition of all modalities in N sub-ensembles for each given context would not prevent sub-partitions, that would correspond to additional details or "hidden variables", that are forbidden by our basic postulate. This corresponds mathematically to Bell's or Kochen-Specker's theorems, and all their variants, which basically show the inadequacy of partition-based probabilities. This problem obviously vanishes when projectors are used, and then from Gleason's theorem no other choice is left than Born's rule. It is worth emphasizing also that Bell's or Kochen-Specker's theorems consider discrete sets of contexts, whereas Gleason's theorem is based upon the interplay between the continuum of contexts, and the quantized number of modalities accessible in a given context. This feature also fits perfectly with the CSM ideas.

We note that some recent derivations of Born's law [START_REF] Zurek | Quantum Theory of the Classical: Quantum Jumps, Born's Rule, and Objective Classical Reality via Quantum Darwinism[END_REF][START_REF] Cabello | The physical origin of quantum nonlocality and contextuality[END_REF][START_REF] Masanes | The Measurement Postulates of Quantum Mechanics are Redundant[END_REF] dismiss Gleason's theorem, on the basis that its hypotheses are either too strong (extracontextuality) or unjustified (projective probabilities). More precisely, refs. [START_REF] Zurek | Quantum Theory of the Classical: Quantum Jumps, Born's Rule, and Objective Classical Reality via Quantum Darwinism[END_REF][START_REF] Cabello | The physical origin of quantum nonlocality and contextuality[END_REF][START_REF] Masanes | The Measurement Postulates of Quantum Mechanics are Redundant[END_REF] argue for the non-relevance of Gleason's theorem to QM, in opposition to the CSM view. Quoting [START_REF] Masanes | The Measurement Postulates of Quantum Mechanics are Redundant[END_REF]: "As mentioned in the introduction, Gleason's theorem and many other derivations of the Born rule assume the structure of quantum measurements. That is, the correspondence between measurements and orthonormal bases {ϕ i }, or more generally, positive-operator valued measures. But in addition to this, they assume that the probability of an outcome ϕ i does not depend on the measurement (basis) it belongs to."

In [START_REF] Masanes | The Measurement Postulates of Quantum Mechanics are Redundant[END_REF] this additional assumption (which is physically true) is called "non-contextuality", that is clearly misleading, clashing with the terminology used in the Kochen-Specker theorem. As written above, a better name is "non-contextual assignment of probabilities", and the best name is just extracontextuality, that has deep physical roots. This is made clear by associating projectors to extravalence classes, clearly distinguishing the physical result (the modality) and the mathematical construction (the projector). To answer the remark about "assuming the structure of quantum measurements", we do posit the projective structure of quantum probabilities [START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF], not as a deduction but as a duly justified inference [START_REF]Our approach can be related to an "Inference to the Best Explanation[END_REF]. In the CSM approach the mathematical formalism works because physics tells the rules, and not the opposite. Therefore in our approach Gleason's hypotheses have a deep physical content, linking contextual quantization and extracontextuality of modalities. Since these features are required from empirical evidence, the QM formalism provides a good answer to a well-posed question.

VI. ALGEBRAIC SCHEME FOR QUANTUM MEASUREMENTS.

A consequence of our approach is that usual textbook quantum mechanics, which is limited to type-I operator algebra as introduced initially by Murray and Von Neumann [START_REF] Murray | On Rings of Operators[END_REF], is not universal because it does not include the context. This issue was already discussed by Von Neumann [START_REF] Neumann | On infinite direct products[END_REF], and again later in the framework of algebraic quantum theory [START_REF] Landsman | For a review on quantum measurements in the algebraic framework[END_REF]. Nevertheless, as discussed in these articles, it is possible to get a full picture by including the context in the formalism, taking into account that its number of degrees of freedom is unbounded, which makes its algebra of operators non-type I [START_REF] Neumann | On infinite direct products[END_REF][START_REF] Landsman | For a review on quantum measurements in the algebraic framework[END_REF]. Fig. 3 displays such a generic scheme, including the system (plus ancillas) and context, separated by a (movable) cut. The full scheme is then universal, but the mathematical description including type II or III algebra does not allow arbitrary quantum superpositions at the context level -in agreement with empirical evidence. Then a quantum measurement proceeds as follows:

• Before the measurement the modality is associated with the following (density) operator in context C 1 :

|ψ i ψ i | ⊗ ρ (C1) i
Specifying the modality requires to give both |ψ i ψ i | and ρ (C1) i because the projector |ψ i ψ i | specifies only an extravalence class of modalities.

• After the measurement carried out in context C 2 , but before reading out the result, the sectorized state (statistical mixture) is

j p j |φ j φ j | ⊗ ρ (C2) j
This form is completely generic from a mathematical point of view because the context is unbounded, and it can be justified in several possible ways: sectorization in the non-type-I algebra, loss of offdiagonal elements of the reduced density matrix, flow of information to the environment, loss of interference, loss of the ability to create entanglement in a projective measurement... They all lead to the same results, as discussed e.g. in [START_REF] Coles | Unification of different views of decoherence and discord[END_REF].

• After reading out the measurement result k in context C 2 , the new modality can be updated and it is associated with the operator:

|φ k φ k | ⊗ ρ (C2) k
This defines a new pre-measurement modality, and |φ k φ k | may evolve unitarily until the next measurement is performed.

Summarizing, the non-unitary step in the measurement is due to the fact that the whole unbounded context is involved in a transient way; this is not an additional ingredient, but a required part of the full (non type-I) formalism. Looking at |ψ as the "state of the system", as done usually, is misleading because the vector (or projector) is associated with an extravalence class of modalities. The basic CSM tenet, that the modality belongs to both the system and the context, appears explicitly here under a mathematical form.

As a conclusion, usual type I QM provides a description of (idealized) isolated quantum systems. A state vector or projector is "incomplete" because it is not associated with an actual modality, but with an extravalence class of modalities, belonging to different contexts. From a physical point of view, the modality belongs jointly to a quantum system, and to a specified context. From a mathematical point of view, the behavior of modalities can be studied using type-I QM, where Born's rule applies as a consequence of Gleason's theorem. On the other hand, the description of (unbounded) contexts requires a non type-I formalism. Overall, these combined tools provide a consistent picture of quantum measurements within a unified quantum framework.

Annex 1 : Proof of the geometrical lemmas.

Here we show that for two vectors u, v in the northern hemisphere with h(u) > h(v), one has f (u) ≥ f (v).

For this purpose we define D u , the great circle going through u and cutting the equator at two points corresponding to vectors orthogonal to u. By convention D u is called the "descent through u", and u is obviously the "northern vector" in D u . Then one prooves the two lemmas:

Basic lemma : One has f (u) ≥ f (u ) for any u in D u . Proof : Consider a vector u, and another vector u within D u . Let v (resp. v ) be a vector in D u orthogonal to u (resp. u ). Adding a vector w perpendicular to the D u plane, {u, v, w} and {u , v , w} are two orthonormal basis. By definition of f one has f

(u) + f (v) + f (w) = f (u ) + f (v ) + f (w) therefore f (u) = f (u ) + f (v ) since f (v) = 0 because v is on the equator. Since f (v ) ≥ 0 one has f (u) ≥ f (u ).
Piron's lemma : Consider u, v such that h(u) > h(v). Then there is a series of N vectors w n such that w 0 = u, w N = v, and each w n is within D wn-1 , i.e. in the descent through the previous vector of the series. Proof : It relies on a smart geometrical construction due to Piron [START_REF] Piron | Foundations of Quantum Physics[END_REF]. It is convenient to project the northern hemisphere on a plane tangent at the pole p, using a projection from the center of the sphere. The different latitudes are then concentric circles centered on p, and the equator is projected at infinity. The descent through u is a straight line, tangent at u to the circle corresponding to the latitude of u. Then there are two cases :

-If u and v have the same longitude, one takes u = w 0 , v = w 2 , and there exists w 1 with a latitude between those of w 0 and w 2 , located on D u = D w0 , and such that w 2 is on D w1 (this is clear by looking at the previous projection, u and v are on the same line coming from p).

-If u and v have different longitudes, one can take u = w 0 , v = w N , and build the other vectors w n by progressively rotating between the two circles associated to the two latitudes. When these latitudes get closer, N becomes larger, and it tends to infinity for two different longitudes with almost the same latitude (again this is clear from a drawing). This proves the lemma.

Therefore the basic lemma relates u and u within the descent through u, and Piron's lemma relates u and v from a succession of descents through the vectors in the series w n . As a conclusion, one deduces from the two lemmas that for u, v in the northern hemisphere with h(u) > h(v), one has f (u) ≥ f (v).

Annex 2 : Proof of the magical lemma. For mathematical reasons related to continuity [24], it is assumed that g is defined on [0,1] except an at most countable set K of points, and that the hypotheses (i) g(0)=0, (ii) a < b =⇒ g(a) < g(b), (iii) a + b + c = 1 =⇒ g(a) + g(b) + g(c) = 1 are valid under these same conditions. One has then g(1) = 1, since g(0) = 0 and g(a) + g(b) + g(c) = 1. Considering the rational numbers r and s, and a 0 outside K, one gets: g(ra 0 ) + g(sa 0 ) + g(1 -ra 0 -sa 0 ) = 1 g(0) + g(ra 0 + sa 0 ) + g(1 -ra 0 -sa 0 ) = 1 =⇒ g(ra 0 ) + g(sa 0 ) = g(ra 0 + sa 0 ) = g((r + s)a 0 ) =⇒ g(ra 0 ) = rg(a 0 )

Taking the limit r → 1/a 0 one gets g(1) = g(a 0 )/a 0 = 1 and thus g(a 0 ) = a 0 and g(ra 0 ) = ra 0 .

FIG. 1 :

 1 FIG.1:The four modalities {ui, vj, x l , w k } belong to four different contexts, and ui (resp. vj) is extravalent with x l (resp. w k ). Then all probabilities represented by dashed lines are equal according to Theorem 2.

The word "context" includes the actual settings of the device, e.g. measurement of Sz rather than Sx: the context must be factual, not contrafactual. On the other hand all devices able to measure Sz are equivalent as a context, in a (Bohrian) sense that they all define the same conditions for predicting the future behaviour of the system.

We omit the free evolution of the system; if it is present, the result of a new measurement can still be predicted with certainty, but in another context that can be deduced from the free evolution. Mutatis mutandis, this is equivalent to full repeatability.

In ref.[START_REF] Auffèves | Recovering the quantum formalism from physically realist axioms[END_REF] extravalent modalities in different contexts are considered to be the same modality, transferred from a context to another. This is however not satisfactory, since a modality belongs to a specific context (and system). The notions of extracontextualy and extravalence are therefore more suitable to distinguish modalities and "state vectors", as explained in ref.[

[START_REF] Auffèves | Extracontextuality and extravalence in quantum mechanics[END_REF]].[START_REF]Our approach can be related to an "Inference to the Best Explanation[END_REF] Note that extravalent modalities appear only if N ≥ 3, this has an obvious geometrical interpretation in relation with Gleason's theorem (see below).

In order to make sense of Theorem 2, it is essential to distinguish between modalities and vectors in an Hilbert space, that will correspond to extravalence classes of modalities (see below). This issue is also essential for understanding Gleason's hypotheses.

This means a positive semidefinite Hermitian operator with unit trace. It describes a pure state if it is a rank one projector.

Acknowledgments

The authors thank Franck Laloë and Roger Balian for many useful discussions, and Nayla Farouki for continuous support.

System

Cut Context (unbounded) Type I (countable basis)

Type II or III (uncountable basis) Unitary equivalence Loss of unitary equivalence Superpositions Sectorization + updating o- -----------X---------------------- -------------------X--------------FIG. 3: Generic scheme including the system, a possible ancilla, and the context. The number of degrees of freedom in the context is unbounded, which makes its algebra non type-I. The cut separates a type-I system algebra, where usual QM applies, from the type-II or III context algebra where there is no more unitary equivalence of representations.