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Context. Investigations of the magnetism of the quiet Sun are hindered byextremely weak polarization signals in Fraunhofer spectral lines. Photon noise,straylight, and the systematically different sensitivity of the Zeeman effectto longitudinal and transversal magnetic fields result in controversial resultsin terms of the strength and angular distribution of the magnetic field vector.  Aims. The information content of Stokes measurements close to the diffractionlimit of the 1.5 m GREGOR telescope is analyzed. We took the effects of spatialstraylight and photon noise into account.  Methods. Highly sensitive full Stokes measurements of a quiet-Sun region atdisk center in the deep photospheric Fe I lines in the 1.56 {\mu}m region wereobtained with the infrared spectropolarimeter GRIS at the GREGOR telescope.Noise statistics and Stokes V asymmetries were analyzed and compared to asimilar data set of the Hinode spectropolarimeter (SOT/SP). Simple diagnosticsbased directly on the shape and strength of the profiles were applied to theGRIS data. We made use of the magnetic line ratio technique, which was testedagainst MHD simulations.  Results. About 80% of the GRIS spectra of a very quiet solar region showpolarimetric signals above a 3{\sigma} level. Area and amplitude asymmetriesagree well with small-scale surface dynamo MHD simulations. The magnetic lineratio analysis reveals ubiquitous magnetic regions in the ten to hundred Gaussrange with some concentrations of kilo-Gauss fields.  Conclusions. The GRIS spectropolarimetric data at a spatial resolution of0.40" are so far unique in the combination of high spatial resolution scans andhigh magnetic field sensitivity. Nevertheless, the unavoidable effect ofspatial straylight and the resulting dilution of the weak Stokes profiles meansthat inversion techniques still bear a high risk of misinterpretating the data.

Probing deep photospheric layers of the quiet Sun with high magnetic sensitivity