  Quantum theory introduces a cut between the observer and the observed system,
but does not provide a definition of what is an observer. Based on an
informational definition of observer, Grinbaum has recently predicted an upper
bound on bipartite correlations in the Clauser-Horne-Shimony-Holt (CHSH) Bell
scenario equal to 2.82537, which is slightly smaller than the Tsirelson bound
of standard quantum theory, but is consistent with all the available
experimental results. Not being able to exceed Grinbaum's limit would support
that quantum theory is only an effective description of a more fundamental
theory and would have a deep impact in physics and quantum information
processing. Here we present a test of the CHSH Bell inequality on photon pairs
in maximally entangled states of polarization in which a value 2.82759+-0.00051
is observed, violating Grinbaum's bound by 4.3 standard deviations and
providing the smallest distance with respect to Tsirelson's bound ever
reported, namely, 0.00084+-0.00051. This sets a new lower experimental bound
for Tsirelson's bound, strengthening the value of principles that predict
Tsirelson's bound as possible explanations of all natural limits of
correlations, and has important consequences for cryptographic security,
randomness certification, characterization of physical properties in
device-independent scenarios, and certification of quantum computation.
