Each grey line is a linear fit to the EVPAs measured at the four ALMA Band 3 and Band 6 SPWs, yielding the RM in each epoch, and the extrapolated intercept at the Y-axis is $\chi_0$. Note the large offset in $\chi_0$ between the 3 mm and 1.3 mm bands, despite the consistency of $\chi_0$ at 1.3 mm across 6 days.

We report a tentative detection of  CP at 1.3 mm in the range  $(-1.0 \pm 0.6)$% to $(-1.5 \pm 0.6)$%.
This is consistent with the first detection with the SMA  from observations carried out in 2005–2007 <cit.> and with a more recent ALMA study based on 2016 observations <cit.>.
This result suggests that the handedness of the mm-wavelength CP is stable over timescales larger than 12 yr.
Interestingly, historical VLA data (from 1981 to 1999) between 1.4 and 15 GHz  show that  the emission is circularly polarized at the 0.3% level and is consistently left-handed  <cit.>, possibly extending the stability of the CP sign to 40 years.
Such a remarkable consistency of the sign of CP over (potentially four) decades suggests a stable magnetic field configuration (in the emission and conversion region).

Similarly to the RM, we also note a weak anti-correlation  between LP and CP (although more observations are needed to confirm it).

We do not detect  CP at 3 mm ($<0.06\% $, $3\sigma$ upper limit).
<cit.> and  <cit.> find that, once one combines the cm and mm measurements,  the CP fraction as a function of frequency should be characterized by a power law with $\nu^{0.35}$. Using the measurements at 1.3 mm, this shallow power-law would imply a CP fraction at the level of $\sim 0.7-1.1$% at 3 mm,  which would have been readily observable.
The non detection of CP at 3 mm suggests that the CP spectrum may not be monotonic.

Although the origin of the CP is not well understood,
since a relativistic synchrotron plasma is expected to produce little CP, <cit.> suggest that the observed CP is likely generated close to the event horizon by the Faraday conversion which transforms LP into CP via thermal electrons that are mixed with the relativistic electrons responsible for the linearly polarized synchrotron emission <cit.>.
In this scenario, while the high degree of order in the magnetic field necessary to produce LP $\sim$ 7% at 1.3 mm naturally leads to a high CP  in a synchrotron source, the absence of CP at 3 mm is consistent with the low LP measured.
See  <cit.> for a detailed discussion of potential origins for the CP emission.

A final caveat is that based on the analysis presented in <ref> and Appendix <ref>, the physical interpretations above should be considered as tentative.

Flux-density variability

We do not report significant variability  in total intensity and polarized intensity,
which is about 10% in six days (comparable to the absolute flux-scale uncertainty in ALMA Band 6).
<cit.> and <cit.> report more  significant variability in all polarization parameters based on intra-day light curves in all four Stokes parameters. This type of analysis is beyond the scope of this paper, and will be investigated elsewhere.

M87 RM measurements with ALMA.

Date      I        LP     EVPA     $\chi_0$     RM     Beamsize     Project Code

[Jy]        [%]     [deg]     [deg]     [$10^5$ rad m$^{-2}$]

7c3 mm

2015/09/19             2.17$\pm$0.11          1.37$\pm$ 0.03                       30.68$\pm$0.74          41.7$\pm$3.1                    -0.201$\pm$0.054     053             2013.1.01022.S[Stokes I, Q, and U were extracted from the images using the CASA task .  was used for all the other experiments.]

2015/11/11             1.93$\pm$0.10          1.30$\pm$ 0.03                       21.47$\pm$0.69           3.9$\pm$2.7                   0.318$\pm$0.049       015          2015.1.01170.S$^{\rm a}$

2016/10/04             1.85$\pm$0.10          2.39$\pm$0.03                       33.35$\pm$0.36           107.4$\pm$1.4               -1.227$\pm$0.023             043          2016.1.00415.S

7c1.3 mm

2017/04/05              1.28$\pm$0.13         2.42$\pm$ 0.03                        -7.78$\pm$0.37                -14.6$\pm$2.9                0.64$\pm$0.27             15            2016.1.01154.V

2017/04/06              1.31$\pm$0.13         2.16$\pm$ 0.03                       -7.61$\pm$0.39                 -23.6$\pm$3.1                1.51$\pm$0.29              18            2016.1.01154.V

2017/04/10              1.33$\pm$0.13         2.74$\pm$ 0.03                       0.11$\pm$0.32                  3.5$\pm$2.5               -0.32$\pm$0.24            15            2016.1.01154.V

2017/04/11              1.31$\pm$0.13         2.71$\pm$ 0.03                       -0.63$\pm$0.31                 3.7$\pm$2.4                -0.41$\pm$0.23              10           2016.1.01154.V

2018/04/21              1.11$\pm$0.11         2.29$\pm$ 0.03                       27.18$\pm$0.38                 70.6$\pm$3.0                      -4.11$\pm$0.28              09           2017.1.00841.V

2018/04/22              1.18$\pm$0.12         1.71$\pm$ 0.03                      26.42$\pm$0.52                  68.9$\pm$4.0                      -4.02$\pm$0.39              09           2017.1.00841.V

2018/04/25              1.14$\pm$0.11         2.21$\pm$ 0.03                      36.12$\pm$0.39                   68.4$\pm$3.0                -3.05$\pm$0.29              09           2017.1.00841.V

2018/09/25              1.16$\pm$0.12         0.78$\pm$ 0.04                       –                     –            –      035           2017.1.00608.S[The lower LP estimated for this project is likely caused by a systematic offset between Stokes $QU$ and $I$ (see <ref>). 2017.1.00608.S was not used in the analysis.]

§ DISCUSSION

In this section, we review general polarization properties of AGN comparing the two (1.3 mm and 3 mm) frequency bands, different AGN classes, and depolarization mechanisms   ( <ref>); then we
interpret the Faraday properties  derived for M87 in the context of existing accretion and jet models as well as a new two-component  polarization model ( <ref>);
and finally we discuss additional constraints on the Sgr A* polarization model from a comparison of 1.3 mm and 3 mm observations ( <ref>).

§.§ Polarization degree and Faraday rotation in AGN

§.§.§ 1.3 mm vs. 3 mm

Synchrotron emission opacity
The total intensity spectral indexes  for  the AGN sources in the sample vary in the range $\alpha$=[–0.7,–0.3] at 3 mm and $\alpha$=[–1.3,–0.6] at 1.3 mm, Cen A being the only exception, with $\alpha$=–0.2 (see Tables <ref> and <ref> and Appendix <ref>).
This contrasts with the flat spectra  ($\alpha$=0) typically found at longer cm wavelengths in AGN cores <cit.>, corresponding to optically-thick emission.
In addition, we  observe a spectral  steepening  (with $\Delta \alpha=[-0.2,-0.4]$) between 3 mm and 1.3 mm; although one should keep in mind the caveat of time variability,
since the observations in the two frequency bands were close in time (within ten days) but not simultaneous.
Such spectral steepening can naturally be explained by decreased opacity of the synchrotron emission  at higher frequencies in a standard jet model <cit.>.

LP degree
We detect LP in the range 0.9–13% at 3 mm and 2–15% at 1.3 mm (excluding the unpolarized sources NGC 1052 and Cen A).
At 1.3 mm, the median fractional polarization is 5.1%, just slightly higher than the median LP at 3 mm, 4.2%, yielding a ratio of 1.2.
If we consider only the sources observed in both bands, then the ratio goes slightly up to 1.3 (or 1.6 including also Sgr A*).
Despite the low statistics, these trends are marginally consistent with results from previous single-dish surveys with the IRAM 30-m telescope <cit.> and  the Plateau de Bure Interferometer or PdBI <cit.>. In particular, <cit.> find an LP ratio of 1.6 between 1 mm and 3 mm based on  simultaneous, single-epoch observations of a sample of 22 radio-loud ($F>1$ Jy) AGN,
while  <cit.> find an LP ratio of 2.6 based on long-term  monitoring, non-simultaneous observations of 29 AGN.
<cit.> find similar numbers from a sample of 73 AGN observed
as part of the IRAM/PdBI calibration measurements  during   standard interferometer science operations[The polarimetric data analysis is based on Earth rotation polarimetry and is antenna-based, i.e. executed for each antenna separately. Therefore, no interferometric polarization images are available from this study.].
The comparison of these statistics at both  wavelengths suggests  a general higher degree of polarization at 1 mm as compared to 3 mm.
This finding can be related either to a smaller size of the emitting region   and/or to a higher ordering  of  the  magnetic-field  configuration <cit.>.
In fact, according to the standard jet model, the size of the core region decreases as a power-law of the observing frequency, which could help explain the higher LP observed at 1 mm.  Alternatively, the more ordered magnetic-field  configuration  could be related  to a large-scale (helical) magnetic-field structure along the jets.

Faraday RM
Among the six sources observed both  at 3 mm and 1.3 mm, we have RM detections at the two  bands only in  3C273, where
the estimated value at 3 mm is significantly lower than at 1.3 mm.
For the remaining sources with RM
detections at 3 mm (NRAO 530, OJ 287, and 3C279)
and  at 1.3 mm (J1924-2914 and 4C 01.28), their $3\sigma$ upper limits, respectively at 1.3 mm and 3 mm,  still allow a larger RM at the higher frequency band.

A different  'in-band' RM  in the 3 mm and 1.3 mm bands can be explained either with
(i)  the  presence of an internal Faraday screen or multiple external  screens in the beam; or with
(ii)  a different opacity of the synchrotron emission  between the two bands.
Case (i) will cause a non-$\lambda^2$ behaviour of the EVPA and a non-trivial relation between the  'in-band' RM determined at only two narrow radio bands.
Evidence for non-$\lambda^2$ behaviour of the EVPA can be possibly seen in OJ 287, 4C 01.28, and J0006-0623 at 1.3 mm (Fig. <ref>) and J0510+1800 at 3 mm (Fig. <ref>).
In order to estimate $B$ or $n_e$ from the RM (see Eq. <ref>), one would need to sample densely the EVPA over a broader frequency range and perform a more sophisticated  analysis, using techniques like
the Faraday RM synthesis or Faraday tomography <cit.>.
This type of analysis is beyond the scope of this paper and can be investigated in a future study (we refer to   <ref> for evidence of internal Faraday rotation and     <ref> for an example of a multiple component Faraday model for the case of M87).
Since the spectral index analysis shows that  the AGN in the sample become more optically thin at 1 mm, the  observed  differences in the 'in-band' RM  at 3 mm and 1.3 mm  can be likely explained with synchrotron opacity effects alone (with the caveat of time variability since the observations are near-in-time but not simultaneous).

It is also interesting to note that we also see a sign reversal  between the RM measured at 3 mm and 1.3 mm for 3C 273.
RM sign reversals require  reversals in $B_{||}$ either over time  (the observations in the two bands were not simultaneous) or across the emitting region (the orientation of the magnetic field is different in the 3 mm and 1 mm regions).
With the data in hand we cannot distinguish between time variability
or spatial incoherence of the magnetic field (we refer to  <ref> for a discussion on possible origins of RM sign reversals in AGN).

§.§.§ Blazars vs. other AGN

We find that blazars are more strongly polarized than other AGN in our sample,  with a median LP $\sim$7.1% vs. 2.4% at 1.3 mm, respectively.
Furthermore, blazars have approximately an order-of-magnitude  lower RM values  (on average)   than other AGN, with a median value of
$\sim 0.07 \times 10^5$   at 1.3 mm (with the highest values of $\sim 0.4 \times 10^5$  exhibited by J1924-2914), whereas for other AGN we find a median value of $\sim 0.4 \times 10^5$ at 1.3 mm[In computing the median we exclude the unpolarized Cen A and NGC 1052 for which we cannot measure a RM.] (with the highest values $> 10^5$   exhibited by M87 and 3C 273).

<cit.> used the Combined Array for Millimeter Astronomy (CARMA) and  the SMA to observe at 1.3 mm two low-luminosity AGN (LLAGN), M81 and M84, finding upper limits to LP of 1%–2%.  Similarly,  <cit.> used CARMA to observe the LLAGN 3C 84 at 1.3 and 0.9 mm, measuring an LP in the 1%–2% range, and a very high RM of $\sim (9\pm2) \times 10^5$.
These low values of LP (and high values of RM) are comparable to what we find in M87, which is also classified as a LLAGN <cit.>.

When put together, these results suggest that blazars have different polarization properties at mm wavelengths from all other AGN, including LLAGN, radio galaxies, or regular QSOs[Similar conclusions were reached from VLBI imaging studies of large AGN samples at cm wavelengths <cit.>.].
These mm polarization differences can be understood in the context of the viewing angle unification scheme of AGN.
A smaller  viewing angle implies a stronger Doppler-boosting  of the synchrotron emitting plasma in the jet, which in turn  implies a  higher polarization fraction for  blazars. Furthermore, their face-on geometry    allows the observer to reach the innermost radii of the nucleus/jet and reduces the impact of  the ‘scrambling’ of  linearly polarized radiation  by averaging different polarization components within the source (e.g. Faraday and beam depolarization – see next section), also resulting in  higher LP (and lower RM).

§.§.§ Depolarization in radio galaxies

In the previous section we point out that radio galaxies and LLAGN exhibit  lower polarization degree than blazars.
In particular, the radio galaxies Cen A and NGC 1052 do not show appreciable polarized intensity (LP$<$0.2%) at 1.3 mm.
We suggest several  depolarization mechanisms that may be at play in these radio galaxies (and potentially other LLAGN):

* Faraday depolarization due to a thick torus or a dense accretion flow.

* Bandwidth depolarization due to a strong  magnetic field.

* Beam depolarization due to a disordered magnetic field.

* Thermal (non-synchrotron) emission.

In the following, we elaborate on these mechanisms.

a. Faraday depolarization due to a thick torus or a dense accretion flow
Radio galaxies are often surrounded by an obscuring torus. The cold gas in the torus can be photo-ionized by UV photons from the inner accretion disk  and  the mixture of thermal and non-thermal material    could be responsible for the strong depolarization of the inner regions  via Faraday rotation – and one speaks of   Faraday depolarization.
In the case of NGC 1052,
<cit.> created synthetic radio maps of the jets using  special-relativistic hydrodynamic (SRHD) simulations and suggested that an obscuring torus can explain some of the observed properties of these jets.
In fact, the presence of a massive ($\sim 10^7 \, M_\odot$) and dense ($> 10^7$ cm$^{-3}$) molecular torus has been recently demonstrated with ALMA observations <cit.>.
A clumpy  torus is also known to surround the nucleus of Cen A <cit.>, as also suggested by our 1.3 mm map (see Fig. <ref>, bottom-right panel).
Therefore the presence of a thick torus of cold gas could naturally explain the low polarization degree in both radio galaxies.
A similar mechanism can be at play  in LLAGN whose radio emission is thought to be powered by synchrotron radiation from a geometrically thick, hot accretion flow <cit.>.

b. Bandwidth depolarization due to a strong magnetic field
A large homogeneous magnetic field implies an intrinsically large homogeneous RM (see Eq. <ref>),
resulting in the source to appear unpolarized in broadband observations – and one speaks of bandwidth depolarization.
In  NGC 1052, GMVA imaging at 86 GHz helped constrain the magnetic field at Schwarzschild radius () scales in the range  360–70000 G <cit.>,  providing evidence of an extremely high magnetic field near the SMBH. Coupled with its high inclination <cit.>, such a strong magnetic field would result in  the source to appear unpolarized in ALMA broadband observations.

c. Beam depolarization due to a tangled magnetic field
If the magnetic field in the emitting regions or in a foreground Faraday screen is tangled or generally disordered on scales much smaller than the observing beam, magnetic field regions with similar  polarization degrees but opposite signs will cancel out and the net observed polarization degree would be  significantly decreased – and one speaks of beam depolarization.
We do not have evidence of such tangled magnetic field for any of the low LP sources in our sample, which will require high-resolution polarization imaging with the GMVA or the EHT.

d. Thermal (non-synchrotron) emission
Multi-wavelength (MWL) studies in Cen A show that its SED is moderately inverted up to the infrared, possibly indicating a dust contribution at mm wavelengths <cit.>.
Using VLBI imaging at 229 GHz with the EHT, Janssen et al. (submitted) measured a flux of  $\sim 2$ Jy in the VLBI core, indicating that the EHT filters out $\sim 65$% of the emission seen by ALMA.
While we cannot exclude contribution from thermal emission to the total flux measured by ALMA, the flux measured with the EHT must necessarily be associated with non-thermal emission. We therefore conclude that dust emission is an unlikely explanation for the lack of LP at 1.3 mm.

An improved data analysis including spectro-polarimetry could be helpful to measure the actual RM in both NGC 1052 and Cen A and thus assess which is the dominant depolarization mechanism among the ones discussed above (this is however beyond  the scope of this paper).
As a final note, an interesting insight may come from a comparison between mm and IR wavelengths, where both  NGC 1052 <cit.> and Cen A <cit.> are highly polarized. These characteristic are similar to Cygnus A, where the low polarized core at mm-wavelengths and the high polarized core at IR wavelengths may be the signature of an ordered magnetic field in the plane of the accretion disk supporting the accretion flow and/or jet formation <cit.>.

§.§ Physical origin of the rotation measure in M87

We can now use   the polarimetric and Faraday properties of the mm emission from M87  reported in  <ref>,  to constrain properties of accretion models onto the M87 SMBH.

Models aiming to explain the origin of the RM in M87 should match the following key observed features (see  <ref> for a full list of findings):

* Low LP and high RM.
M87 has a rather low LP ($\sim 2.3\%$ mean at 1.3 mm) when compared to Sgr A* and other blazars in the sample (see <ref>), while the measured RM can be as high as a few times $10^5$ .

* RM sign reversals.
Observations on different dates yield large differences in the measured RM values, which can be either positive or negative  (in both the 3 mm and 1.3 mm bands).
This requires sign flips in $B_{||}$ over time and/or across the emitting region.

* Rapid RM time variability.
In Apr 2017, the RM magnitude appears to vary significantly (from non-detection  up to $1.5 \times 10^5$ ) in just 4-5 days.
This suggests the presence of small-scale fluctuations in the emitting source and/or the Faraday screen.

* $\lambda^2$ scaling.
Plots in Figures <ref> and <ref> clearly display a $\lambda^2$ dependence of the EVPA at 1.3 mm and 3 mm on specific days (although this is not always the case).

The MWL spectral energy distribution (SED) of the M87 core is best explained
by emission from advection dominated/radiatively inefficient accretion flow
<cit.> or from a jet <cit.>,
or emission from a combination of these two components <cit.>. In the hybrid models, the low-frequency radio emission is produced by the jet while the optically-thin mm/submm (and X-ray) emission can either come from the jet base or the inner accretion flow.

The traditional  approach  adopted  in  previous  studies was to  assume that
the large scale ($r \sim~100$ ) accretion flow itself may act as a Faraday screen and that the  core emission region lies entirely behind the same portion of the Faraday screen
(e.g., the core emission size is small compared to the scale of any fluctuations in large scale flow).
In the framework of semi-analytic RIAF/ADAF models, the RM magnitude
has then been used  to estimate mass accretion rates onto black holes in Sgr A*  <cit.> and in 3C 84 <cit.>.
A similar approach has been used in M87, yielding estimates of the accretion rate in the range from $\dot{M} < 9 \times 10^{-4}$ <cit.> to $\dot{M}\sim [0.2,1] \times 10^{-3}$ <cit.>, where the quoted values are either upper limits or depend on specific model assumptions (e.g. black hole spin or the exact location of the Faraday screen).
From the largest RM that we measured,
$ 4 \times 10^5$  (on Apr 2018), using Equation (9) in <cit.>, we would infer a mass accretion rate of $\dot{M} = 7.7 \times 10^{-8} |RM|^{2/3} \sim 4 \times 10^{-4}$  assuming an inner boundary to the Faraday screen of  $R_{\rm RM, in}=21$  (as suggested by ).

While our estimates of mass accretion rates are consistent with previous similar estimates and upper limits from the ADAF/RIAF models, the observed properties listed above, especially the time variable RM and its sign reversals, provide new constraints.
In particular, the timescale of the RM variability can be set by  the rotating medium  dynamical time ($\propto \sqrt{R^3/GM}$), thus
constraining the radius at which the RM originates.
The rapid variability observed in Apr 2017 implies that the RM should occur much closer to the SMBH (within a few ) than  assumed in previous mass accretion models, which in turn suggests the possibility of a co-location of the emitting and rotating medium.
In the alternative, the Faraday screen could be at further distance and  the observed variability could be ascribed to rapid fluctuations in the emitting source.
Therefore, both  a turbulent accretion flow acting  as a Faraday screen or a varying compact source with an external screen can  explain the observed time-variability.
Finally, the accretion flow is not the only possible source of Faraday rotation.
Since  simulations show that relativistic jets can have a “spine-sheath” structure <cit.>, the jet sheath can provide a magnetized screen surrounding the jet, and indeed it has been also suggested as a plausible source of Faraday rotation in AGN  <cit.>.
Therefore, either (or both) the accretion flow and/or the jet can in principle be the sources of the mm emission and/or  the Faraday rotation.

All the scenarios described above imply a more complicated physical origin of the Faraday rotation than is usually assumed in traditional semi-analytic models that use the RM to infer a mass accretion rate. We conclude that, unlike the case of Sgr A*, the RM in M87 may not provide an accurate estimate of the mass accretion rate onto the black hole.

In what follows, we review clues on the location
of the Faraday screen using observational constrains from  ALMA    ( <ref>)
as well as information on horizon scales from the EHT
( <ref>).

§.§.§ Location of the Faraday screen: internal vs. external

We distinguish between two general cases:  internal and  external Faraday rotation.

* Internal Faraday rotation:
the accretion flow or jet can simultaneously be the source of synchrotron radiation and the Faraday screen.

* RM rapid time variability and sign reversals.
Recent time-dependent GRMHD simulations of the M87 core <cit.>
show that turbulence within the accretion flow is able to change $B_{||}$ in both amplitude and orientation, resulting in significant RM fluctuations and sign reversals on the dynamical time at $R \simeq 2.5-5$ , corresponding to short timescales of a few days for M87 (consistent  with properties #ii and #iii).

* Beam Depolarization.
An internal Faraday screen could cause
beam depolarization of the synchrotron emission. This has been theoretically predicted by GRMHD simulations of the M87 core emission,  which yield  low  values of LP (typically  in the range 1%—3%) and  large Faraday RM ($\gtrsim 10^5$ )  <cit.>, broadly consistent with the  observed feature #i.
We however note that the beam depolarization could be also caused by external Faraday rotation in an  inhomogeneous screen.

* External Faraday rotation:
the emission region lies entirely behind (and it is not inter-mixed with) the Faraday screen.

* $\lambda^2$ scaling of the EVPA.
A $\lambda^2$ dependence is typically adopted as observational evidence of the fundamental assumption on the location of the Faraday screen as external relative to the background source.
Although it can be argued that EVPA  variations in a narrow frequency range could  be  approximated  to  be linear
(at  1.3/3 mm the fractional $\lambda^2$ bandwidth is only 16/32% of the central wavelength),
good linear fits of EVPA  vs.  $\lambda^2$  are also obtained from lower-frequency observations, including the range 2, 5 and 8 GHz  <cit.>, 8, 15, 22 and 43 GHz <cit.>, and 24, 43 and 86 GHz <cit.>.

* RM sign reversals and helical magnetic fields.
Polarimetric images with the VLBA at 43 GHz have revealed magnetic field vectors wrapped around the core <cit.>, suggesting that toroidal fields might be dominant on scales of hundreds of .
Helical magnetic fields threading the jet may be produced by the differential rotation either in the BH ergosphere or in the innermost regions of the accretion disk  <cit.>.
If toroidal fields are dominant in the sheath, one would expect transverse RM gradients across the jet, with opposite signs of the RM from one edge to the other, as   shown in a handful cases where VLBI images resolve the jet width <cit.>.
Systematic changes in the signs of these gradients, leading to RM sign reversals in unresolved measurements,
can be explained with a number of models,
including the magnetic "tower" model <cit.>,
or  the "striped" jet model <cit.>.
Nevertheless, it remains difficult to explain the rapid fluctuations observed in Apr 2017 with these models.

A long-term  monitoring   with beam-matched simultaneous observations at multiple frequency bands would be required to assess the frequency-dependence of the RM and to conclusively discriminate between internal and external Faraday rotation.
Clear evidence of $\lambda^2$ scaling in a wider frequency range   would be evidence of the external scenario, while deviations from $\lambda^2$ would support the internal scenario.
A time cadence from a few days to a few months would allow us to assess whether the RM sign flips are stochastic (favoring the internal scenario), or systematic (favoring the external scenario).

§.§.§ Two-component polarization model for M87

Diagram of the two-component model, comprised of a compact (blue) and extended (red) polarized emission regions with corresponding small-scale and large-scale Faraday screens.  The small-screen, which may be external (shown) or internal (not shown) acts only on the compact component, which is observed by the EHT.  ALMA observes the combined emission from both components.

EHT estimates of the  flux in the compact core, i.e., that arising on horizon scales, can at most account for $50\%$ of that measured by ALMA, both in total intensity <cit.> and polarized intensity <cit.> emission.  Natural origins for the additional emission are the larger-scale structures in the jet.
The additional components may encompass many scales, be discrete features (e.g., HST-1), or be a combination thereof.
In order to interpret these differences revealed by the EHT and ALMA, we adopt the simplest version of a multi-scale model permissible – a two-component model comprised of a variable compact region and static extended region (see Figure <ref>).  We find that this is sufficient to harmonize the polarimetric properties observed by both the EHT and ALMA in Apr 2017, including the interday variability in the ALMA RMs   and the  EVPA variation of the compact core as observed by the EHT.

Both the compact and extended components of the two-component model consist of total intensity, spectral index, linearly polarized flux, and polarization angle.  We consider both internal and external Faraday screen models for the compact component.  In both cases, the Faraday screen for the extended component is assumed to be external.  A model likelihood is constructed using the integrated EHT Stokes $I$, $Q$, and $U$ ranges presented in Table 7 in Appendix H2 of <cit.>, and the ALMA core Stokes $I$, $Q$, and $U$ values for the individual SPWs in Table <ref>, assuming Gaussian errors.  This likelihood is then sampled with the EMCEE python package <cit.> to obtain posterior probability distributions for the model components.  For more details regarding the model, priors, and fit results, see Appendix <ref>.

Across days, only the LP and EVPA of the compact component is permitted to vary.  This is consistent with the extended component being associated with much larger physical structures and required by the polarimetric variability observed by the EHT <cit.>.  There is no evidence that variability in any other component of the two-component model is required: despite static Faraday screens, permitting the polarization of the compact component to vary is capable of reproducing the rapid changes in the ALMA RMs.  In this picture, the observed RMs arise in part from the wavelength-dependent competition between the two components, and thus are not directly indicative of the properties of either Faraday screen.

Posteriors implied by the April 5, 6, 10, and 11, 2017 ALMA and EHT observations for the   RMs measured on ALMA (extended) and EHT (compact) scales  in the two component model when it is assumed that the compact Faraday screen is external (lower left, blue) and internal (upper right, red) to the emission region.  Contours are shown for the 50%, 90%, and 99% quantiles.  For comparison, the $2\sigma$ range of April 5, 6, 10, and 11, 2017 ALMA RMs reported in Table <ref> are shown by the black crosses and gray bands.

Nevertheless, via this model we are able to separately constrain the RMs that are observed on ALMA and EHT scales; these are shown in Figure <ref>.
Specifically, while the RM of the large-scale Faraday screen is comparable to the Apr 2017 values reported in Table <ref>,
that associated with the compact component
is not directly constrained by the ALMA measurements and can be factors of many larger: at 95% confidence, the compact RM is between $-5.4\times10^5~{\rm rad~m^{-2}}$ and $-2.9\times10^5~{\rm rad~m^{-2}}$.
Interestingly, the estimated range is consistent with
the RM inferred from low-inclination GRMHD models of the M87 core <cit.>
and comparable to  the Apr 2018 values reported in Table <ref>.
This consistency  suggests the possibility that in Apr 2018 ALMA may be   seeing the core RM (e.g., as a consequence of a different beating of the  two components).
This hypothesis can be directly tested with the 2018 EHT observations which, unlike the 2017 ones, covered the full frequency spacing of ALMA (212–230 GHZ; see  <ref>), and  are therefore expected to directly measure the resolved RM of the core. This will in turn allow to quantify the interplay between compact and extended components,
and potentially explain the time variability observed with ALMA.

§.§ Constraints on Sgr A*  model  from  polarization and Faraday properties at 3 mm and 1.3 mm

Measurements of Faraday rotation at radio/mm wavelengths, either towards Sgr A* itself <cit.> or the nearby pulsar PSR J1745-2900 <cit.>, have been used to probe the magnetized accretion flow in Sgr A* on scales from tens of  out to the Bondi radius ($\sim10^5$ $R_{Sch}$).
Using the same semi-analytic RIAF/ADAF models introduced in  <ref> <cit.>, from the measured RM values at 1.3 mm we obtain an accretion rate of order $10^{-8}$  (assuming $R_{\rm RM, in}=10$ ), with a maximum variation of approximately 20% across the observing week.

We have derived for the first time the polarization and Faraday properties of Sgr A* both at 3 mm and 1.3 mm in a time-window of ten days.
Since the synchrotron photosphere in the accretion flow  moves outwards with decreasing frequency (because of increased opacity),
the polarized emission at 3 mm and 1.3 mm is expected to arise from different locations with potentially different magnetic field structures.
Any variation of the intrinsic EVPA or RM with frequency  can therefore provide interesting insights on the polarized source and magnetic field structure.
In  <ref> we infer that the intrinsic polarization vector is rotated  between
the 3 mm ($\chi_o \sim +135$   or –45 assuming a full 180 wrap) and the 1.3 mm  ($\chi_o \sim$ [–15,–19]) bands
and that the RM magnitude at 3 mm is about half of the RM value measured at 1.3 mm over a three-days separation.

From earlier VLBI measurements, we know that the emission at mm wavelengths must come from very closely situated regions of the black hole, with an intrinsic (i.e. unscattered) size of  $\sim$120 $\mu$as (or 12 ) at 3.5 mm <cit.> and $\sim$50 $\mu$as (or 5 ) at 1.3 mm <cit.>. Therefore the radius of the 3.5 mm source is 2.4 times larger than the 1.3 mm source, i.e. very close to the ratio of the RM values measured with ALMA at the two wavelengths  (see RM paragraph in  <ref>).
Taken at face value,
this result suggests that about half of  the Faraday rotation at 1.3 mm may occur between the 3 mm photosphere and the 1.3 mm source.

Although this result would be extremely constraining for the model of Sgr A*, we should point out two caveats:
i) possible presence of multiple components;
ii) potential RM time variability.
We explain these caveats below.

Multiple components
In addition to the  mini-spiral,  which is however unpolarized (see  <ref>)  and thus should not contribute significantly to the polarized flux,
the presence of a relativistic compact jet has been proposed based on theoretical modeling of the source SED, in particular to explain the radio emission in Sgr A* <cit.>.
In addition,  the only available VLBI polarimetry study at 1.3 mm has shown that the polarization structure of Sgr A* is complex <cit.>
and can be in principle different at the two wavelengths.
Therefore, we cannot completely exclude the presence of an additional jet component to the accretion flow or a more complex morphology for the intrinsic polarization.
Nevertheless, we argue that  the stability of LP, RM, and CP (including their sign) observed in Sgr A* over more than a decade, unlike the case of M87, favors the presence of one single dominating polarized component.

RM time variability
At 1.3 mm, the RM changes by $+1.5 \times10^5$ rad/m$^2$ ($\sim$ 30%) between the Apr 6/7 and Apr 11.
Assuming the same rate of time variability at 3 mm and 1.3 mm, such variability  is likely not  responsible for a factor of two difference over three days.
Likewise, the large offset in $\chi_o$ observed at 3 mm and 1.3 mm  is unlikely a consequence of time variability, given the $\chi_0$ consistency  on Apr 6–11.
Larger variations in RM were however recorded by <cit.> and <cit.> on timescales from hours to months (see for example Figure 12 in  ).
Since the observations in the two frequency bands  were close in time but not simultaneous,  we cannot definitely exclude time variability as the origin of the observed difference in RM magnitude  at 3 mm and 1.3 mm.

Future simultaneous measurements over a wider wavelength range (including  3 mm and 1.3 mm)  will allow us to separate time variability and source structure effects.

§ CONCLUSIONS

We have determined and analysed the polarization and Faraday properties of Sgr A*, the nucleus of M87, and  a dozen radio-loud AGN,
observed with ALMA during the 2017 VLBI campaign in the 3 mm and 1 mm bands in concert with the GMVA and the  EHT, respectively.

Our main findings can be summarised as follows:

* The AGN sources in our sample are highly polarized, with linear polarization degrees in the range 2–15% at 1.3 mm and 0.9–13% at 3 mm.
The radio galaxies NGC 1052 and Cen A are the only exceptions with LP$<$0.2%.

* The AGN sources have negative spectral indexes varying in the range $\alpha=[-1.3,-0.2]$,  in contrast with the flat spectra  ($\alpha$=0) typically found at longer cm wavelengths in AGN cores.
We also observe  a spectral steepening between the 3 mm and the 1.3 mm bands, which
can naturally be explained by decreased opacity of the synchrotron emission  at higher frequencies in a standard jet model <cit.>.

* We find marginal evidence for a general higher degree of polarization and RM magnitude
in the 1 mm band as compared to the 3 mm band (a
trend which is consistent with  single-dish surveys).
The increase of polarized intensity at higher frequency may be the result of an increased magnetic-field order in the inner portions of jets and/or to the smaller size of the high-frequency emitting regions.
The increase of RM with frequency can be explained by opacity effects: emission at higher frequencies is generated in, and propagates along, regions with higher magnetic fields and plasma densities <cit.>.
Given the small-number statistics (eight AGN observed at 3 mm, eleven   at 1.3 mm, and six  in both bands) and the caveat of time variability (in a time window of ten days), simultaneous observations of a larger AGN sample at multiple frequency bands are needed  to confirm these results.

* The blazars  (seven in our sample) have on average the highest level of polarization (LP $\sim$7.1% at 1.3 mm) and an order of magnitude lower RM ($\sim 0.07 \times 10^5$   at 1.3 mm) when compared with other AGN in our sample (with  LP $\sim$2.4% and RM $\sim 0.4 \times 10^5$ , respectively).  These mm polarization differences can be understood in the context of the viewing angle unification scheme of AGN:  blazars'  face-on geometry  implies a stronger Doppler-boosting of the synchrotron emitting plasma in the jet and reduces the effect of Faraday and beam depolarization in the accretion flow, resulting in higher LP (and lower RM).
Future observations of a broader sample of sources are necessary for assessing the statistical significance of these trends.

* We constrain the  circular polarization fraction in the observed AGN to $<$0.3%. For Sgr A* we report CP = [-1.0,-1.5]%, consistent with previous SMA and ALMA studies.
However, we explicitly note that the ALMA observatory does not guarantee a CP level $<$0.6% ($1\sigma$), therefore these measurements should be regarded as tentative detections.

* We derive for the first time
the polarization and Faraday properties of Sgr A* both at 3 mm and 1.3 mm in a time-window of ten days.
The RM magnitude at 3 mm, $(-2.1\pm0.1) \times10^5$ rad/m$^2$, is about half of the RM value measured at 1.3 mm over a three-days separation, suggesting that about half of  the Faraday rotation at 1.3 mm may occur between the 3 mm photosphere and the 1.3 mm source (although we cannot exclude effects related to time variability).

* We report the first unambiguous measurement of Faraday rotation   toward the M87 nucleus at mm wavelengths. At variance with Sgr A*, the M87 RM exhibits significant changes in magnitude and sign reversals.
At 1.3 mm, it spans from positive values
($+1.5\times 10^5$  at a $5\sigma$ level),
to $<3\sigma$ non-detections  in Apr 2017,
to negative values ($-3$ to $-4\times 10^5$  at a $10\sigma$ level) in Apr 2018.
At 3 mm the RM measured values span the range from $-1.2 \rm \ to\  0.3 \times 10^5$  from Sep 2015 to Oct 2016.
The large scatter and time variability revealed by the ALMA measurements
suggest a more complicated physical origin of the Faraday rotation than is usually assumed in models using the RM to infer a mass accretion rate.
We conclude that, unlike the case of Sgr A*, the RM in M87 may not  provide an accurate estimate of the mass accretion rate onto the black hole.

* The observed RM in M87 may result from Faraday rotation internal to the emission region, as commonly found in GRMHD models of turbulent accretion flows or expected in a structured jet, or from a time-varying helical magnetic field threading the jet boundary layer acting as an external Faraday screen.
As an alternative, we put forward a two-component model comprised of a variable compact region  and static extended region.    We find that this simple model is able to simultaneously explain the polarimetric properties observed  in Apr 2017 by both the EHT (on horizon scales) and ALMA (which observes the combined emission from both components).

The ALMA measurements presented in this work provide critical constraints for the calibration and analysis of simultaneously obtained VLBI data.
This is an essential resource for two instruments like the EHT and the GMVA which have the resolving power to reveal polarization structures and measure magnetic field strengths and particle densities on horizon scales (in the case of M87 and Sgr A*) and/or in the inner few parsecs for the AGN.

ALMA is a partnership
of the European Southern Observatory (ESO;
Europe, representing its member states), NSF, and
National Institutes of Natural Sciences of Japan, together
with National Research Council (Canada), Ministry
of Science and Technology (MOST; Taiwan),
Academia Sinica Institute of Astronomy and Astrophysics
(ASIAA; Taiwan), and Korea Astronomy and
Space Science Institute (KASI; Republic of Korea), in
cooperation with the Republic of Chile. The Joint
ALMA Observatory is operated by ESO, Associated
Universities, Inc. (AUI)/NRAO, and the National Astronomical
Observatory of Japan (NAOJ). The NRAO
is a facility of the NSF operated under cooperative agreement
by AUI.

This paper makes use of the following ALMA data:

The authors of the present paper thank the following
organizations and programs: the Academy
of Finland (projects 274477, 284495, 312496, 315721); the
Advanced European Network of E-infrastructures for
Astronomy with the SKA (AENEAS) project, supported
by the European Commission Framework Programme
Horizon 2020 Research and Innovation action
under grant agreement 731016; the Alexander
von Humboldt Stiftung; an Alfred P. Sloan Research Fellowship;
Allegro, the European ALMA Regional Centre node in the Netherlands, the NL astronomy
research network NOVA and the astronomy institutes of the University of Amsterdam, Leiden University and Radboud University;
the Black Hole Initiative at
Harvard University, through a grant (60477) from
the John Templeton Foundation; the China Scholarship
Agencia Nacional de Investigación y Desarrollo (ANID), Chile via NCN19-058 (TITANs) and Fondecyt 3190878;
Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico, projects  U0004-246083, U0004-259839, F0003-272050, M0037-279006, F0003-281692,
104497, 275201, 263356);
the Delaney Family via the Delaney Family John A.
Wheeler Chair at Perimeter Institute; Dirección General
de Asuntos del Personal Académico–Universidad
Nacional Autónoma de México (DGAPA–UNAM,
projects IN112417 and IN112820); the European Research Council Synergy
Grant "BlackHoleCam: Imaging the Event Horizon
of Black Holes" (grant 610058); the Generalitat
Valenciana postdoctoral grant APOSTD/2018/177 and
GenT Program (project CIDEGENT/2018/021); MICINN Research Project PID2019-108995GB-C22;
Gordon and Betty Moore Foundation (grants GBMF-
3561, GBMF-5278); the Istituto Nazionale di Fisica
Nucleare (INFN) sezione di Napoli, iniziative specifiche
TEONGRAV; the International Max Planck Research
School for Astronomy and Astrophysics at the
Universities of Bonn and Cologne; the Jansky Fellowship
program of the National Radio Astronomy Observatory
Joint Princeton/Flatiron and Joint Columbia/Flatiron Postdoctoral Fellowships, research at the Flatiron Institute is supported by the Simons Foundation;
the Japanese Government (Monbukagakusho:
MEXT) Scholarship; the Japan Society for
the Promotion of Science (JSPS) Grant-in-Aid for JSPS
Research Fellowship (JP17J08829); the Key Research
Program of Frontier Sciences, Chinese Academy of
Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJSSW-
SYS008, ZDBS-LY-SLH011); the Leverhulme Trust Early Career Research
Fellowship; the Max-Planck-Gesellschaft (MPG);
the Max Planck Partner Group of the MPG and the
CAS; the MEXT/JSPS KAKENHI (grants 18KK0090,
JP18K13594, JP18K03656, JP18H03721, 18K03709,
18H01245, 25120007); the Malaysian Fundamental Research Grant Scheme (FRGS) FRGS/1/2019/STG02/ UM/02/6; the MIT International Science
and Technology Initiatives (MISTI) Funds; the Ministry
of Science and Technology (MOST) of Taiwan (105-
2112-M-001-025-MY3, 106-2112-M-001-011, 106-2119-
M-001-027, 107-2119-M-001-017, 107-2119-M-001-020,
and 107-2119-M-110-005, MOST 108-2112-M-001-048 and MOST 109-2124-M-001-005); the National Aeronautics and
Space Administration (NASA, Fermi Guest Investigator
grant 80NSSC20K1567, NASA Astrophysics Theory Program grant 80NSSC20K0527, and Hubble Fellowship grant
HST-HF2-51431.001-A awarded by the Space Telescope
Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555, ); the National
Institute of Natural Sciences (NINS) of Japan; the National
Key Research and Development Program of China
(grant 2016YFA0400704, 2016YFA0400702); the National
Science Foundation (NSF, grants AST-0096454,
AST-0352953, AST-0521233, AST-0705062, AST-0905844, AST-0922984, AST-1126433, AST-1140030,
DGE-1144085, AST-1207704, AST-1207730, AST-1207752, MRI-1228509, OPP-1248097, AST-1310896, AST-1337663, AST-1440254, AST-1555365, AST-1615796, AST-1715061, AST-1716327, AST-1716536,
OISE-1743747, AST-1816420, AST-1903847,
AST-1935980, AST-2034306); the Natural Science
Foundation of China (grants 11573051, 11633006,
11650110427, 10625314, 11721303, 11725312, 11933007, 11991052, 11991053); a fellowship of China Postdoctoral Science Foundation (2020M671266); the Natural
Sciences and Engineering Research Council of
Canada (NSERC, including a Discovery Grant and
the NSERC Alexander Graham Bell Canada Graduate
Scholarships-Doctoral Program); the National Youth
Thousand Talents Program of China; the National Research
Foundation of Korea (the Global PhD Fellowship
Grant: grants NRF-2015H1A2A1033752, 2015-
R1D1A1A01056807, the Korea Research Fellowship Program:
NRF-2015H1D3A1066561, Basic Research Support Grant 2019R1F1A1059721); the Netherlands Organization
for Scientific Research (NWO) VICI award
(grant 639.043.513) and Spinoza Prize SPI 78-409; the
New Scientific Frontiers with Precision Radio Interferometry
Fellowship awarded by the South African Radio
Astronomy Observatory (SARAO), which is a facility
of the National Research Foundation (NRF), an
agency of the Department of Science and Technology
(DST) of South Africa; the Onsala Space Observatory
(OSO) national infrastructure, for the provisioning
of its facilities/observational support (OSO receives
funding through the Swedish Research Council under
grant 2017-00648) the Perimeter Institute for Theoretical
Physics (research at Perimeter Institute is supported
by the Government of Canada through the Department
of Innovation, Science and Economic Development
and by the Province of Ontario through the
Ministry of Research, Innovation and Science); the Spanish
Ministerio de Economía y Competitividad (grants
PGC2018-098915-B-C21, AYA2016-80889-P, PID2019-108995GB-C21); the State
Agency for Research of the Spanish MCIU through
the "Center of Excellence Severo Ochoa" award for
the Instituto de Astrofísica de Andalucía (SEV-2017-
0709); the Toray Science Foundation; the Consejería de Economía, Conocimiento, Empresas y Universidad of the Junta de Andalucía (grant P18-FR-1769), the Consejo Superior de Investigaciones Científicas (grant 2019AEP112);
the US Department
of Energy (USDOE) through the Los Alamos National
Laboratory (operated by Triad National Security,
LLC, for the National Nuclear Security Administration
of the USDOE (Contract 89233218CNA000001);
the Italian Ministero dell’Istruzione Università e Ricerca
through the grant Progetti Premiali 2012-iALMA (CUP
C52I13000140001); the European Union’s Horizon 2020
research and innovation programme under grant agreement
No 730562 RadioNet; ALMA North America Development
Fund; the Academia Sinica; Chandra TM6-
17006X; the GenT Program (Generalitat Valenciana)
Project CIDEGENT/2018/021; NASA NuSTAR award 80NSSC20K0645; Chandra award DD7-18089X.
This work used the
Extreme Science and Engineering Discovery Environment
(XSEDE), supported by NSF grant ACI-1548562,
and CyVerse, supported by NSF grants DBI-0735191,
DBI-1265383, and DBI-1743442. XSEDE Stampede2 resource
at TACC was allocated through TG-AST170024
and TG-AST080026N. XSEDE JetStream resource at
PTI and TACC was allocated through AST170028.
The simulations were performed in part on the SuperMUC
cluster at the LRZ in Garching, on the
LOEWE cluster in CSC in Frankfurt, and on the
HazelHen cluster at the HLRS in Stuttgart. This
research was enabled in part by support provided
by Compute Ontario (http://computeontario.ca), Calcul
Quebec (http://www.calculquebec.ca) and Compute
Canada (http://www.computecanada.ca).

We thank
the staff at the participating observatories in the GMVA  and EHT, correlation
centers, and institutions for their enthusiastic support.

The GMVA is coordinated by the VLBI group at the Max-Planck-Institut für Radioastronomie (MPIfR) and consists of telescopes operated by MPIfR, IRAM, Onsala, Metsahovi, Yebes, the Korean VLBI Network, the Green Bank Observatory and the VLBA.

APEX is a collaboration between the MPIfR (Germany),
ESO, and the Onsala Space Observatory (Sweden). The
SMA is a joint project between the SAO and ASIAA
and is funded by the Smithsonian Institution and the
Academia Sinica. The JCMT is operated by the East
Asian Observatory on behalf of the NAOJ, ASIAA, and
KASI, as well as the Ministry of Finance of China, Chinese
Academy of Sciences, and the National Key R&D
Program (No. 2017YFA0402700) of China. Additional
funding support for the JCMT is provided by the Science
and Technologies Facility Council (UK) and participating
universities in the UK and Canada. The
LMT is a project operated by the Instituto Nacional
de Astrófisica, Óptica, y Electrónica (Mexico) and the
University of Massachusetts at Amherst (USA). The
IRAM 30-m telescope on Pico Veleta, Spain is operated
by IRAM and supported by CNRS (Centre National de
la Recherche Scientifique, France), MPG (Max-Planck-
Gesellschaft, Germany) and IGN (Instituto Geográfico
Nacional, Spain). The SMT is operated by the Arizona
Radio Observatory, a part of the Steward Observatory
of the University of Arizona, with financial support of
operations from the State of Arizona and financial support
for instrumentation development from the NSF.
The SPT is supported by the National Science Foundation
through grant PLR- 1248097. Partial support is
also provided by the NSF Physics Frontier Center grant
PHY-1125897 to the Kavli Institute of Cosmological
Physics at the University of Chicago, the Kavli Foundation
and the Gordon and Betty Moore Foundation grant
GBMF 947. The SPT hydrogen maser was provided on
loan from the GLT, courtesy of ASIAA. The EHTC has
received generous donations of FPGA chips from Xilinx
Inc., under the Xilinx University Program. The EHTC
has benefited from technology shared under open-source
license by the Collaboration for Astronomy Signal Processing
and Electronics Research (CASPER). The EHT
project is grateful to T4Science and Microsemi for their
assistance with Hydrogen Masers. This research has
made use of NASA’s Astrophysics Data System. We
gratefully acknowledge the support provided by the extended
staff of the ALMA, both from the inception of
the ALMA Phasing Project through the observational
campaigns of 2017 and 2018. We would like to thank
A. Deller and W. Brisken for EHT-specific support with
the use of DiFX.
We thank Sergio Martin for the help interpreting the Sgr A* spectrum in SPW=2.
We acknowledge the significance that
Maunakea, where the SMA and JCMT EHT stations
are located, has for the indigenous Hawaiian people.
MP acknowledges support by the Spanish Ministry of Science through Grants PID2019-105510GB-C31, and PID2019-107427GB-C33, and from the Generalitat Valenciana through grant PROMETEU/2019/071.

Projects and sources observed in  the 3 mm Band.

Date      Project     Target      Pol. Cal.     Other sources     UT range

2017 Apr 02      2016.1.01116.V         OJ287      4C 01.28      J0510+1800     06:55:08 –  15:19:43

2017 Apr 03      2016.1.00413.V         Sgr A*      NRAO 530     J1924–2914, 4C 09.57    20:52:28  –  04:43:54

2017 Apr 04      2016.1.01216.V         3C273      3C279      —    00:24:57  –  05:32:46

Projects and sources observed in the 1.3 mm Band.

Date      Project     Target      Pol. Cal.     Other sources     UT range

2017 Apr 05                   04/22:12 –  05/09:13

2016.1.01114.V      OJ287        3C279     4C 01.28, M87    04/22:12 –  05/03:22

2016.1.01154.V      M87           3C279     4C 01.28, OJ287     05/03:24 –  05/07:18

2016.1.01176.V      3C279       3C279     M87      05/07:19 –  05/09:13

2017 Apr 06                   06/00:18 –  06/16:19

2016.1.01154.V      M87           3C279     3C273      06/00:18 –  06/08:02

2016.1.01404.V      Sgr A*       3C279    NRAO 530, J1924–2914      06/08:03 –  06/14:40

2016.1.01290.V      NGC 1052       3C279     J0132-1654, J0006-0623     06/14:51 –  06/16:19

2017 Apr 07                   07/03:45  –  07/20:47

2016.1.01404.V      Sgr A*       J1924–2914     NRAO 530     07/03:45 –  07/14:31

2016.1.01290.V      NGC 1052       J1924–2914      J0132-1654, 3C 84$^a$      07/19:23 –  07/20:47

2017 Apr 10                   09/23:02 –  10/10:02

2016.1.01114.V      OJ287        3C279     4C 01.28, M87      09/23:02 –  10/03:49

2016.1.01176.V      3C279       3C279     Cen A, M87       10/03:51 –  10/06:21

2016.1.01198.V      Cen A       3C279    —      10/06:23 –  10/10:02

2017 Apr 11                   10/21:44 –  11/10:31

2016.1.01114.V      OJ287        3C279     4C 01.28     10/21:44 –  11/00:22

2016.1.01154.V      M87           3C279     —      11/00:23 –  11/05:03

2016.1.01176.V      3C279       3C279      M87      11/05:05 –  11/08:45

2016.1.01404.V      Sgr A*      3C279     J1924–2914      11/08:46 –  11/14:03

aFlagged before data analysis (see text in Appendix <ref> and <ref>).

§ PROJECTS OBSERVED  DURING THE 2017 VLBI CAMPAIGN

Tables <ref> and <ref>  list the projects observed in the 3 mm and 1.3 mm bands, ordered by date of execution.
Each row reports the observing date, the ALMA project code, the science target, the source used as polarization calibrator, other sources observed in the project, and the duration of each observation.
In Table <ref>, each row-group refers to an individual VLBI run or ”Track”, which includes observations of different projects carried out during the same night.
The calibration of EHT projects was done per  track (and not per project; see ).
Two sources listed in Table <ref>, 3C 84 and  J0006-0623, both observed on Apr 07, were excluded from the analysis presented in this paper:
3C 84 was observed with an elevation below 25, while J0006-0623 was observed for just $\sim2$ min  close to an elevation of 25;
the resulting calibrated data display critical phase and amplitude scatter and hence were flagged before data analysis (see  <ref>).

Frequency-averaged imaging parameters of GMVA sources (at a representative frequency of 93 GHz).

Source     T$_{\rm on \ source}$     RMS(I)     RMS(Q)     RMS(U)     RMS(V)     Synthesized Beam

[h]    [mJy]     [mJy]     [mJy]     [mJy]     $\theta_M$[] $\times \theta_m$[] (P.A.[$^{\circ}$])

7cApr 2

OJ287                                  2.584           0.31          0.34           0.46         0.17      4.7  $\times$ 2.7 (-86.2$^{\circ}$)

4C 01.28                                0.269          0.17          0.30           0.36         0.14      4.9  $\times$ 2.4 (-86.8$^{\circ}$)

J0510+1800                             0.363           0.31          0.29           0.52         0.12      5.8  $\times$ 2.5 (-70.1$^{\circ}$)

7cApr 3

J1924-2914                              0.270          0.16          0.65           0.13         0.13      5.5  $\times$ 2.5 (-75.2$^{\circ}$)

NRAO 530                             0.479             0.02          0.02           0.02         0.03      4.8  $\times$ 2.4 (-83.5$^{\circ}$)

Sgr A*                                 2.643           0.80          0.09           0.08         0.04      5.0  $\times$ 2.7 (-81.1$^{\circ}$)

4C 09.57                              0.133            0.09          0.08           0.14         0.07      6.1  $\times$ 2.7 (72.0$^{\circ}$)

7cApr 4

3C273                                    1.396         0.48          0.26           0.54         0.13      5.0  $\times$ 3.4 (-86.7$^{\circ}$)

3C279                                 0.215            0.37          0.20           0.15         0.18      5.0  $\times$ 3.4 (-85.8$^{\circ}$)

Frequency-averaged imaging parameters of EHT sources (at a representative frequency of 221 GHz).

Source     T$_{\rm on-source}$     RMS(I)     RMS(Q)     RMS(U)     RMS(V)     Synthesized Beam

[h]    [mJy]     [mJy]     [mJy]     [mJy]     $\theta_M$[] $\times \theta_m$[] (P.A.[$^{\circ}$])

7cApr 5

M87                                     1.645          0.18          0.05           0.05         0.02      2.0  $\times$ 1.0 (-85.5$^{\circ}$)

3C279                                  1.068           0.24          0.09           0.07         0.16      2.2  $\times$ 0.9 (-80.9$^{\circ}$)

OJ287                                   1.406            0.13          0.16           0.15         0.11      2.0  $\times$ 1.1 (88.7$^{\circ}$)

4C 01.28                               0.230           0.12          0.11           0.08         0.12      2.0  $\times$ 0.9 (87.0$^{\circ}$)

7cApr 6

J0006-0623                            0.045            0.47          0.12           0.15         0.13      2.2  $\times$ 1.4 (-81.1$^{\circ}$)

J0132-1654                            0.059            0.06          0.06           0.06         0.06      2.3  $\times$ 1.4 (87.8$^{\circ}$)

NGC 1052                              0.373            0.05          0.03           0.03         0.03      2.7  $\times$ 1.3 (80.3$^{\circ}$)

Sgr A*                                   2.529         0.44          0.18           0.33         0.08      2.2  $\times$ 1.3 (-77.5$^{\circ}$)

J1924-2914                            0.269            0.10          0.07           0.08         0.11      2.2  $\times$ 1.3 (-82.5$^{\circ}$)

NRAO 530                              0.269            0.04          0.06           0.05         0.05      2.2  $\times$ 1.3 (-76.4$^{\circ}$)

M87                                       1.613        0.24          0.06           0.06         0.02      2.2  $\times$ 1.5 (-69.4$^{\circ}$)

3C279                                   0.430          0.30          0.13           0.08         0.13      2.2  $\times$ 1.3 (-78.4$^{\circ}$)

3C273                                  0.403           0.19          0.10           0.09         0.10      2.3  $\times$ 1.4 (-75.0$^{\circ}$)

7cApr 7

NGC 1052                               0.200           0.11          0.04           0.04         0.04      2.6  $\times$ 1.0 (-76.3$^{\circ}$)

J0132-1654                             0.056           0.08          0.07           0.07         0.06      3.0  $\times$ 1.0 (-72.8$^{\circ}$)

NRAO 530                              0.403            0.08          0.03           0.03         0.05      2.1  $\times$ 0.9 (-89.6$^{\circ}$)

J1924-2914                            0.312            0.18          0.04           0.04         0.08      2.1  $\times$ 0.9 (89.7$^{\circ}$)

Sgr A*                                4.109          0.25          0.14           0.13         0.04      2.1  $\times$ 0.9 (-88.6$^{\circ}$)

7cApr 10

Cen A                                   1.401          0.11          0.07           0.07         0.12      2.3  $\times$ 0.9 (-79.0$^{\circ}$)

M87                                     0.454          0.25          0.06           0.07         0.04      2.0  $\times$ 1.0 (-88.9$^{\circ}$)

OJ287                                 1.083            0.15          0.14           0.18         0.08      2.0  $\times$ 1.1 (-82.5$^{\circ}$)

3C279                                 1.120           0.29          0.12           0.08         0.16      2.1  $\times$ 0.8 (-85.2$^{\circ}$)

4C 01.28                              0.289            0.15          0.14           0.10         0.09      2.2  $\times$ 0.9 (80.0$^{\circ}$)

7cApr 11

Sgr A*                                    1.934        0.35          0.24           0.20         0.06      1.2  $\times$ 0.7 (-85.1$^{\circ}$)

J1924-2914                            0.244            0.29          0.10           0.10         0.14      1.2  $\times$ 0.7 (89.9$^{\circ}$)

3C279                                  1.705            0.23          0.11           0.08         0.11      1.2  $\times$ 0.7 (-86.6$^{\circ}$)

M87                                     1.831          0.16          0.04           0.04         0.03      1.2  $\times$ 0.8 (79.3$^{\circ}$)

OJ287                                  0.804           0.21          0.15           0.11         0.17      1.2  $\times$ 0.9 (59.6$^{\circ}$)

4C 01.28                              0.110            0.26          0.17           0.14         0.23      1.5  $\times$ 0.8 (67.9$^{\circ}$)

§ POLARIMETRIC IMAGES

Tables <ref>  and <ref> report the main imaging parameters for each source observed on each day of the 2017 VLBI campaign  in Band 3  and  Band 6, respectively.
These parameters include the on-source time, the RMS achieved in each Stokes parameter, and the synthesized beamsize.
The on-source time is computed after full calibration and flagging of bad data (see <ref>).
The RMS does not simply scale as $\sqrt{T_{\rm on-source}}$ but depends on several parameters such as source structure, number of observing antennas, array configuration, weather, and details of the VLBI scheduling blocks (e.g. low-elevation scans).
The synthesized beamsize changes by a factor of two  at 1.3 mm due to the changing array configuration during the observing week
(see <ref>).
The resulting images are  dynamic range limited and showcase different structures on different days (depending on the array beamsize).

The full suite of polarization images for all the sources observed in the VLBI campaign are shown in Figures <ref>–<ref>.
In particular, Figure <ref> displays 1.3 mm images of Sgr A* in the three days of the EHT observations, Figures <ref> and <ref> display the 1.3 mm images of M87 and 3C279 on the four days of the EHT observations.  Figures <ref> and <ref> report 1.3 mm maps for all the other AGN sources observed with the EHT for three days and one/two days, respectively. Finally, Figure <ref> shows 3 mm maps of the sources observed with the GMVA.

In each plot,   the black vectors showcase the orientation of the EVPAs, while their length is linearly proportional to the polarized flux. The EVPAs are plotted every 8 pixels (i.e. are spaced by 16 at 1.3 mm and 4 at 3 mm) for all sources, except for M87, where the EVPAs are plotted  every 4 pixels (i.e. are spaced by 08), in order to sample more uniformly the jet.
Note that the EVPAs are not Faraday-corrected and that the magnetic field vectors should be rotated by 90.

Polarization images of Sgr A* at 1.3 mm (see Figure <ref> for a description of the plotted quantities).
The apparently lower quality of the image in the right panel is due to the fact that the observations on Apr 11 have about half the beam size, and  are therefore less sensitive to the extended emission in the mini-spiral, when compared to the other two days (the beamsizes are shown as  ovals in the lower left corner of each panel; see values reported in Table <ref>).
Note that there are several tiny EVPAs plotted across the mini-spiral, apparently locating regions with polarized flux above the  image RMS noise cutoff ($5\sigma$). The LP and EVPA errors are however dominated by the systematic leakage (0.03% of I onto QU), which is not added to the images.  Once these systematic errors are added, the LP flux in those points falls below the $3\sigma$ detection threshold. Therefore we do not claim detection of polarized emission outside of the central core in Sgr A*.

Polarization images of M87 at 1.3 mm (see Figure <ref> for a description of the plotted quantities).
The apparently different jet structures (in total and polarized intensity) across days  are due to different beamsizes (shown as  ovals in the lower left corner of each panel; see values reported in Table <ref>).

Polarization images of 3C279 at 1.3 mm (see Figure <ref> for a description of the plotted quantities).

Polarization images of selected AGNs at 1.3 mm (see Figure <ref> for a description of the plotted quantities).

Polarization images of selected AGNs at 1.3 mm (see Figure <ref> for a description of the plotted quantities).
Note that for NGC 1052 and Cen A no EVPA could be reliably derived owing to their low level of LP.
We also note that although we detect  polarized flux  in Cen A above the  image RMS noise cutoff ($5\sigma$), once  the systematic leakage (0.03% of I onto QU) is added to thermal noise, the LP flux would fall below the $3\sigma$ detection threshold. Therefore we do not claim detection of polarized emission in Cen A.

Polarization images of all targets observed at 3 mm (at a central frequency of 93 GHz).
The beamsizes are shown as an oval in the lower left corner of each panel (see values reported in table <ref>).
See Figure <ref> for a description of the plotted quantities.

§ COMPARATIVE ANALYSIS ACROSS MULTIPLE FLUX-EXTRACTION METHODS

Some of our targets (chiefly Sgr A*  and  M87, but also a few other AGNs; see Appendix <ref>)  reveal extended emission at arc-seconds scales, which is differently   resolved out by the different observing array configurations. Here we assess the impact of such extended emission in the flux values  extracted in the visibility domain for  the compact cores.
For this purpose,
we compare the parameters derived with uvmultifit with those derived with two  imaged-based methods:  the sum of the nine central pixels  in the clean model image () and  the integrated flux (intf)  from Gaussian fitting with the casa task [  provides in output also the peak flux, but the integrated flux was preferred because its values were more consistent with the estimates from the other two methods.] (see <ref>). Table <ref> reports the results from this comparative analysis between the image-based and the $uv$-based methods,
showing the ratio of the four Stokes parameters, the LP, and the RM (3x3/uvmf and intf/uvmf, respectively),
and the difference of the EVPA in degrees (3x3-uvmf and intf-uvmf, respectively). Three sources (Cen A, NGC 1052, and J0132-1654) are excluded from this statistics due to their weak polarized signal.

The difference between three flux-extraction methods for EHT tracks: uvmultifit (uvmf);  the sum of the nine central pixels (3x3) of the model image; and the integrated flux (intf) from imfit. The two upper rows correspond to the comparison between uvmf and 3x3, while the two bottom rows correspond to that of uvmf and intf. Different panels show the ratio between the flux estimates in each Stokes parameter,  in LP and RM.  The panel devoted to the EVPA shows instead the difference in degrees.
The statistics are reported in the lower-right corner of each panel group, where the first value in each line refers to the median value of the distribution, the value in parenthesis is the Median Absolute Deviation (MAD), and the value in squared brackets reports on the percentage of cases farther than three times the MAD from the median.

Overall, the analysis clearly shows that  Flux(3x3) $<$ Flux(uvmf) $<$ Flux(intf)
when considering the median of the results. On the one hand, one can interpret this as the 3$\times$3 method being the least affected and intf being the most affected by extended emission (with a weak dependency on the  array-configuration resolution). On the other hand, the 3$\times$3 method can also be more affected by phase and amplitude calibration errors (which will remove flux from the phase-center into the sidelobes), resulting in less recovered flux.
Despite this systematic deviation, the median offsets are compatible with no significant difference within the MAD values. In Stokes $I$, both the offset and MAD are $\leq 0.4$% across the methods, which is negligible when compared to the absolute uncertainty of ALMA's flux calibration (10% in Band 6). In the case of Stokes $Q$ and $U$, and resulting  LP, the systematic offset and MAD between methods can be up to 1%.  With the LP measured in our sample, these differences are generally comparable (in absolute terms) to the 0.03% of Stokes $I$ leakage onto $Q$ and $U$.
The EVPA shows a MAD value of  $\sim0.1\,$deg, which results in a MAD value of up to 10% in the RM (note these are comparable or better than the observed statistical uncertainties – see Tables <ref> and <ref>).
The poorer results for Stokes $V$ are induced by the low level of CP in the sample (although the MAD is still comfortably around 4%).

Table <ref> reports a fraction of “outliers” in the distributions (those cases that are 5$\sigma$ away from the sample median). These are generally associated with the sources with the most prominent extended emission (other outliers are associated to parameters estimated at low-significance values). In order to better assess the magnitude of these outliers, in Figure <ref> we show a detailed comparison between the $uv$-based and the 3x3 image-based methods for M87 (upper panels) and Sgr A* (lower panels), respectively.
Analogously to Table <ref>, individual panels
show the ratio of the four Stokes parameters, the LP, and the RM (3x3/uvmf),
and the difference of the EVPA in degrees (3x3-uvmf);
the results are reported per day and SPW.

In the case of M87, the two methods exhibit an excellent agreement, except for Stokes $U$. This is due to a combination of faint $U$ emission at the core  and appreciable emission from knots A and B.
The discrepancy in Stokes $U$ results generally in $\Delta \rm EVPA \lesssim 0.4$  comparable with the uncertainties quoted in Table <ref>, with one single case of $\Delta \rm EVPA \sim 0.6$ (on Apr 11 SPW=1).
Given the consistency in RM (within $1\sigma$), this worst-case scenario discrepancies in EVPA do not affect the results of the analysis of this paper.

In the case of Sgr A*, the discrepancies are much more pronounced in most parameters.
Despite the prominence of the mini-spiral (see <ref> and Fig. <ref>),
Stokes $I$ values actually agree within less than 1% between the two methods. It is however noteworthy  that Flux(3x3)$>$Flux(uvmf) (i.e., the opposite trend with respect to Table <ref>).
This inverted trend is due to a flux-decrement in the visibility amplitudes at around 25–30 k$\lambda$ affecting most prominently  uvmfit  on Apr 11 (this can be assessed by inspecting amplitude vs. baseline-length uv-plots – not shown here – for the parallel hands on all days).
Unlike Stokes $I$, Stokes $QU$ are heavily affected, up to 20–30% in the worst cases,
resulting in LP differences up to 10% and EVPA offsets up to 2.
These large deviations in Stokes $QU$ are systematic since all four SPW appear to deviate in each day by an approximate amount. This in turn results in
consistent RM values within $1\sigma$. We note that we do not observe the same systematic offset either in M87 or J1924-2914, which were also calibrated using 3C279 as the polarization-calibrator (on Apr 6 and 11), hence we discard calibration issues as being responsible for these systematics.
One possible explanation is differential Stokes $I$ leakage onto $Q$ and $U$ from extended unpolarized emission (see the spurious EVPA vectors matching the spiral-arms in Fig. <ref>). Note also that such unpolarized emission extends farther than the inner third of the image field of view, where beam-squash <cit.> combined with leakage may induce significant deviations.

Comparison between image-based and $uv$-based flux-extraction methods for the two sources with the most prominent  extended emission in our sample: M87 (top panels), and Sgr A* (bottom panels). The two methods being compared are: uvmultifit (uvmf) and the sum of the nine central pixels (3x3) of the model image.
All parameters are compared via ratios, except  for the EVPA (showing the difference in degrees). The errorbar in each data-point is the combination in quadrature of the statistical error from each flux extraction method.
The vertical dotted line and the shaded region show the median and MAD from Table <ref>. The labels in the Y-axis indicate  the observing day in April 2017 (5, 6, 7, 10, 11) and the observing SPW (s1, s2, s3, s4); the RM panel shows only the days.
Non-detections in the images are indicated with a cross.
Note that the plotted uncertainties are not those of individual measurements but ratios or differences between the two methods. Therefore the errors on LP and EVPA for M87 on Apr 10/11 appear comparable to other days despite the large error bars and/or non detections in Stokes $U$ (i.e. the errors in U displayed do not propagate in the LP and EVPA plots).

Given the results from this comparative analysis, we conclude that, for the purpose of the polarimetric analysis conducted in this paper, the uv-fitting method provides sufficiently accurate  flux values of Stokes $IQU$ in all cases,
although for Sgr A* we observe deviations in Stokes $QU$
when comparing the two flux extraction methods.

Comparison of the three flux-extraction methods for EHT targets.

method$^{a}$     I$^{b}$     Q$^{b}$     U$^{b}$     V$^{b}$     LP$^{b}$     EVPA$^{c}$ []     RM$^{b}$

3x3     0.9995(0.0007)$^{d}$     0.989(0.008)     0.998(0.004)     1.00(0.04)     0.993(0.006)     0.0(0.1)     1.06(0.08)

0%$^{d}$     2%     1%     5%     10%     10%     0%

intf     1.001(0.004)     1.00(0.01)     0.996(0.006)     1.00(0.04)     1.001(0.004)     0.04(0.09)     1.0(0.1)

0%     2%     17%     10%     5%     6%     0%

aThis table compares uvmultifit (uvmf) with: the sum of the nine central pixels (3x3) of the model image; and the integrated flux (intf) from imfit.
bThe  values  reported in columns are ratios between the method referred in the first column and uvmf, with the latter being in the denominator.
cThe EVPA column shows the angle difference (in deg) between the methods.
d In each instance, three values are displayed: the median value of the distribution; the Median Absolute Deviation or MAD (the value in brackets);  the percentage of cases farther from the median than five times the MAD in quadrature with the measurement error (the value in the second row).

§ COMPARISON OF STOKES PARAMETERS WITH THE AMAPOLA POLARIMETRIC GRID SURVEY

For the purposes of absolute flux calibration, ALMA monitors the flux of bright sources (mainly blazars or QSOs) spread over the full range in right ascension (the Grid) by observing them together with solar system objects, the so-called Grid-Survey (GS), with a period of approximately 10 days. These observations are  executed with the ACA, in Bands 3, 6, and 7.  Since full-polarization mode is adopted, it is possible to retrieve polarimetry information from the GS sources.
This is done with  AMAPOLA[<http://www.alma.cl/$\sim$skameno/AMAPOLA/>], a set of CASA-friendly Python scripts used to reduce the full-Stokes polarimetry of GS observations with ACA.
Some of our targets are part of the  GS, including: 3C273 (J1229+0203); 3C279 (J1256-0547); 4C 01.28 (J1058+0133); 4C 09.57 (J1751+0939); J0510+1800; OJ287 (J0854+2006); J0006-0623; J1733-1304; and J1924-2914.
Although the reported values by AMAPOLA are used for observation planning and the two arrays cover different $uv$-ranges, it is still useful to make a comparison with this database to assess any systematics or clear inconsistent variability within a week's time-frame.

Figures <ref> and <ref> show the observed polarimetry parameters for the 2017 VLBI sessions (data points and errorbars), specifically $Q$, $U$, and $V$ Stokes parameters, LP, and EVPA. The shaded $\pm1\sigma$ regions highlight the time variance of the same parameters as measured by AMAPOLA (an inflection in the trend means a time of GS observation). The colour-coding is such that blue refers to Band 3 measurements, green to Band 6, and red to Band 7 ones.
These figures show that most Band 3 ALMA-VLBI measurements  fall within the blue regions, while most Band 6 measurements fall in-between the blue and red regions, indicating that our measurements are broadly consistent with the AMAPOLA trends.  Some potential conflicts can still be consistent with the inter-GS-cadence variability or differential time-variability between frequency bands (as also observed in some cases in the AMAPOLA monitoring).
We conclude that, despite the different array specifications and data-reduction schemes, the ALMA-VLBI and AMAPOLA results are consistent between each other.

Comparison with time between the polarimetric results obtained for all the sources observed in the ALMA-VLBI campaign and those retrieved from the AMAPOLA polarimetric analysis of Grid Survey data. Each row shows a parameter (from top to bottom: Stokes Q, U, and V, LP, and EVPA) while each column corresponds to a source (from left to right: 3C279, OJ287, J1924-2914, and 4C 01.28; see Figure <ref> for more).
Only sources with entries in the ALMA archive (close in time to the observations, i.e., between end of March and April 2017) are displayed.
The measured flux values during the ALMA-VLBI observations are indicated as data points and corresponding errorbars (blue and green for Band 3 and 6 observations, respectively).
The shaded regions indicate AMAPOLA's $\pm1\sigma$ uncertainty in Band 3 (97.5 GHz; blue shade) and Band 7 (343.4 GHz; red shade), respectively.
These are obtained from the ACA GS and their time evolution (lines) are obtained by interpolating between these measurements.

Same as Fig. <ref>, but for different sources (from left to right: 3C273, J0006-0623, NRAO 530, 4C 09.57). We note that source J0006-0623 was observed on Apr 7 but the calibrated data were of poor quality and were flagged before analysis (see  <ref>).

§ STOKES PARAMETERS PER ALMA FREQUENCY BAND (SPW)

We report the polarimetric quantities (Stokes $IQU$, LP, EVPA)  per SPW in Tables <ref> (GMVA sources) and <ref> (EHT sources).
The Stokes parameters were fitted directly on the visibilities using   (see <ref>).  Uncertainties are assessed with MC simulations, as the standard deviation of 1000 MC simulations for each Stokes parameter.
The quoted uncertainties  include in quadrature the fitting error and
the Stokes $I$ leakage onto Stokes $QU$ (0.03% of $I$),
as recommended by the ALMA observatory.
The LP uncertainty is dominated by such  systematic error, except for the weakest sources J0132-1654 and NGC 1052, for which the thermal noise starts to dominate.

In order to correct for the LP bias in the low SNR regime,
we estimate a debiased LP as $\sqrt{LP^2-\sigma_{LP^2}}$, where
$LP =\sqrt{Q^2+U^2}/I$.

$\sigma_{LP}$ can be estimated using propagation of errors:

\begin{equation*}
\begin{aligned}
\sigma_{LP}^2 I^2  = &
{\frac{(Q \sigma_Q)^2 +  (U \sigma_U)^2 + 2 QU \sigma_{QU}}{Q^2+U^2}} + \\
& [(\frac{Q}{I})^2 + (\frac{U}{I})^2 ] \sigma^2_I - 2 \frac{Q}{I}  \sigma_{QI} - 2 \frac{U}{I}  \sigma_{UI}
\end{aligned}
\end{equation*}

where $\sigma_{I}$, $\sigma_{Q}$, $\sigma_{U}$ are the uncertainties in $I$, $Q$ and $U$, respectively.

Assuming  $\sigma_{QU} \sim \sigma_{QI} \sim \sigma_{UI} \sim  0$, then

\begin{equation*}
\begin{aligned}
\sigma_{LP}  = \frac{1}{I} \sqrt{{\frac{(Q \sigma_Q)^2 +  (U \sigma_U)^2}{Q^2+U^2}}+
[(\frac{Q}{I})^2 + (\frac{U}{I})^2 ] \sigma^2_I}.
\end{aligned}
\end{equation*}

The LP values quoted in Tables <ref>, <ref>, <ref>, and <ref> have this debiased correction applied.
The latter does not affect most of the sources studied here, but it is especially important for the low-polarization sources such as NGC 1052 and Cen A.

The absolute flux-scale calibration systematic error  (5% and 10% of Stokes $IQU$ fluxes in Band 3 and Band 6, respectively), is not included in  Tables <ref> and <ref>.

The analysis on Stokes $V$ is performed separately (see Appendix <ref> and Tables <ref>, <ref>).

Polarization parameters of GMVA sources per frequency band (spw) and per day.

Frequency     I     Q     U     LP     EVPA

[GHz]     [mJy]     [mJy]     [mJy]     [%]     [deg]

6cApr 2

86.3                      6242.41$\pm$0.26           -409.6$\pm$1.9           -351.2$\pm$1.9            8.643$\pm$0.030     -69.69$\pm$0.10

88.3                      6170.33$\pm$0.26           -410.2$\pm$1.9           -347.5$\pm$1.9            8.711$\pm$0.030     -69.87$\pm$0.10

98.3                      5787.47$\pm$0.28           -398.9$\pm$1.7           -327.8$\pm$1.7            8.919$\pm$0.030     -70.30$\pm$0.10

100.3                      5674.69$\pm$0.27           -392.4$\pm$1.7           -324.2$\pm$1.7            8.970$\pm$0.030     -70.214$\pm$0.093

86.3                      3259.30$\pm$0.57           -127.5$\pm$1.0           22.6$\pm$1.0            3.976$\pm$0.030     84.97$\pm$0.21

88.3                      3208.82$\pm$0.59           -126.2$\pm$1.0           31.9$\pm$1.0            4.054$\pm$0.030     82.91$\pm$0.20

98.3                      2996.03$\pm$0.62           -120.30$\pm$0.90           46.96$\pm$0.91            4.311$\pm$0.030     79.33$\pm$0.20

100.3                      2962.36$\pm$0.63           -121.52$\pm$0.90           43.17$\pm$0.90            4.353$\pm$0.030     80.22$\pm$0.21

6c4C 01.28

86.3                      5039.680$\pm$0.072           90.9$\pm$1.5           -195.6$\pm$1.5            4.281$\pm$0.030     -32.54$\pm$0.20

88.3                      4970.067$\pm$0.055           93.2$\pm$1.5           -192.6$\pm$1.5            4.306$\pm$0.030     -32.09$\pm$0.20

98.3                      4745.126$\pm$0.053           92.5$\pm$1.4           -194.1$\pm$1.4            4.533$\pm$0.030     -32.26$\pm$0.20

100.3                      4665.997$\pm$0.054           92.1$\pm$1.4           -192.1$\pm$1.4            4.566$\pm$0.030     -32.18$\pm$0.18

6cApr 3

6cSgr A*

86.3                      2574.84$\pm$0.87           9.28$\pm$0.77           -3.64$\pm$0.77            0.388$\pm$0.030     -10.7$\pm$2.2

88.3                      2480.44$\pm$0.82           12.51$\pm$0.74           -2.36$\pm$0.74            0.513$\pm$0.030     -5.3$\pm$1.7

98.3                      2514.64$\pm$0.74           18.44$\pm$0.75           16.77$\pm$0.75            0.991$\pm$0.030     21.10$\pm$0.86

100.3                      2504.04$\pm$0.72           15.20$\pm$0.75           21.41$\pm$0.75            1.048$\pm$0.030     27.31$\pm$0.81

86.3                      5273.93$\pm$0.18           -10.6$\pm$1.6           -245.0$\pm$1.6            4.650$\pm$0.030     -46.25$\pm$0.19

88.3                      5244.90$\pm$0.18           -12.7$\pm$1.6           -248.9$\pm$1.6            4.752$\pm$0.030     -46.46$\pm$0.18

98.3                      4984.67$\pm$0.19           -12.0$\pm$1.5           -248.2$\pm$1.5            4.986$\pm$0.030     -46.39$\pm$0.17

100.3                      4920.17$\pm$0.21           -12.3$\pm$1.5           -244.6$\pm$1.5            4.978$\pm$0.030     -46.44$\pm$0.18

6cNRAO 530

86.3                      2857.621$\pm$0.036           7.11$\pm$0.86           24.37$\pm$0.86            0.889$\pm$0.030     36.8$\pm$1.0

88.3                      2826.545$\pm$0.035           6.60$\pm$0.85           24.53$\pm$0.85            0.899$\pm$0.030     37.4$\pm$1.0

98.3                      2649.116$\pm$0.034           4.24$\pm$0.80           24.59$\pm$0.80            0.943$\pm$0.030     40.08$\pm$0.92

100.3                      2618.629$\pm$0.035           3.71$\pm$0.79           24.62$\pm$0.79            0.952$\pm$0.030     40.73$\pm$0.92

6c4C 09.57

86.3                      2920.272$\pm$0.085           65.23$\pm$0.88           -98.94$\pm$0.88            4.057$\pm$0.030     -28.30$\pm$0.21

88.3                      2898.304$\pm$0.073           65.59$\pm$0.87           -96.53$\pm$0.88            4.028$\pm$0.030     -27.91$\pm$0.22

98.3                      2805.452$\pm$0.071           61.49$\pm$0.84           -96.65$\pm$0.85            4.084$\pm$0.030     -28.76$\pm$0.21

100.3                      2775.60$\pm$0.48           60.67$\pm$0.84           -96.55$\pm$0.84            4.107$\pm$0.030     -28.93$\pm$0.22

6cApr 4

86.3                      13309.65$\pm$0.19           70.9$\pm$4.0           1612.7$\pm$4.0            12.129$\pm$0.030     43.746$\pm$0.071

88.3                      13168.27$\pm$0.19           67.6$\pm$4.0           1596.6$\pm$4.0            12.136$\pm$0.030     43.791$\pm$0.069

98.3                      12671.78$\pm$0.19           52.6$\pm$3.8           1541.8$\pm$3.8            12.175$\pm$0.030     44.021$\pm$0.070

100.3                      12575.88$\pm$0.19           49.9$\pm$3.8           1533.0$\pm$3.8            12.198$\pm$0.030     44.067$\pm$0.069

86.3                      10066.53$\pm$0.38           -13.8$\pm$3.0           -419.4$\pm$3.0            4.167$\pm$0.030     -45.95$\pm$0.20

88.3                      10013.63$\pm$0.36           -12.3$\pm$3.0           -405.9$\pm$3.0            4.054$\pm$0.030     -45.87$\pm$0.21

98.3                      9698.37$\pm$0.31           -3.3$\pm$2.9           -370.9$\pm$2.9            3.826$\pm$0.030     -45.25$\pm$0.23

100.3                      9645.14$\pm$0.31           3.4$\pm$2.9           -375.2$\pm$2.9            3.891$\pm$0.030     -44.74$\pm$0.23

Polarization parameters of EHT sources per frequency band (spw) and per day.

Frequency     I     Q     U     LP     EVPA

[GHz]     [mJy]     [mJy]     [mJy]     [%]     [deg]

6cApr 5

6c4C 01.28

213.1                      3572.28$\pm$0.11           144.5$\pm$1.1           -151.7$\pm$1.1           5.867$\pm$0.030     -23.20$\pm$0.14

215.1                      3617.41$\pm$0.11           145.8$\pm$1.1           -153.3$\pm$1.1           5.848$\pm$0.030     -23.23$\pm$0.15

227.1                      3429.88$\pm$0.11           138.9$\pm$1.0           -146.7$\pm$1.0           5.890$\pm$0.030     -23.28$\pm$0.14

229.1                      3419.77$\pm$0.11           141.9$\pm$1.0           -147.0$\pm$1.0           5.976$\pm$0.030     -23.01$\pm$0.14

213.1                      4455.60$\pm$0.15           -214.2$\pm$1.3           -342.2$\pm$1.3           9.060$\pm$0.030     -61.027$\pm$0.092

215.1                      4476.37$\pm$0.17           -214.8$\pm$1.3           -339.6$\pm$1.3           8.976$\pm$0.030     -61.161$\pm$0.092

227.1                      4220.58$\pm$0.15           -205.0$\pm$1.3           -314.9$\pm$1.3           8.902$\pm$0.030     -61.53$\pm$0.10

229.1                      4198.43$\pm$0.15           -203.6$\pm$1.3           -325.3$\pm$1.3           9.139$\pm$0.030     -61.022$\pm$0.093

213.1                      1336.29$\pm$0.13           30.07$\pm$0.40           -7.70$\pm$0.40           2.323$\pm$0.030     -7.17$\pm$0.36

215.1                      1325.68$\pm$0.13           29.94$\pm$0.40           -8.10$\pm$0.40           2.338$\pm$0.030     -7.57$\pm$0.38

227.1                      1236.46$\pm$0.11           29.62$\pm$0.37           -8.80$\pm$0.37           2.500$\pm$0.030     -8.29$\pm$0.35

229.1                      1227.58$\pm$0.11           29.81$\pm$0.37           -8.74$\pm$0.37           2.530$\pm$0.030     -8.15$\pm$0.35

213.1                      9202.50$\pm$0.11           -8.0$\pm$2.8           1214.2$\pm$2.8           13.195$\pm$0.030     45.187$\pm$0.066

215.1                      9144.74$\pm$0.11           -7.2$\pm$2.7           1206.7$\pm$2.7           13.195$\pm$0.030     45.168$\pm$0.065

227.1                      8845.31$\pm$0.12           -7.9$\pm$2.7           1169.0$\pm$2.7           13.216$\pm$0.030     45.193$\pm$0.063

229.1                      8774.06$\pm$0.12           -6.9$\pm$2.6           1161.8$\pm$2.6           13.241$\pm$0.030     45.172$\pm$0.066

6cApr 6

6cSgr A*

213.1                      2631.24$\pm$0.32           -130.65$\pm$0.79           -113.68$\pm$0.79           6.581$\pm$0.030     -69.49$\pm$0.13

215.1                      2629.81$\pm$0.33           -128.22$\pm$0.79           -118.33$\pm$0.80           6.636$\pm$0.030     -68.65$\pm$0.13

227.1                        2533.72$\pm$0.27*           -105.98$\pm$0.76           -144.59$\pm$0.76           7.076$\pm$0.030     -63.12$\pm$0.13

229.1                      2625.81$\pm$0.28           -106.00$\pm$0.79           -156.22$\pm$0.79           7.188$\pm$0.030     -62.07$\pm$0.13

213.1                      3342.261$\pm$0.089           -28.0$\pm$1.0           -202.0$\pm$1.0           6.101$\pm$0.030     -48.95$\pm$0.14

215.1                      3313.239$\pm$0.093           -28.5$\pm$1.0           -199.2$\pm$1.0           6.072$\pm$0.030     -49.07$\pm$0.14

227.1                      3179.165$\pm$0.088           -30.5$\pm$1.0           -190.4$\pm$1.0           6.064$\pm$0.030     -49.55$\pm$0.14

229.1                      3156.545$\pm$0.087           -30.65$\pm$0.95           -191.3$\pm$1.0           6.138$\pm$0.030     -49.55$\pm$0.14

213.1                      426.29$\pm$0.15           7.28$\pm$0.19           4.11$\pm$0.19           1.963$\pm$0.044     14.68$\pm$0.64

215.1                      420.00$\pm$0.16           7.19$\pm$0.20           4.32$\pm$0.20           1.999$\pm$0.047     15.48$\pm$0.65

227.1                      409.56$\pm$0.16           7.16$\pm$0.20           4.59$\pm$0.20           2.076$\pm$0.049     16.34$\pm$0.66

229.1                      404.98$\pm$0.16           6.62$\pm$0.19           4.05$\pm$0.19           1.919$\pm$0.048     15.68$\pm$0.74

6cNGC 1052

213.1                      437.43$\pm$0.13           0.36$\pm$0.14           0.17$\pm$0.14           0.090$\pm$0.033     13$\pm$11

215.1                      438.85$\pm$0.11           0.56$\pm$0.15           0.39$\pm$0.15           0.154$\pm$0.033     17.0$\pm$6.0

227.1                      414.12$\pm$0.10           0.47$\pm$0.14           0.46$\pm$0.14           0.159$\pm$0.033     21.9$\pm$6.3

229.1                      415.236$\pm$0.062           0.05$\pm$0.14           0.30$\pm$0.14           0.074$\pm$0.033     39$\pm$18

213.1                      1361.85$\pm$0.16           27.72$\pm$0.41           -6.24$\pm$0.41           2.088$\pm$0.030     -6.34$\pm$0.43

215.1                      1349.84$\pm$0.15           27.65$\pm$0.41           -6.70$\pm$0.41           2.108$\pm$0.030     -6.83$\pm$0.41

227.1                      1269.57$\pm$0.13           26.95$\pm$0.38           -8.34$\pm$0.38           2.224$\pm$0.030     -8.61$\pm$0.38

229.1                      1257.40$\pm$0.13           26.70$\pm$0.38           -8.28$\pm$0.38           2.224$\pm$0.030     -8.61$\pm$0.39

213.1                      2044.52$\pm$0.32           213.57$\pm$0.63           139.74$\pm$0.65           12.482$\pm$0.031     16.598$\pm$0.071

215.1                      2023.69$\pm$0.33           212.00$\pm$0.63           137.19$\pm$0.64           12.480$\pm$0.031     16.452$\pm$0.072

227.1                      1944.0$\pm$1.1           205.43$\pm$0.62           130.51$\pm$0.62           12.520$\pm$0.033     16.215$\pm$0.073

229.1                      1931.6$\pm$1.1           204.15$\pm$0.61           134.03$\pm$0.62           12.645$\pm$0.033     16.646$\pm$0.074

213.1                      9571.40$\pm$0.12           72.2$\pm$2.9           1240.7$\pm$2.9           12.984$\pm$0.030     43.337$\pm$0.068

215.1                      9517.18$\pm$0.13           72.4$\pm$2.9           1234.1$\pm$2.9           12.990$\pm$0.030     43.324$\pm$0.067

227.1                      9180.18$\pm$0.12           69.2$\pm$2.8           1193.4$\pm$2.8           13.022$\pm$0.030     43.345$\pm$0.066

229.1                      9168.61$\pm$0.11           69.3$\pm$2.8           1193.1$\pm$2.8           13.035$\pm$0.030     43.338$\pm$0.066

6cNRAO 530

213.1                      1655.066$\pm$0.071           -8.41$\pm$0.50           36.58$\pm$0.50           2.267$\pm$0.030     51.46$\pm$0.38

215.1                      1669.741$\pm$0.079           -8.95$\pm$0.51           37.51$\pm$0.51           2.309$\pm$0.030     51.70$\pm$0.38

227.1                      1567.831$\pm$0.072           -8.86$\pm$0.48           36.81$\pm$0.48           2.414$\pm$0.030     51.76$\pm$0.37

229.1                      1554.353$\pm$0.070           -8.41$\pm$0.47           36.72$\pm$0.47           2.424$\pm$0.030     51.44$\pm$0.36

213.1                      7744.46$\pm$0.11           -58.5$\pm$2.3           -191.7$\pm$2.3           2.587$\pm$0.030     -53.49$\pm$0.33

215.1                      7707.72$\pm$0.11           -61.7$\pm$2.3           -186.8$\pm$2.3           2.553$\pm$0.030     -54.15$\pm$0.34

227.1                      7421.16$\pm$0.10           -68.4$\pm$2.2           -153.4$\pm$2.2           2.262$\pm$0.030     -57.01$\pm$0.38

229.1                      7357.33$\pm$0.10           -66.8$\pm$2.2           -145.2$\pm$2.2           2.170$\pm$0.030     -57.36$\pm$0.39

6cApr 7

213.1                      3243.110$\pm$0.083           -26.8$\pm$1.0           -191.3$\pm$1.0           5.957$\pm$0.030     -48.99$\pm$0.15

215.1                      3219.06$\pm$0.10           -27.3$\pm$1.0           -190.0$\pm$1.0           5.963$\pm$0.030     -49.09$\pm$0.15

227.1                      3073.91$\pm$0.11           -27.90$\pm$0.92           -181.62$\pm$0.93           5.978$\pm$0.030     -49.37$\pm$0.14

229.1                      3048.69$\pm$0.12           -28.11$\pm$0.92           -180.17$\pm$0.92           5.982$\pm$0.030     -49.44$\pm$0.15

213.1                      418.13$\pm$0.18           6.66$\pm$0.21           5.00$\pm$0.21           1.990$\pm$0.051     18.43$\pm$0.73

215.1                      417.76$\pm$0.19           6.99$\pm$0.23           4.97$\pm$0.23           2.055$\pm$0.055     17.75$\pm$0.75

227.1                      402.18$\pm$0.20           6.82$\pm$0.22           4.37$\pm$0.23           2.014$\pm$0.056     16.29$\pm$0.80

229.1                      394.71$\pm$0.21           6.12$\pm$0.23           4.77$\pm$0.23           1.965$\pm$0.058     18.93$\pm$0.84

6cNRAO 530

213.1                      1620.07$\pm$0.10           -7.36$\pm$0.49           37.57$\pm$0.49           2.363$\pm$0.030     50.55$\pm$0.35

215.1                      1604.64$\pm$0.11           -7.57$\pm$0.49           37.41$\pm$0.49           2.379$\pm$0.030     50.72$\pm$0.36

227.1                      1536.162$\pm$0.068           -7.65$\pm$0.47           36.73$\pm$0.47           2.443$\pm$0.030     50.89$\pm$0.37

229.1                      1526.938$\pm$0.067           -7.37$\pm$0.46           37.77$\pm$0.46           2.520$\pm$0.030     50.51$\pm$0.35

6cSgr A*

213.1                      2418.25$\pm$0.25           -124.16$\pm$0.73           -113.45$\pm$0.73           6.954$\pm$0.030     -68.79$\pm$0.12

215.1                      2404.74$\pm$0.27           -121.16$\pm$0.72           -117.97$\pm$0.72           7.032$\pm$0.030     -67.88$\pm$0.12

227.1                        2318.41$\pm$0.22*           -100.51$\pm$0.70           -140.51$\pm$0.70           7.451$\pm$0.030     -62.79$\pm$0.11

229.1                      2403.69$\pm$0.23           -100.83$\pm$0.72           -148.69$\pm$0.72           7.472$\pm$0.030     -62.07$\pm$0.11

6cNGC 1052

213.1                      398.01$\pm$0.20           -0.31$\pm$0.14           0.87$\pm$0.14           0.231$\pm$0.035     54.9$\pm$4.5

215.1                      397.25$\pm$0.18           -0.10$\pm$0.15           0.70$\pm$0.15           0.178$\pm$0.037     49.6$\pm$6.0

227.1                      369.99$\pm$0.25           0.21$\pm$0.14           0.12$\pm$0.14           0.069$\pm$0.038     13$\pm$20

229.1                      358.94$\pm$0.35           -0.46$\pm$0.14           0.44$\pm$0.14           0.177$\pm$0.039     68.1$\pm$7.9

6cApr 10

6c4C 01.28

213.1                      3668.38$\pm$0.16           154.6$\pm$1.1           -102.5$\pm$1.1           5.055$\pm$0.030     -16.77$\pm$0.16

215.1                      3677.75$\pm$0.17           153.4$\pm$1.1           -102.8$\pm$1.1           5.020$\pm$0.030     -16.91$\pm$0.17

227.1                      3514.62$\pm$0.14           147.8$\pm$1.1           -99.7$\pm$1.1           5.074$\pm$0.030     -17.00$\pm$0.18

229.1                      3512.42$\pm$0.13           152.2$\pm$1.1           -99.6$\pm$1.1           5.179$\pm$0.030     -16.60$\pm$0.17

213.1                      4319.36$\pm$0.10           -167.8$\pm$1.3           -253.8$\pm$1.3           7.044$\pm$0.030     -61.73$\pm$0.12

215.1                      4333.61$\pm$0.10           -166.9$\pm$1.3           -251.3$\pm$1.3           6.962$\pm$0.030     -61.79$\pm$0.12

227.1                      4119.79$\pm$0.10           -158.1$\pm$1.2           -235.9$\pm$1.2           6.894$\pm$0.030     -61.91$\pm$0.12

229.1                      4105.17$\pm$0.10           -162.0$\pm$1.2           -243.2$\pm$1.2           7.119$\pm$0.030     -61.83$\pm$0.12

6cCen A

213.1                      5710.166$\pm$0.085           1.8$\pm$1.7           0.5$\pm$1.7           0.040$\pm$0.030     6$\pm$30

215.1                      5677.690$\pm$0.090           3.1$\pm$1.7           2.3$\pm$1.7           0.070$\pm$0.030     18$\pm$15

227.1                      5621.837$\pm$0.094           3.2$\pm$1.7           4.7$\pm$1.7           0.101$\pm$0.030     27.9$\pm$9.1

229.1                      5628.36$\pm$0.10           0.2$\pm$1.7           3.9$\pm$1.7           0.071$\pm$0.030     43$\pm$14

213.1                      1382.93$\pm$0.26           37.15$\pm$0.42           -0.24$\pm$0.42           2.688$\pm$0.030     -0.19$\pm$0.32

215.1                      1371.95$\pm$0.25           37.12$\pm$0.42           -0.08$\pm$0.42           2.706$\pm$0.030     -0.08$\pm$0.31

227.1                      1285.70$\pm$0.23           35.68$\pm$0.39           0.47$\pm$0.39           2.777$\pm$0.030     0.37$\pm$0.31

229.1                      1271.90$\pm$0.23           35.24$\pm$0.39           0.05$\pm$0.39           2.770$\pm$0.030     0.03$\pm$0.31

213.1                      8750.09$\pm$0.11           216.5$\pm$2.6           1264.2$\pm$2.6           14.659$\pm$0.030     40.141$\pm$0.057

215.1                      8699.92$\pm$0.11           215.5$\pm$2.6           1257.5$\pm$2.6           14.665$\pm$0.030     40.137$\pm$0.058

227.1                      8415.64$\pm$0.11           209.4$\pm$2.5           1220.6$\pm$2.5           14.717$\pm$0.030     40.133$\pm$0.058

229.1                      8373.72$\pm$0.11           208.6$\pm$2.5           1216.4$\pm$2.5           14.739$\pm$0.030     40.135$\pm$0.061

6cApr 11

6cSgr A*

213.1                      2388.59$\pm$0.31           -41.33$\pm$0.72           -169.55$\pm$0.72           7.307$\pm$0.030     -51.85$\pm$0.12

215.1                      2383.72$\pm$0.30           -37.26$\pm$0.72           -169.35$\pm$0.72           7.275$\pm$0.030     -51.21$\pm$0.12

227.1                        2266.69$\pm$0.26*           -13.97$\pm$0.68           -170.91$\pm$0.68           7.565$\pm$0.030     -47.34$\pm$0.11

229.1                      2362.91$\pm$0.27           -12.14$\pm$0.71           -181.89$\pm$0.71           7.714$\pm$0.030     -46.91$\pm$0.12

6c4C 01.28

213.1                      3643.15$\pm$0.20           158.8$\pm$1.1           -87.2$\pm$1.1           4.972$\pm$0.030     -14.38$\pm$0.18

215.1                      3632.17$\pm$0.20           157.7$\pm$1.1           -88.5$\pm$1.1           4.980$\pm$0.030     -14.65$\pm$0.17

227.1                      3503.45$\pm$0.21           152.0$\pm$1.1           -88.4$\pm$1.1           5.018$\pm$0.030     -15.10$\pm$0.17

229.1                      3485.19$\pm$0.22           152.9$\pm$1.1           -87.0$\pm$1.1           5.047$\pm$0.030     -14.82$\pm$0.18

213.1                      4366.345$\pm$0.094           -148.7$\pm$1.3           -274.2$\pm$1.3           7.143$\pm$0.030     -59.23$\pm$0.12

215.1                      4356.28$\pm$0.10           -151.4$\pm$1.3           -271.9$\pm$1.3           7.143$\pm$0.030     -59.56$\pm$0.12

227.1                      4170.47$\pm$0.36           -149.1$\pm$1.3           -258.1$\pm$1.3           7.146$\pm$0.030     -60.01$\pm$0.12

229.1                      4152.92$\pm$0.11           -145.7$\pm$1.2           -259.6$\pm$1.2           7.168$\pm$0.030     -59.65$\pm$0.12

213.1                      3299.39$\pm$0.19           -35.7$\pm$1.0           -156.7$\pm$1.0           4.870$\pm$0.030     -51.42$\pm$0.18

215.1                      3289.89$\pm$0.19           -36.8$\pm$1.0           -155.1$\pm$1.0           4.845$\pm$0.030     -51.68$\pm$0.18

227.1                      3155.4$\pm$1.0           -37.30$\pm$0.95           -147.9$\pm$1.0           4.834$\pm$0.030     -52.08$\pm$0.18

229.1                      3146.13$\pm$0.17           -38.04$\pm$0.95           -149.99$\pm$0.95           4.918$\pm$0.030     -52.12$\pm$0.17

213.1                      1393.36$\pm$0.11           35.91$\pm$0.42           -1.16$\pm$0.42           2.579$\pm$0.030     -0.94$\pm$0.33

215.1                      1380.69$\pm$0.11           36.33$\pm$0.42           -1.06$\pm$0.42           2.634$\pm$0.030     -0.85$\pm$0.33

227.1                      1290.61$\pm$0.10           36.30$\pm$0.39           -0.68$\pm$0.39           2.813$\pm$0.030     -0.54$\pm$0.31

229.1                      1278.47$\pm$0.10           36.23$\pm$0.39           -0.31$\pm$0.39           2.833$\pm$0.030     -0.25$\pm$0.30

213.1                      8365.96$\pm$0.23           208.5$\pm$2.5           1223.0$\pm$2.5           14.830$\pm$0.030     40.163$\pm$0.059

215.1                      8317.40$\pm$0.23           207.2$\pm$2.5           1217.3$\pm$2.5           14.846$\pm$0.030     40.168$\pm$0.058

227.1                      8000.38$\pm$0.25           201.7$\pm$2.4           1180.8$\pm$2.4           14.974$\pm$0.030     40.154$\pm$0.058

229.1                      7972.54$\pm$0.25           200.5$\pm$2.4           1178.2$\pm$2.4           14.991$\pm$0.030     40.171$\pm$0.058

6l*The flux of Sgr A* at 227 GHz (spw=2) is systematically $\sim$5% lower than at 229 GHz (spw=3),

6lowing to the presence of spectral absorption lines (see Appendix <ref>).

6lGiven its flat spectral index, $F_{spw=2}=F_{spw=3}$ should be assumed for Sgr A*.

§ FARADAY RM PLOTS

We display the EVPA data  as a function of $\lambda^2$  and their RM fitted models
in Figures <ref>, <ref>,  <ref>, and <ref>; EVPAs are measured  in each of the four ALMA SPWs.
Let us first focus on Figure <ref>, showing the
Faraday RM of Sgr A* (upper panels), M87 (middle panels), and 3C 279 (lower panels).
The EVPAs measured in Sgr A* reveal  a remarkably precise $\lambda^2$ dependence both at $\lambda$3 mm (left panel) and $\lambda$1.3 mm  (second to fourth panels).
It is also remarkable that Sgr A* showcases a very consistent slope across all days, while in M87 the slope appears to change sign from Apr 5/6 to Apr 10/11.
3C 279 was used as polarization calibrator on Apr 5, 6, 10, and 11, and its measured RM (consistent with zero across all days) demonstrates the stability of the polarization measurements on M87 and on Sgr A* on the same days.
On Apr 7, the polarization calibrator was J1924-2914, which also shows consistent RM across days  (see Figure <ref>, upper panels).
Collectively, they demonstrate the stability of the polarization measurements in Apr 2017 on Sgr A*, M87, and, by extension, on the remaining AGN observed at $\lambda$1.3 mm (displayed in Figure <ref>) and at $\lambda$3 mm (displayed in Figure <ref>).

We notice that, for some of the targets, the RM fits are almost perfectly consistent with the measured EVPAs; in particular, the plots for  Sgr A*, 3C 279  (Fig. <ref>), 3C 273, J1924-2914 (Fig. <ref>),  and M87 (Fig. <ref>), look very unlikely given the  error bars.
In fact, a standard $\chi^2$ analysis for these sources would give values close to 0
while for 4 points/2-parameter fit, we would expect  $\chi^2\sim$2.
This apparently unexpected behaviour  can be explained by the fact that the EVPA uncertainties displayed in the RM plots are not just the thermal errors (which would naturally introduce scatter in the measurements)  but also include a systematic error (0.03% of $I$ into $QU$ errors), which in fact  dominates the total error budget (especially for the strongest sources with high Stokes $I$).
We assessed that once  such  systematic error is removed,
the EVPA data points are no longer consistent with the line to within their (thermal-noise-only) uncertainties. The fact that the thermal-only error bars are too small, but the error bars in the plots of this paper that include the systematic 0.03% Stokes $I$ leakage into the $QU$ error budget are too large, suggests that there is a real systematic error in the EVPA measurements but that it is smaller than the ALMA standard value. The fact that we are being conservative in our error estimates should ensure that we are not over-interpreting our measurements.

Finally, we notice that, for some of the targets and/or on specific days, there are $>1\sigma$ deviations between the observed EVPA and the RM fit-predicted values (e.g., OJ287,  4C 01.28, J0006-0623, J0510+1800; see selected panels in  Figs. <ref> and <ref>).
This may suggest either the presence of an additional systematic error not accounted for in our error budget or that the assumption of the EVPA $\lambda^2$-dependence is not valid in some cases.
In fact, in  <ref>, <ref>, and  <ref> we discuss the possibility that some of the observed Faraday rotation may be partly internal,  which would imply a more complex RM model than assumed in our analysis.
Without a wider wavelength coverage, we cannot distinguish between the two cases.  Therefore, we will not  consider any additional systematic error in the RM analysis presented in this paper.

Faraday RM of Sgr A* (upper panels), M87 (middle panels), and 3C 279 (lower panels).
The EVPA as a function of wavelength-squared is presented with $1\sigma$ error bars for each SPW and for each day.
The EVPAs are measured in the 1.3 mm band, except for the  upper left panel, presenting the EVPAs measured at 3 mm in Sgr A*.
The error bars are derived
adding in quadrature to the thermal uncertainty  of the Q and U maps ($1\sigma$ image RMS)  a systematic error of 0.03% of Stokes I (error bars for Sgr A* are smaller than the displayed points).
The  line is a linear fit to the data giving the RM reported inside the box.
Plots in each row span the same vertical axis range in degrees  in order to highlight differences in slope (the upper left panel is an exception).
It is remarkable that Sgr A* showcases a very consistent slope across days, while in M87 the slope appears to change sign.
Note also the different EVPA values between Apr 6/7 and Apr 11 in Sgr A* and
between Apr 5/6 and Apr 10/11 in M87 and 3C279.

The measured RM in 3C 279 (used as polarization calibrator on Apr 5, 6, 10, and 11) is consistent with zero across all days, demonstrating the stability of the polarization measurements on M87 and on Sgr A* on the same days.

RM fits for remaining 2017 EHT targets with EVPA measurements at 1.3 mm (see Table <ref>).
Plots in each row span the same vertical axis range for the same source in order to highlight differences in slope.

RM fits for the 2017 GMVA targets with EVPA measurements at 3 mm (see Table <ref>).

RM of M87 from observations carried out in Sep & Nov 2015 and Oct 2016 at 3 mm (top row) and in Apr  2018 at 1.3 mm (bottom row).
Plots in both rows span the same vertical axis range (25 and 8 respectively) in order to highlight differences in slope (but note the different EVPA values in different panels).

§ CIRCULAR POLARIZATION AND ASSESSMENT OF THE ALMA POLARIMETRY

A reliable detection of Stokes V in interferometric observations with linear-polarization feeds (i.e., the case of ALMA) strongly depends on the correct estimate of two instrumental quantities. On the one hand, the relative phase, $\Delta$, between the X and Y polarizers (i.e., the cross-polarization phase) at the reference antenna <cit.>. On the other hand, the imaginary parts of the Dterms that describe the polarization leakage of the individual ALMA elements <cit.>.
Following the standard ALMA calibration procedure, $\Delta$ is estimated by assuming a negligible Stokes V in the polarization calibrator. With this assumption, and neglecting also the effects from polarization leakage, the cross-polarization correlations ($XY^*$ and $YX^*$) between two ALMA antennas observing a polarized point source are

\begin{equation}
XY^* = e^{-j\Delta} p \sin{\left[2(\phi - \psi)\right]} ~~ \mathrm{and} ~~ YX^* = e^{j\Delta} p \sin{\left[2(\phi - \psi)\right]},
\label{CrossPolEq}
\end{equation}

where the data are assumed to be already corrected for phase and amplitude gains, $\psi$ and $\phi$ are the feed angle of the antennas[The parallactic angle plus the rotation of the receiver cartridge with respect to the antenna mount.] and the EVPA of the observed source (respectively), and $p$ is the calibrator's linearly-polarized flux density ($p= LP \times I$).

The only complex quantity that appears in Eqs. <ref> is given by the factor $e^{j\Delta}$. Hence, the $XY^*$ (and $YX^*$) visibilities may change their amplitudes as a function of time (via the changes in $\psi$), but their phases will remain constant and equal to $\Delta$. The CASA-based calibration algorithm for ALMA (provided in the  task) takes advantage of this fact, and estimates the value of $\Delta$ from the fit of $XY^*$ and/or $YX^*$ (over different values of $\psi$) to a model with a constant phase. We call the phase estimated in this way as $\Delta_{\mathrm{QA2}}$.

§.§ Effects of an inaccurate ALMA polarization calibration

§.§.§ Polarization calibrator

If the true value of the cross-polarization phase, $\Delta$, is offset from $\Delta_{\mathrm{QA2}}$ by an unknown quantity, $\beta$ (i.e., $\Delta = \beta + \Delta_{\mathrm{QA2}}$), this offset will introduce a leakage-like effect into the polconverted VLBI visibilities, which will be described by the Dterms given in Eq. 13 of <cit.>. If $\beta$ is very small, that equation predicts an ALMA leakage for VLBI which has two remarkable properties: 1) it is pure imaginary, and 2) it is the same for the R and L polarization hands. The value of this Dterm is directly related to $\beta$ via the equation:

$$ D^{VLBI}_R = D^{VLBI}_L \sim j \beta. $$

Therefore, if the QA2 calibration procedure has introduced an offset $\beta$ into the cross-polarization phase, $\Delta_{\mathrm{QA2}}$, we predict a pure imaginary Dterm in the ALMA-VLBI visibilities. The actual Dterms estimated from the ALMA-VLBI observations during the EHT 2017 campaign are reported in <cit.>.

The phase offset $\beta$ may also introduce a spurious V signal into the data, which can be described by the equation <cit.>

\begin{equation}
V^{tot} = V^{const}\cos{\beta} + p\,\sin{\left[2\left(\phi - \psi\right)\right]}\sin{\beta},
\label{SpuriousVEq}
\end{equation}

where $V^{tot}$ is the total V signal recovered from the (QA2-calibrated) data and $V^{const}$ is a constant V signal, independent of $\psi$. If the QA2 Dterm estimates for ALMA are correct, $V^{const}$ is equal to the true Stokes V of the source, (i.e., $V^{const} = V^{true}$).
However, if the ALMA Dterms estimated in the QA2 are offset from their true values, there is another instrumental contribution to $XY^*$ and $YX^*$, which couples to  $V^{const}$ as following

\begin{equation}
\begin{aligned}
XY^* &= p\,\sin{\left[ 2 \left( \phi - \psi \right) \right]} + jV^{true} + \\
&\left( D_X^a + (D_Y^b)^* \right) I
+ \mathcal{O}(p\, D ) + \mathcal{O}(V^{true}\, D ).
\end{aligned}
\label{CrossPolEq3}
\end{equation}

In this equation, $D_X^k$ is the part of the Dterm of the $X$ polarizer of antenna $k$ that remains uncalibrated after the QA2 (and similarly for the $Y$ polarizer). Therefore, if the effects of the antenna Dterms are not fully removed from the data, the calibrator source will show a non-negligible $V^{true}$, while the CASA model assumes it to be null.

In Fig. <ref>, we show the $V^{tot}$ signal (computed as the average real part of $(XY^* - YX^*)/j$ among all ALMA antennas) of the polarization calibrator 3C 279, as a function of parallactic angle, for the epochs where this source was observed. We have averaged the visibilities in time bins of 120 seconds and the data taken with antenna elevations below 30 degrees have been discarded. In the figure, we also show the simple model given by Eq. <ref>, where $V^{const}$ and $\Delta$ are the only two free parameters used in the fit.

The V signals from the polarization calibrator show a clear dependence with parallactic angle, which indicates that there are offsets, $\beta$ in the estimated cross-polarization phases, $\Delta_{\mathrm{QA2}}$. Using the values of $p$ and $\phi$ estimated for 3C 279 on  different days, we fit an X-Y phase offset of $\beta = 0.2-0.5$ degrees in all tracks except April 5, where $\beta \sim 1-2$ degrees. We notice that these ranges assume no bias in the QA2 estimates of the ALMA Dterms.

It is interesting to note that the data depart from the sinusoidal model of Eq. <ref>, especially for the epochs on April 6 and 11, and for observations far from transit. These deviations may be related to other instrumental effects (e.g., the second-order leakage contributions, like $\mathcal{O}(p\,D)$ in Eq. <ref>).

Reconstructed Stokes V of 3C 279 as a function of feed angle,  $\phi$, during the EHT campaign.
Stokes V is computed as the real part of $(XY^* - YX^*)/j$.
Dashed lines show the simplified model of Eq. <ref>, based on cross-polarization phase offset, $\beta$, of the phased-ALMA reference antenna.

From the values of $\beta$ fitted with Eq. <ref>, we can estimate a rough upper bound for the Stokes V of the calibrator, assuming a perfect calibration of the ALMA Dterms.
For 3C 279, our analysis yields V = [27, 4, 4, 2] mJy, corresponding to  CP = [0.30, 0.04, 0.07, 0.02] %, and $\beta$ = [2.2, 0.67, 0.57, 0.35] degrees, on Apr 5, 6, 10, 11, respectively. For J1924-2914, our analysis yields V = 0.7 mJy, corresponding to  CP =  0.02%, and $\beta$ = -0.91 degrees, on Apr 7 (3C279 was not observed on Apr 7).

The estimated values of $V^{const}$ obtained from Eq. <ref> are all of the order of 1 mJy, at most. These values are small compared to the $V^{true}$ coming from the $\beta$ estimates. This is especially true for the epoch of April 5. As discussed at the beginning of this section, a low $V^{const}$ can be explained by the compensating effect of (small) biases in the QA2 estimates of the ALMA Dterms (Eq. <ref>). Such systematics would force $V^{const}$ to be close to zero, regardless of the value of $V^{true}$.

In summary, two clear conclusions can be drawn from Fig. <ref>. On the one hand, there is an offset, $\beta$, in the X-Y cross-polarization phase of the reference antenna. On the other hand, there may be other systematics in the estimates of the antenna Dterms that may compensate the imprint of a true circular polarization of the calibrator (since CASA always assumes a null Stokes V in the calibrator).

§.§.§ Other sources

The Dterms and cross-polarization phases discussed in the previous subsection are applied to all sources in the data. Hence, the V Stokes from the polarization calibrator, which may have introduced a cross-polarization phase offset, $\beta$, and other biases to the Dterm estimates, will be systematically put back (after applying the polarization calibration) into the Stokes V signals of the rest of sources.

Thus, if we find different values of the fractional V Stokes among different sources, there has to be a contribution to their $V^{const}$ values (Eqs. <ref> and <ref>) that is independent of that introduced by the polarization calibration. In the frame of our modelling of the instrumental polarization, such a contribution would likely be related to $V^{true}$, i.e., a true circular polarization associated to the sources.

Reconstructed fractional Stokes V of all sources (but 3C 279), as a function of feed angle, during the EHT campaign.
Stokes V is computed as the real part of $(XY^* - YX^*)/j$.
Continuum lines show a simplified model based on cross-polarization phase offsets at ALMA, which have been fixed to the $\beta$ values shown on top of each figure. Note that Sgr A* is not plotted  because it displays significant intrinsic variability on the timescales (of hours) plotted here  <cit.>.

In Fig. <ref>, we show the fractional Stokes V values for all the sources, with the exception of 3C 279, as a function of feed angle. As with Fig. <ref>, a time binning of 120 seconds has been applied and data with elevations lower than 30 degrees have not been used. We also show the model fitted with Eq. <ref>, where $\beta$ is fixed to the value derived from the 3C 279 data (so the only free parameter is $V^{const}$). The only exception to this modelling is for the epoch on April 7, where J1924-291 was used instead as the polarization calibrator.

Some sources show a clear dependence of $V^{tot}$ with feed angle, being 3C 273 (on April 6) and OJ287 (on April 5) remarkable examples. However, the model prediction for such a variability, based on the cross-polarization phase offsets, $\beta$, estimated from 3C 279, cannot reproduce all the data. A possible explanation for this discrepancy could be, for instance, a small (within 1 degree) variation of $\Delta$ with pointing direction (i.e., antenna elevation and azimuth) and/or an effect related to residual Dterms (see Eq. <ref>). A deep analysis of these possibilities is out of the scope of this paper and should indeed be carried out at the Observatory level.

In any case, the fractional circular polarizations shown in Fig. <ref> are very different among sources, which is a good indicative that these are not dominated by the Dterm systematics (at least, to a first-order approximation). Some sources do show the sinusoidal dependence with feed angle, whereas others (like M87) are dominated by $V^{const}$. Since we do not know the exact effects related to residual Dterms, it is not possible to derive $V^{true}$ from the estimated $V^{const}$.
A robust conclusion, however, is that there is circular polarization detected in most of the sources (with amplitudes of the order of a few 0.1%) and that, to our understanding, the instrumental effects, alone, cannot explain the results for all sources in a self-consistent way.

Frequency-averaged circular polarization fraction of GMVA targets (at a representative frequency of 93 GHz).

Source     Day     I      V$_{meas}$     V$_{true}$     CP     LP

[2017]      [Jy]    [mJy]      [mJy]      [%]      [%]

OJ287                  Apr 2                    5.97$\pm$0.30             -4$\pm$36                        -3                 -0.05$\pm$0.60                  8.811$\pm$0.030

J0510+1800                  Apr 2                    3.11$\pm$0.16             -4$\pm$19                        -4                 -0.14$\pm$0.60                  4.173$\pm$0.031

4C 01.28$^a$                  Apr 2                    4.86$\pm$0.24             0$\pm$29                       1.0                  0.02$\pm$0.60                  4.420$\pm$0.030

Sgr A*                  Apr 3                    2.52$\pm$0.13             -0$\pm$15                        -0.5                 -0.02$\pm$0.60                  0.734$\pm$0.030

J1924-2914                  Apr 3                    5.11$\pm$0.26             0$\pm$31                        5                 0.09$\pm$0.60                  4.841$\pm$0.031

NRAO 530$^a$                  Apr 3                    2.74$\pm$0.14             0$\pm$16                       0.0                  0.00$\pm$0.60                  0.919$\pm$0.031

4C 09.57                  Apr 3                    2.85$\pm$0.14             -9$\pm$17                        -10                 -0.34$\pm$0.60                  4.069$\pm$0.030

3C279$^a$                  Apr 4                    12.93$\pm$0.65             -0$\pm$78                       -10                  -0.1$\pm$1.2                  12.159$\pm$0.030

3C273                  Apr 4                    9.86$\pm$0.49             11$\pm$59                        14                 0.14$\pm$0.60                  3.984$\pm$0.029

7l$^a$The polarization calibrator is assumed to have Stokes V=0

7lfor polarization calibration purposes <cit.>.

Frequency-averaged circular polarization fraction of EHT targets (at a representative frequency of 221 GHz).

Source     Day     I      V$_{meas}$     V$_{true}$     CP     LP

[2017]      [Jy]    [mJy]      [mJy]      [%]      [%]

3C279$^a$              Apr 5                    8.99$\pm$0.90             0$\pm$54                        27                 0.30$\pm$0.60                  13.210$\pm$0.030

3C279$^a$              Apr 6                    9.36$\pm$0.94             0$\pm$56                        4                 0.04$\pm$0.60                  13.010$\pm$0.030

3C279$^a$              Apr 10                    8.56$\pm$0.86             0$\pm$51                        6                 0.07$\pm$0.60                  14.690$\pm$0.030

3C279$^a$              Apr 11                    8.16$\pm$0.82             0$\pm$49                        2                 0.02$\pm$0.60                  14.910$\pm$0.030

M87                  Apr 5                    1.28$\pm$0.13             -1.5$\pm$7.7                        -2                 -0.15$\pm$0.60                  2.420$\pm$0.030

M87                  Apr 6                    1.31$\pm$0.13             -4.4$\pm$7.9                        -5                 -0.34$\pm$0.60                  2.160$\pm$0.030

M87                  Apr 10                    1.33$\pm$0.13             -3.5$\pm$8.0                        -4                 -0.28$\pm$0.60                  2.730$\pm$0.030

M87                  Apr 11                    1.34$\pm$0.13             -5.4$\pm$8.0                        -6                 -0.41$\pm$0.60                  2.710$\pm$0.030

Sgr A*                  Apr 6                    2.63$\pm$0.26             -40$\pm$16                        -40                 -1.51$\pm$0.61                  6.870$\pm$0.030

Sgr A*                  Apr 7                    2.41$\pm$0.24             -27$\pm$15                        -27                 -1.14$\pm$0.61                  7.230$\pm$0.030

Sgr A*                  Apr 11                    2.38$\pm$0.24             -24$\pm$14                        -24                 -1.01$\pm$0.60                  7.470$\pm$0.030

J1924-2914                  Apr 6                    3.25$\pm$0.32             0$\pm$19                        0.1                 0.00$\pm$0.60                  6.090$\pm$0.030

J1924-2914$^a$                   Apr 7                    3.15$\pm$0.31             0$\pm$19                        0.7                 0.02$\pm$0.60                  5.970$\pm$0.030

J1924-2914                  Apr 11                    3.22$\pm$0.32             2$\pm$19                        2                 0.05$\pm$0.60                  4.870$\pm$0.030

OJ287                  Apr 5                    4.34$\pm$0.43             17$\pm$26                        20                 0.46$\pm$0.60                  9.020$\pm$0.030

OJ287                  Apr 10                    4.22$\pm$0.42             5$\pm$25                        5                 0.12$\pm$0.60                  7.000$\pm$0.030

OJ287                  Apr 11                    4.26$\pm$0.43             -1$\pm$26                        0.6                 0.01$\pm$0.60                  7.150$\pm$0.030

4C 01.28                  Apr 5                    3.51$\pm$0.35             10$\pm$21                        9                 0.25$\pm$0.60                  5.890$\pm$0.030

4C 01.28                  Apr 10                    3.59$\pm$0.36             3$\pm$22                        3                 0.09$\pm$0.60                  5.080$\pm$0.030

4C 01.28                  Apr 11                    3.57$\pm$0.36             -1$\pm$21                        0.2                 0.01$\pm$0.60                  5.000$\pm$0.030

NRAO 530                  Apr 6                    1.61$\pm$0.16             -0$\pm$10                        -0.4                 -0.02$\pm$0.60                  2.350$\pm$0.030

NRAO 530                  Apr 7                    1.57$\pm$0.16             0.5$\pm$9.4                        0.8                 0.05$\pm$0.60                  2.430$\pm$0.030

J0132-1654                  Apr 6                    0.420$\pm$0.040             -0.7$\pm$2.5                        -0.8                 -0.19$\pm$0.60                  1.990$\pm$0.050

J0132-1654                  Apr 7                    0.410$\pm$0.040             -0.1$\pm$2.5                        -0.2                 -0.04$\pm$0.60                  2.000$\pm$0.050

NGC 1052                  Apr 6                    0.430$\pm$0.040             0.2$\pm$2.6                        0.2                 0.05$\pm$0.60                  0.110$\pm$0.030

NGC 1052                  Apr 7                    0.380$\pm$0.040             0.4$\pm$2.3                        0.3                 0.08$\pm$0.60                  0.150$\pm$0.040

Cen A                  Apr 10                    5.66$\pm$0.57             -2$\pm$34                        -2                 -0.04$\pm$0.60                  0.060$\pm$0.030

3C273                  Apr 6                    7.56$\pm$0.76             10$\pm$45                        9                 0.12$\pm$0.60                  2.390$\pm$0.030

J0006-0623                  Apr 6                    1.99$\pm$0.20             -1$\pm$12                        -3                 -0.16$\pm$0.60                  12.530$\pm$0.030

7l$^a$The polarization calibrator is assumed to have Stokes V=0

7lfor polarization calibration purposes <cit.>.

§ SPECTRAL INDICES OF TOTAL INTENSITY

We compute the total intensity spectral index for all the sources observed in the VLBI campaign at 3 mm and 1.3 mm.
For each source, the spectral index $\alpha$, defined as $I(\nu) \propto \nu^{\alpha}$, is derived "in-band", performing a weighted least-squares fit across the four flux-density values estimated with uvmultifit  in each SPW, i.e. at frequencies of 213, 215, 227, 229 GHz in the 1.3 mm band, and 86, 88, 98, and 100 GHz in the 3 mm band, respectively.

For Sgr A*, the spectral index of the compact core is around 0 both at 3 mm ($\alpha = 0.01\pm 0.1$) and 1.3 mm ($\alpha=[-0.03,-0.15]\pm 0.06$).

For M87, the spectral index of the compact core at 1.3 mm is negative ($\alpha$=[–1.2,–1.1]).
Cycle 0 ALMA observations at 3 mm  at a comparable angular resolution ($2.6'' \times 1.4''$) yields a much flatter $\alpha$=[-0.2,-0.3] <cit.>, consistent with VLA measurements at radio frequency bands (8.4–43 GHz).
VLBA observations revealed a flat-spectrum ($\alpha$=0) compact core <cit.>.
The steeper spectrum measured in the 1.3 mm band suggests that a spectral break must occur between 3 mm and 1.3 mm.