Dear Editor,

we appreciate the referee's comments on a more detailed discussion on an
assessment of the systematic errors of our measurement which tries to closely
approach Tsirelson's bound, and the suggestion to elaborate more on the nature
of Grinbaum's bound.

We tried to address both issues in the revised version of the manuscript.

Regarding the call for a systematic error analysis, we perhaps like to to
point out that most of the systematic errors considered in the two papers
cited by the referee, the CODATA paper (P. Mohr et al., Rev. Mod. Phys. 84,
1527 (2012)) or the recent EDM paper (The ACME Collaboration, Science 343, 269 (2014)), are a
consequence of the indirect measurement technique necessary for some of the
quantities. Our measurement of the Bell quantity S is as direct a measurement
as possible, notwithstanding the experimental loopholes we are not able to
address in the setup (and thus require a fair sampling assumption).
However, we did look into possible systematics that could affect our error estimation and final value.
As made explicit in Eq.(3) of the manuscript, the quantity that is directly
observed is the number of coincidence counts in a fixed time and for a given
position of the polarization filters.

The various coincidence counts are a function of the actual photon pair rate,
the detector efficiency, dead time of the detectors, the time
uncertainty of the electronic counter, and an alignment uncertainty of the
polarization filters in corresponding measurements.
In the current version of the manuscript, we elaborated on all these factors
to support our initial claim that 
the uncertainty in the value we present is indeed dominated by Poissonian
counting statistics. Specifically, we addressed the following sources for
uncertainties:

Poissonian counting statistics - This originally quoted uncertainty due to the
limited number of events contributes about 4.9e-4 uncertainty to the value of
S. We did verify that this uncertainty corresponds to the standard deviation
of a few subsets we divided the original data in, but we would have no reason to
believe that time translation invariance of the generation process is not
valid. This number did change compared to the previous version of the
manuscript when reviewing the statistical analysis, from 0.00082 to 0.00049,
because we overestimated contributions from different coincidence
counts.

Detector efficiency - it is reasonable to assume for Silicon avalanche
photodetectors that they remains stable over a period much longer than the
realization of the presented measurement, approximatively one week. The really
relevant time span of stability that would affect the result of the violation
S is the cycling time it takes us to evaluate all settings and their
complements, e.g. the time difference between a corresponding H and V
polarization measurement. This time is 8 minutes, and we have no indication of
a drift of detector efficiencies over that time scale. Another effect on
detection efficiencies would be the dead time of our devices (about 1.6
microseconds), but this is only
a second order effect as the single event rate is approximatively 5000 counts
per second, and does not vary significantly as a function of the measurement
settings, or over time.

Detector dead time - the Silicon avalanche photodetectors we use in the
experiment have a dead time of 1.6e-6 s. Fluctuations of the total acquisition
time due to the dead time are proportional to the square root of the the
number of single counts. Propagating this uncertainty to the calculated value
of S, we obtain a correction in the order of 5.4e-7, two order of magnitude
smaller than the quoted error.

Timing uncertainty - the counting intervals of 60 s are controlled by a
hardware component in a microcontroller, giving a time uncertainty of at most
100 ns. A code review of this element revealed a systematic underestimation of
the integration time of 1.33 microseconds, but no jitter other than the clock
uncertainty of the controller. The 100 ns is an extremely conservative
estimation of this jitter, originated in the PLL in the microcontroller. This
contributes a factor in the order of 1e-10 to the final value of S, orders of
magnitude smaller than the error associated with the statistics of the
process. The systematic overestimation of the integration time is not affecting
the result, as all integration intervals were the same 60 seconds. A much more
prominent influence on the absolute timing interval has probably the crystal
oscillator determining the timing interval, which is specified to 50ppm
absolute (not affecting the result of S), and its temperature dependency
(specified as below 20ppm/K). Near the timing device, we logged a
temperature fluctuation of less than 0.5K in an hour over the experiment, or
less than 0.1K in the relevant time interval between two corresponding
measurements (like a H/V pair). The realistic timing uncertainty due to
temperature fluctuations around the oscillator (thermally shielded from the
ambient air by equipment) is much lower than the corresponding 2ppm
from the estimation above. An in-situ measurement of the oscillator frequency
against an atomic clock in a temperature profile comparable to the time when the
Bell measurements were taken (i.e., with similar daily variations in
temperature in the lab) revealed a frequency uncertainty below 0.1ppm over the
day. The effect of this on the value of S is around 2.8e-9.

The systematic errors just presented on a longer time scale than the 8 minutes
of corresponding H/V-type measurements can also have a biasing effect to the
observed value. This biasing effect is expected to be of the same order of
magnitude of the errors and all with the same sign, and will reduce the observed
value of S.

A similar argument applies for the systematic errors in the angular position
of the polarization filters: any error in the reproducibility of
the angular position reduces the observed violation. With the current angular
uncertainty of 0.1 degrees, the error introduced is of the order of 1.2e-4.

So in summary, the uncertainties are still dominated by statistical
uncertainties in our experiment, followed by the orientation uncertainty of the
polarizing filters. As the measurement of S is a very direct one, and any long
term efficiency drifts that do not affect the ratio between corresponding
complementary outcomes like H and V only would lower the value of the observed
S, we believe that systematic errors in our setup would only lower the degree
of violation.

As mentioned above, the overestimation of the Poisson contribution was
significant; the resulting uncertainty in the quoted value for S changed form
8.2e-4 to 5.1e-4. This resulted in a reduction of the distance from the
Grinbaum bound from 2.72 sigma to 4.38 sigma.

We did detail a number of uncertainties we considered in the revised
manuscript and revised the quoted uncertainty.

The other issue the referee had with the manuscript was the lack of
explanation of the Grinbaum bound. We completely rewrote the introductory
part, and included a brief outline of Grinbaum's idea, and also tried to
integrate it better in a wider context of other principles.


With this, we feel to have addressed the issues raised by the reviewer, and
were able to convince the editor and reviewer that this is not merely another
Bell test, but a significant observational contribution in the landscape of
principles governing quantum correlations that warrants publication in Physical
Review Letters.

With Best Regards on behalf of all authors,

Christian Kurtsiefer