# A robust, fiber-coupled scanning probe magnetometer using electron spins at
the tip of a diamond nanobeam

Yufan Li  [  Gesa Welker  [  Richard Norte  [  Toeno van der Sar  [
<EMAIL_ADDRESS>
###### Abstract

Fiber-coupled sensors are well suited for sensing and microscopy in hard-to-
reach environments such as biological or cryogenic systems. We demonstrate
fiber-based magnetic imaging based on nitrogen-vacancy (NV) sensor spins at
the tip of a fiber-coupled diamond nanobeam. We incorporated angled ion
implantation into the nanobeam fabrication process to realize a small ensemble
of NV spins at the nanobeam tip. By gluing the nanobeam to a tapered fiber, we
created a robust and transportable probe with optimized optical coupling
efficiency. We demonstrate the imaging capability of the fiber-coupled
nanobeam by measuring the magnetic field generated by a current-carrying wire.
With its robust coupling and efficient readout at the fiber-coupled interface,
our probe could allow new studies of (quantum) materials and biological
samples.

QN] Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft
University of Technology, Delft 2628CJ, The Netherlands [3me] Department of
Precision and Microsystems Engineering, Faculty of Mechanical Engineering,
Delft University of Technology, Delft 2628CD, The Netherlands QN] Department
of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of
Technology, Delft 2628CJ, The Netherlands QN] Department of Quantum
Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology,
Delft 2628CJ, The Netherlands [3me] Department of Precision and Microsystems
Engineering, Faculty of Mechanical Engineering, Delft University of
Technology, Delft 2628CD, The Netherlands QN] Department of Quantum
Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology,
Delft 2628CJ, The Netherlands

##  1 Introduction

Visualizing magnetic phenomena with high spatial resolution has proven to be a
powerful tool in both fundamental physics and applied sciences. Magnetometry
with nitrogen-vacancy (NV) centers in diamond stands out for its nanoscale
imaging capability 1, 2, 3, operability from cryogenic to above room
temperature, and its large range of application areas4, 5, 6, 7, 8. Advances
in NV magnetometry in the past decade have led to breakthroughs especially in
revealing the nanoscale physics of condensed matter systems 4, 9, 10, 11, 12.
However, the free-space optics generally used for optical interrogation of the
NV spins are challenging to realize in cryogenic, intra-cellular, or other
hard-to-reach environments. As such, realizing robust all-fiber-based NV
probes with efficient optical readout could enable new measurements in low-
temperature (quantum) or biological systems.

Achieving high spatial resolution and sensitivity in scanning-probe imaging
requires the sensor to be in close proximity to the target sample, to have a
small size, and to provide efficient addressing and readout. Fiber-coupled NV
probes based on diamond microcrystals placed onto the end of cut fibers have
enabled micron-scale magnetic imaging 13, 14, 15. However, reaching higher
spatial resolution requires confining the NV sensor spins to a nanoscopic
volume located at the tip of a scanning-probe assembly.

Here, we realize a fiber-coupled diamond nanobeam sensor with an ensemble
($\sim$1000) of NV spins deterministically implanted into the nanobeam tip. By
attaching the beam to a tapered optical fiber using ultraviolet-curing optical
glue 16, we realize a robust and transportable nanobeam-fiber assembly with
optimized optical coupling efficiency. We demonstrate through-fiber microscopy
by monitoring the photoluminescence at probe-sample contact, enabled by the
robust nature of the assembly. The through-fiber detected photoluminescence
enables navigation over the sample surface, control over the fiber-sample
distance, and scanning-probe magnetometry without free-space optics. We
demonstrate the magnetic imaging capabilities of the probe by measuring the
magnetic field generated by an omega-shaped current-carrying wire on a chip.

##  2 Results and Discussion

###  2.1 Fabrication and assembly of the fiber-coupled diamond nanobeam

We fabricate the diamond nanobeams using a combination of anisotropic and
isotropic reactive ion etching (Methods) developed in ref. 17. To realize
diamond nanobeams with NV centers at their tips, here we incorporate angled
ion implantation into the fabrication flow (fig. 1a). We start with an
electronic-grade, single-crystal diamond (Element-Six, nitrogen concentration
$<$5\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$) and have 14N ions implanted
(INNOViON Corp., $50\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$,
$1\text{\times}{10}^{13}\text{\,}{\mathrm{cm}}^{-2}$) after the anisotropic
diamond etch that defines the beam sidewalls (fig. 1a, iii). To obtain NV
centers at the nanobeam tips, we implant the ions at a grazing angle of
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$ (fig. 1b) into the end facets while
the top surface of the beam is shielded by the SiNx hard mask. Given the
$500\times$500\text{\,}{\mathrm{nm}}^{2}$$ surface area of the nanobeam end
facet, we expect $\sim 2.5\times 10^{4}$ implanted nitrogen ions at the
nanobeam tip. SRIM simulations18 predict an average implantation depth of
$60\text{\,}\mathrm{n}\mathrm{m}$ underneath the nanobeam end facet.

To realize free-hanging NV-nanobeam sensors, we undercut the nanobeams via
isotropic reactive ion etching (fig. 1c, d) 19, 17, 20, remove the mask
materials via acid cleaning, and then vacuum anneal at
$800\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to form NV centers (see Methods).
We confirm the formation of an NV ensemble at the nanobeam tip by imaging the
nanobeam photoluminescence with a home-built scanning confocal microscope.
(fig. 1e). A distinct bright spot indicates the desired presence of the NV
sensor spins at the end of the beam.

Figure 1: Fabrication of diamond nanobeams with deterministically implanted NV
centers at the nanobeam tip. (a) Fabrication flow: (i) Deposition and
patterning of a SiNx hard mask (dark grey) on a single-crystal diamond
substrate (white); (ii) Anisotropic diamond etch to transfer the mask to the
diamond and create the nanobeam sidewalls; (iii) Nitrogen ion implantation at
80 degrees with respect to the surface normal, parallel to the beam direction
and into the end facet, as illustrated in (b). The sidewalls facing the
implantation are marked in red; (iv) Sidewall protection with Al2O3 (light
grey); (v) Quasi-isotropic diamond etch to undercut the nanobeams. (vi)
Removal of all mask materials with hydrofluoric acid. (b) Side-view schematic
of the angled implantation. (c) SEM image of the undercut diamond nanobeam
(false-colored in purple) after fabrication. The nanobeams are aligned along
the $\langle 110\rangle$ orientation of the diamond lattice. The inset shows a
zoomed-in image of the nitrogen-implanted end facet, false-colored in red. (d)
Optical microscope image of a diamond nanobeam, with the red arrow indicating
the implantation orientation. (e) Photoluminescence map in the black dashed
rectangle area in (d), measured by scanning a focused green laser over the
diamond substrate using a home-built scanning confocal microscope. The bright
spot indicates NV centers concentrated at the end of the nanobeam as expected.

Attaching the nanobeam robustly to the fiber is crucial for realizing a sensor
that is transportable to different setups and that does not break upon probe-
sample contact. Here we attach the nanobeam to the fiber using a thin layer of
optical glue (Norland Optical Adhesive 86H) 16. To do so, we mount the diamond
with the free-hanging nanobeams (fig. 1c) onto a slip-stick positioner
(Mechonics MX-35) and bring a nanobeam into contact with a blunt glass tip
(tip radius $\sim$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) attached to a
holder (fig. S1). We drive the positioner such that the tip pushes the
nanobeam sideways until it breaks off and sticks to the tip (fig. 2a, i). We
then remove the diamond substrate, replace it with a silicon carrier chip, and
temporarily place the nanobeam on the chip edge (fig. 2a, ii).

To apply the glue, we replace the blunt tip with the tapered fiber and the
nanobeam carrier chip with a chip carrying a small droplet of glue. We dip the
fiber into the droplet and then pull it out at a speed of about
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$ (fig.
2a, iii). This speed ensures that a thin layer of glue forms on the fiber
without excessive droplets, ready for attaching the nanobeam.

We glue the fiber to the nanobeam in a way that aims to maximize the optical
coupling efficiency: We re-mount the nanobeam-carrier chip and bring the glue-
covered fiber into contact with the nanobeam lying on the chip edge (fig. 2a,
iv). At this stage, the large contact area between nanobeam and carrier causes
the beam to remain stuck to the carrier chip. This enables adjusting the
relative position between fiber and nanobeam while monitoring the NV
photoluminescence that we excite and detect through the fiber (fig. 2b). When
the detected photoluminescence is maximal, we fix the fiber position and
illuminate the fiber-nanobeam assembly with a UV lamp (Thorlabs CS2010) to
cure the glue. After curing, the bonding between the fiber and the nanobeam is
sufficiently strong to overcome the adhesion to the carrier, such that we can
detach the glued assembly (fig. 2a, v). White-light and through-fiber-excited-
photoluminescence images show the resulting fiber-coupled nanobeam sensor with
the NV centers embedded at the tip (fig. 2c).

Figure 2: Assembling the fiber-coupled nanobeam sensor with optimized photon
collection efficiency. (a) Assembly workflow using optical glue: (i) A blunt
glass tip is used to break the nanobeams off the diamond substrate; (ii) The
blunt tip places the nanobeam at the edge of a silicon carrier; (iii) Optical
glue is applied to the tapered fiber by dipping the fiber tip into a glue
droplet on another carrier chip, and then retracting the fiber; (iv) The glue-
covered tapered fiber is brought into contact with the nanobeam on the edge,
and the glue is cured with UV illumination after optimizing the coupling
between the fiber and the nanobeam; (v) The glued fiber-nanobeam is retracted
from the carrier edge. (b) Simplified schematics of the optical setup. Two
long-pass filters at $600\text{\,}\mathrm{nm}$ and $700\text{\,}\mathrm{nm}$
are placed in front of the photodiode to reduce background photoluminescence.
(c) Microscope image of the glued fiber-nanobeam assembly under ambient
illumination (left, the fiber is false colored in blue) and with through-fiber
$532\text{\,}\mathrm{nm}$ laser excitation (right, with the red arrow
indicating the end of the beam). Both images are taken with a
$650\text{\,}\mathrm{nm}$ long-pass filter in front of the camera to block the
excitation laser.

###  2.2 Characterization of through-fiber NV photoluminescence readout

Figure 3: Characterizing the through-fiber NV photoluminescence readout. (a)
Electron spin resonance (ESR) signal of the NV centers in the nanobeam
measured through the fiber, measured with 30 µW of laser excitation (measured
in free space in front of the fiber coupler) at zero external magnetic field,
under $600\text{\,}\mathrm{nm}$ \+ $700\text{\,}\mathrm{nm}$ long-pass
filtering. A Lorentzian fit (red curve) yields a contrast of 10.0(2)% and a
$7.1(3)\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ zero-field full width half
maximum. (b) Optical saturation of measured photoluminescence, measured under
$600\text{\,}\mathrm{nm}$ \+ $700\text{\,}\mathrm{nm}$ long-pass filtering.
Additional neutral density filtering of ND1.5 is applied before the APD, in
order for the PL to stay within the measurement range of our APD. Measured
data (grey cross) are fitted with eq. 1 (red curve). (c) Optical saturation
measured under $600\text{\,}\mathrm{nm}$ long-pass filtering, and the fit
using eq. 1 (orange curve). Additional neutral density filtering of ND3 is
applied. The non-zero linear background extracted from the fit is plotted as
the purple dashed line.

We characterize the electron spin resonance (ESR) spectrum of the fiber-
coupled nanobeam sensor by exciting and detecting the NV photoluminescence
through the fiber (fig. 2b). We drive the ESR by applying a microwave current
through a stripline on a chip positioned
$\sim$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ below the nanobeam. We find
that adding a $700\text{\,}\mathrm{nm}$ long-pass filter in the detection path
yields a 10.0(2)% ESR contrast (fig. 3a), which is 60% larger than when
filtering with only a $600\text{\,}\mathrm{nm}$ long-pass filter (see SI).
This indicates a strong background photoluminescence in the
$600\sim$700\text{\,}\mathrm{nm}$$ range.

To determine the origin of this background, we measure the photoluminescence
intensity $I$ as a function of laser excitation power $P$ under two filtering
conditions: $600\text{\,}\mathrm{nm}$ \+ $700\text{\,}\mathrm{nm}$ long-pass
filters for the first measurement (fig. 3b), and a single
$600\text{\,}\mathrm{nm}$ long-pass filter for the second measurement (fig.
3c). We fit both datasets to the model

$I(P)=I_{\mathrm{sat}}\frac{P}{P+P_{\mathrm{sat}}}+kP+I_{0}.$ (1)

Here, the first term models the optical saturation of NV centers 21,
characterized by a saturation power $P_{\mathrm{sat}}$ and an asymptotic
intensity $I_{\mathrm{sat}}$. The second term accounts for background
photoluminescence, produced by e.g. the fiber or the glue, which we assume to
be linear with laser power with proportionality constant $k$ 20. The final
term $I_{0}$ represents the dark count rate of our detector.

For the $600\text{\,}\mathrm{nm}$ \+ $700\text{\,}\mathrm{nm}$ filtered
measurement, the fit yields
$k=$-0.01(7)\text{\times}{10}^{6}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mW}}^{-1}$$,
or $k=0$ if we bound $k$ to $k>0$ (fig. 3b). We conclude that the detected
fiber or glue autoluminescence above $700\text{\,}\mathrm{nm}$ is
insignificant compared to the NV luminescence, indicating potential for
reaching the single-spin regime. For the $600\text{\,}\mathrm{nm}$ filtered
measurement, the fit yields
$k=0.016\pm$0.015\text{\times}{10}^{6}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mW}}^{-1}$$
(fig. 3c). As such, the ratio between the NV signal and the background is
still $I_{\mathrm{sat}}/kP_{\mathrm{sat}}\approx 37\gg 1$ for the low powers
$P\ll P_{\mathrm{sat}}$ used in our measurements. Therefore, this small linear
background PL cannot explain the observed difference in ESR contrast between
the two filtering regimes (fig. 3a and SI). Combined with further analysis on
the measured ESR contrasts (SI), we conclude that the background
photoluminescence originates predominantly from neutral NV centers in the
diamond nanobeam. Increasing the NV- to NV0 ratio, for instance by diamond
surface treatments 22, is therefore of primary importance for future
improvements of our fiber coupled sensors.

###  2.3 Demonstration and analysis of scanning magnetometry

We assess the magnetometry performance of our fiber-coupled nanobeam sensor by
measuring the magnetic field generated by a DC electric current in a
microstrip on a chip (fig. 4a, Methods). We mount the chip vertically with
respect to the nanobeam and apply an external magnetic field $B_{0}$ along one
of the four possible crystallographic orientations of the NV centers to
isolate their ESR transition. To measure the field generated by the DC
current, we first position the beam at about
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ above an edge of the strip. While we
expected the ESR resonance to shift with respect to its value at $B_{0}$
because of the microstrip field, we surprisingly observed that the ESR dip
splits (fig. 4c). By measuring the NV ESR spectrum in a line scan across the
strip, we observe the evolution of this split peak in the changing strip field
(fig. 4e). A strongly shifting dip and a weakly shifting dip are clearly
visible.

Figure 4: Characterizing the NV distribution in the nanobeam by imaging the
magnetic field of a current-carrying wire. (a) Side view of the measurement
scheme. The nanobeam is scanned above a
$100\text{\,}\mathrm{n}\mathrm{m}$-thick,
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide gold strip on a silicon
substrate. $I_{\mathrm{dc}}=$20\text{\,}\mathrm{m}\mathrm{A}$$ is applied
through the gold strip along with the microwave signal used to drive ESR. An
external magnetic field $B_{0}=$7.31(6)\text{\,}\mathrm{m}\mathrm{T}$$ is
applied along one of the four crystallographic NV orientations, at
$35\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the beam axis
($\langle 110\rangle$) and perpendicular to $y$ axis. We use the lower ESR
frequency of this NV orientation in subsequent magnetometry measurements. (b)
Illustration of how a small misalignment $\alpha$ during nitrogen implantation
can lead to NVs distributed along the beam sidewall. (c) ESR spectrum measured
at $x=$0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (left, cyan circle in (a))
and $x=$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (right, cyan square in
(a)). The data (light grey) is fitted with a double Lorentzian (red) to
extract the contribution from the end-facet (orange) and sidewall NVs (blue).
(d) Top view of the measurement scheme. (e) Left: Normalized ESR spectra
measured across the strip (dotted arrow in (d)). Middle: ESR frequencies
extracted via a double Lorentzian fit as in (c). We fit the extracted end-
facet ESR frequencies (orange curve) using a model of the wire magnetic field
(green). Free parameters are the offset and scaling in $x$, the lift height
$h$ and NV orientation $\theta_{z}$ with respect to $z$ axis. The fit yields
$h=$1.8(2)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\theta_{z}=$43(3)\text{\,}\mathrm{\SIUnitSymbolDegree}$$. Right: Calculated
ESR line shape across the strip based on eq. 2. The ESR frequencies at the end
facet (white dashed curve, same as the green curve in the middle panel) are
superimposed on the plot for reference.

The observed splitting of the ESR response indicates the presence of two sub-
ensembles of NV centers in our nanobeam that experience different magnetic
fields. Assuming the strongly shifting dip corresponds to the NVs at the
nanobeam tip (as these NVs should experience the largest magnetic field from
the wire), we can fit the magnetic field extracted from this dip to the
calculated wire field at a distance $h$ above the sample. We find an accurate
match with the data for $h=$1.8(2)\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
From the same fit, we can extract the NV angle $\theta_{z}$ (fig. 4a), as this
angle leads to a spatial asymmetry of the signal. We find
$\theta_{z}=$43(3)\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The difference from
the $35\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle expected from the crystal
orientation indicates a small tilt of the fiber-nanobeam with respect to the
normal of the sample surface. We conclude that, despite the presence of a
weakly shifting ESR resonance, we can accurately extract the surface magnetic
field using the NV ensemble at the tip.

We attribute the weakly shifting resonance to NVs that were spuriously
implanted into an unprotected beam sidewall because of a small misalignment
during implantation (fig. 4b). SRIM simulations of
$50\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ nitrogen implantation at a
$\alpha=$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$ misalignment angle with
respect to the diamond surface yield an average implantation depth of
$\sim$10\text{\,}\mathrm{n}\mathrm{m}$$, sufficient to form a low-density,
optically addressable NV layer along the
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$-long beam sidewall. To check the
validity of this assumption, we model the measured ESR spectra $I(f,x)$ by
integrating the ESR response of NV centers that are partially located at the
end facet and partially distributed homogeneously along the beam sidewall:

$I(f,x)=\frac{1}{1+\beta}\left(I_{0}(f,B(x,h))+\frac{\beta}{L}\int_{h}^{h+L}I_{0}(f,B(x,z))dz\right).$
(2)

Here, $I_{0}(f,B)$ is the normalized PL intensity of a single NV center,
$B(x,z)$ is the magnetic field in the nanobeam,
$h=$1.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ denotes the tip-sample
distance determined previously, and $L=$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ is the length of the nanobeam. We assume all NV centers have equal
contrast and linewidth. We weigh the relative contribution of the sidewall NVs
by a factor $\beta$, which parametrizes their different number and coupling
efficiency to the optical mode in the fiber. This model accurately reproduces
the measured split-dip ESR spectra (fig. 4e right) for $\beta=1.4$.

###  2.4 2D microscopy and magnetometry using a fiber-coupled diamond nanobeam

Figure 5: Magnetic imaging with electron spins in a fiber-coupled diamond
nanobeam. (a) Optical microscope image of the omega-shaped gold strip. The
width of the strip is $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) Through-
fiber microscopy of the omega-shaped strip. At each pixel, the nanobeam is
moved towards the sample. The increased photoluminescence upon contact is
shown in the image. (c) 2D map of the end-facet ESR frequency, with
$B_{0}=$5.0(2)\text{\,}\mathrm{m}\mathrm{T}$$,
$I_{\mathrm{dc}}=$10\text{\,}\mathrm{m}\mathrm{A}$$ and
$h\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. At each pixel, the end-facet
and sidewall ESR frequencies are extracted via a double Lorentzian fit with
the sidewall ESR frequency fixed at
$2.729\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. White dashed lines indicate
the boundary of the gold strip inferred from (b). Both (b) and (c) are post-
corrected for drift based on the known shape of the strip (see SI for the
original drift-distorted image).

To demonstrate the 2D imaging capabilities of our fiber-coupled nanobeam, we
characterize the magnetic field generated by a direct current that is sent
through an omega-shaped, $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$-wide gold
strip (fig. 5a). As our setup does not have atomic force microscope feedback,
it requires an alternative method for maintaining a fixed tip-sample distance
in the presence of thermal drifts and sample tilts. We do so by bringing the
tip in contact with the sample at each pixel and then retracting the tip by a
fixed amount. To detect tip-sample contact, we move the tip towards the sample
until we observe a sharp increase in the detected photoluminescence (SI). We
find that this increase is highly sensitive to the local sample reflectivity,
enabling through-fiber imaging of the sample (fig. 5b) and thus precluding the
need for free-space optics. Crucially, our sensor remains intact under the
repetitive tip-sample contacts, highlighting the robust nature of the glued
nanobeam-fiber assembly.

With height control and navigation in place, we image the magnetic field of
the strip by measuring the ESR frequency of the end-facet NVs (fig. 5c). To do
so, we retract the tip by $\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
after making tip-sample contact, measure the ESR spectrum and extract the end-
facet NV ESR frequency using a double Lorentzian fit as described above (fig.
4c), with the sidewall ESR frequency fixed to
$2.729\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ to increase the robustness of
the fit. The extracted end-facet ESR frequencies (fig. 5c) encode the
projection of the total magnetic field onto the NV axis. The expected near-
uniform magnetic field within the ring and the magnetic field in the opposite
direction outside the ring are clearly observed.

A key parameter for determining the sensitivity of our sensor is the optical
coupling efficiency between the nanobeam and the fiber. Specifically, the
coupling efficiency $\eta$ of the end-facet NVs is the relevant parameter for
our intended measurement regime. An estimation of this efficiency can be made
via the optical saturation behavior of the measured photoluminescence 20,
combined with the estimated ratio between end-facet NVs and sidewall NVs from
the ESR contrast ratio of the two dips (Methods). This results in
$\eta=2\%\text{-}26\%$, with the large uncertainty originating from the
unknown conversion efficiency of implanted nitrogen ions to NV centers. This
result is in reasonable agreement with that of fiber-coupled nanobeams with a
known homogeneous NV density 20.

##  3 Conclusion and Outlook

We have demonstrated a robust, fiber-coupled scanning probe based on diamond
nanobeams, utilizing NV centers for sub-micron resolution 2D magnetic imaging.
Our fabrication approach, which incorporates angled ion implantation into the
quasi-isotropic diamond etch workflow, deterministically positions NV centers
at the end facet of the nanobeam. We attached these nanobeams to tapered
fibers using optical glue to create a robust, transportable probe while
optimizing the fiber-nanobeam interface by maximizing the photoluminescence
detected through the fiber detection efficiency. Using our probe as a scanning
magnetometer, we showed that we can isolate the signal of NV centers at the
end facet even in the presence of the background signal from misaligned NV
centers, and accurately extract the magnetic field variation at the sample
surface. Finally, we demonstrate 2D imaging of the magnetic field generated by
an omega-shaped current carrying wire under sub-micron spatial resolution.
Combined with a method to image the sample surface through reflection-enhanced
signal collection at probe-surface contact, we show that sample navigation and
lift height control can be achieved without the need of free-space optical
components, making our scanning probe ready for all-fiber operation in various
environments.

Reducing the number of sidewall-implanted NVs would improve the signal-to-
background ratio and thereby the magnetometry sensitivity. We anticipate this
can be achieved by performing the angled ion implantation after the atomic
layer deposition step (step iv in fig. 1a): With a
$20\text{\,}\mathrm{n}\mathrm{m}$ layer of alumina uniformly covering the
sample, one could set the implantation energy such that the nitrogen ions
would be implanted into the diamond under the near-normal incidence at the end
facet, but would get trapped in the alumina for the grazing-angle incidence at
the beam sidewalls.

The limiting factor of our spatial resolution is two-fold: First, the NV
ensemble implanted homogeneously into the
($0.5\times$0.5\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$) beam end-facet
limits the spatial resolution. Low-nanoscale imaging therefore requires moving
to single NV regime. With the previous characterization of the negligible
background contribution (fig. 3), we already showed that single NV readout is
possible under $600\text{\,}\mathrm{nm}$ \+ $700\text{\,}\mathrm{nm}$ long-
pass filtering. Second, the reflection-based height control we used in this
work does not allow feedback control of the tip-sample distance below
$h\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. To this end, we are working
towards integrating our fiber probe in a tuning-fork based atomic force
microscope (AFM) system 23, based on gluing the tip of the fiber to the tuning
fork 24 or on keeping the fiber and the tuning fork in mechanical contact with
a dedicated scan head design 25, 26.

One important future application of our fiber-coupled probe is scanning
magnetometry at low temperature, as free space optics is not required. As
such, it could be an excellent platform for exploring the sensing capabilities
of group-IV color centers, which have proven to be more stable against surface
charge noise at low temperatures 27, 28, 29. This potentially enables resonant
optical addressing of target sensor defects in the beam. The nanobeam
structure is also compatible with nanophotonic structuring, holding potential
in further enhancement of excitation and readout efficiency of the color
centers under resonant driving 17, 27. With the above mentioned efforts, our
fiber-coupled probe opens up new avenues for NV-based magnetic sensing in
hard-to-reach environments.

##  4 Methods

###  4.1 Fabrication of the diamond nanobeams

We fabricate the diamond nanobeams using electronic grade single crystal
diamond (Element Six). The fabrication recipe is based on ref.17, where
details on process parameters can be found. Before the fabrication flow, the
diamond substrate is sliced and polished into a
$$2\text{\,}\mathrm{mm}$\times$2\text{\,}\mathrm{mm}$\times$0.05\text{\,}\mathrm{mm}$$
chip (Almax EasyLab). After polishing, we clean the surface with fuming HNO3
and relieve the surface damage induced by mechanical polishing with
consecutive Ar/Cl and O2 etches (Oxford Instruments Plasmalab 100). This
removes a total of $6\text{-}$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ of
diamond from the surface 30.

We start the fabrication by depositing a $200\text{\,}\mathrm{nm}$ layer of
Si3N4 hard mask with inductively-coupled plasma enhanced chemical vapour
deposition (ICPECVD) (Oxford Instruments PlasmaPro 100). Then we spin-coat a
$\sim$400\text{\,}\mathrm{nm}$$ layer of e-beam resist (AR-P 6200-13) and
another $\sim$30\text{\,}\mathrm{nm}$$ layer of conductive Electra-92, and
write the pattern with e-beam lithography (Raith EBPG5200). The pattern is
transferred to the hard mask with an anisotropic CHF3/O2 etch (AMS 100
I-speeder). After cleaning the residual e-beam resist with dimethylformamide
(DMF) and piranha solution (3:1 mixed 96% H2SO4 and 31% H2O2), we
anisotropically etch $\sim$600\text{\,}\mathrm{nm}$$ into the diamond with O2
RIE (Oxford Instruments Plasmalab 100) to create the sidewalls of the
nanobeams.

We then implant 14N ions into the diamond sidewalls as described in the main
text (INNOViON Corp). After implantation, we do another piranha cleaning to
remove contamination on the surface, and deposit
$\sim$20\text{\,}\mathrm{nm}$$ of Al2O3 with atomic layer deposition (ALD)
(Oxford Instruments FlexAL). We then do an anisotropic BCl3/Cl2 RIE (Oxford
Instruments Plasmalab 100) to etch away the Al2O3 on the top surfaces, so that
only the sidewalls of the diamond nanobeams remain protected while the diamond
surfaces around are exposed to O2 plasma in subsequent steps. We do a second
anisotropic O2 RIE which etches another $\sim$250\text{\,}\mathrm{nm}$$ into
the diamond. This is to eliminate the potential effect of the damaged diamond
surface after the ion implantation to the undercut quality. We then proceed
with the quasi-isotropic O2 RIE to undercut the beams and remove the masks
with 40% HF after the beams are fully released. The fabricated diamond chip is
then vacuum annealed at $800\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 2
hours to create the NV centers.

###  4.2 Setup for current measurements

The gold strips for current measurements are patterned on top of a silicon
substrate via evaporation ($5\text{\,}\mathrm{nm}$ titanium +
$100\text{\,}\mathrm{nm}$ gold, Temescal FC-2000) and lift-off. The DC current
is applied using a function generator (Tektronix AFG1062) and read out by a
digital multimeter wired in series with the sample. We apply the RF signal for
driving the NV centers through the same strip, using a bias-T (Mini-Circuits
ZFBT-6GW-FT+) to combine the RF and DC signal. A complete schematic of the
measurement setup can be found in SI.

###  4.3 Vector magnetometry with NV centers

Under an external magnetic field $\mathbf{B}$, the Hamiltonian of an NV center
is written as

$H=DS_{z}^{2}+\gamma\mathbf{B}\cdot\mathbf{S}$ (3)

in which $D=$2.87\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ is the zero-field
splitting of the NV center,
$\gamma=$28\text{\,}\mathrm{GHz}\text{\,}{\mathrm{T}}^{-1}$$ is the
gyromagnatic ratio and $\mathbf{S}=(S_{x},S_{y},S_{z})$ are the $3\times 3$
Pauli matrices for spin-1. The upper and lower ESR frequencies $f_{u}$ and
$f_{l}$ are then the differences between the eigenfrequencies of the
Hamiltonian. By diagonalizing the Hamiltonian, one can determine from the
measured ESR frequencies both the amplitude of the external field $B$, and its
angle with respect to the NV orientation $\theta$:

$B=\frac{1}{\sqrt{3}\gamma}\sqrt{f_{u}^{2}+f_{l}^{2}-f_{u}f_{l}+D^{2}};$ (4)
$\cos^{2}{\theta}=\frac{-(f_{u}+f_{l})^{3}+3(f_{u}^{3}+f_{l}^{3})+2D^{3}}{27D(\gamma
B)^{2}}+\frac{1}{3}.$ (5)

These relations allow us to align the external field $B_{0}$ with the NV
orientation, and precisely determine the value of $B_{0}$ through the measured
ESR spectrum during calibration. In the main text, we also used the inverse of
these relations to calculate the expected ESR frequencies of a given magnetic
field distribution from the DC current, and used this calculation to fit the
experimentally measured ESR frequencies in fig. 4(e).

###  4.4 Estimation of the coupling efficiency for the end-facet NVs

The fraction of end-facet NV contribution in the total measured PL can be
estimated from the contrast ratio extracted from fig. 4c left (where the dips
are most split):

$\beta_{\mathrm{E}}=\frac{I_{\mathrm{E}}}{I_{\mathrm{tot}}}=\frac{C_{\mathrm{E}}}{C_{\mathrm{E}}+C_{\mathrm{S}}}=0.43(7)$
(6)

where the footnotes E and S stand for the end-facet and sidewall contribution
respectively. We then scale the $600\text{\,}\mathrm{n}\mathrm{m}$ filtered
saturation curve in fig. 3c with $\beta_{\mathrm{E}}$ to extract the
saturation intensity of the end-facet NVs

$I_{\mathrm{sat,E}}=\beta_{\mathrm{E}}I_{\mathrm{sat,600nm}}=(5.6\pm
1.1)\times 10^{5}\mathrm{s}^{-1},$ (7)

from which the total collection efficiency of the end-facet NVs can be
estimated as the ratio of the detected saturation intensity and the
theoretically expected saturation emission rate:

$\eta_{\mathrm{E}}=\frac{I_{\mathrm{sat,E}}}{N_{\mathrm{E}}\Gamma},$ (8)

where $N_{\mathrm{E}}$ denotes the number of end-facet NVs, and
$\Gamma=1/($13\text{\,}\mathrm{n}\mathrm{s}$)$ 31 is the expected saturation
photon emission rate of a single NV. In our measurement setup,
$\eta_{\mathrm{E}}$ can be expressed as the product of the fiber-nanobeam
coupling efficiency $\eta_{\mathrm{E,f}}$, ND filtering
$\eta_{\mathrm{ND}}=1\times 10^{-3}$ and the total detection efficiency of the
optical path $\eta_{\mathrm{D}}=0.14(2)$ (including the loss on optical
elements and detection efficiency of the APD).

The major uncertainty in estimating the fiber-nanobeam coupling efficiency
$\eta_{\mathrm{E,f}}$ lies in the NV number $N_{\mathrm{E}}$, due to the large
uncertainty in the conversion efficiency of the implanted nitrogen ions to NV
centers. Taking the data from ref.32 as a reference and further considering
the possible deviation from this dataset caused by the presence of
nanostructures, we assume the conversion efficiency to be between 1%-10%. This
results in $\eta_{\mathrm{E,f}}=2\%\text{-}26\%$.

##  5 Author Imformation

###  5.1 Funding Sources

This work is supported by the Dutch Science Council (NWO) through the NWA
grant 1160.18.208, the NGF Quantum Technology grant 1582.22.038 and the Kavli
Institute of Nanoscience Delft.

###  5.2 Author Contributions

Y.L., R.N. and T.v.d.S. conceived the experiments. Y.L. fabricated the diamond
nanobeam samples and performed the magnetometry measurements. Y.L. and G.W.
developed the gluing workflow and prepared the fiber-coupled nanobeam probes.
Y.L. and T.v.d.S. analyzed the results, and wrote the manuscript with
contributions from all coauthors.

###  5.3 Notes

The authors declare no competing financial interest.

###  5.4 Data availability

All data plotted in the figures are this work are available at zenodo.org with
identifier 10.5281/zenodo.12724082. Additional data related to this paper are
available upon request.

The authors thank N. Codreanu for the help with the diamond nanofabrication
process, and B.G. Simon for helping with diamond preparation and ion
implantation.

Setup for nanobeam manipulation and scanning measurements; Analysis on
background photoluminescence and the effect of filtering wavelengths; Time
trace of measured photoluminescence upon nanobeam-sample contact; Fit results
for the 2D scan.

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††thanks<EMAIL_ADDRESS>
# Information Causality without concatenation

Nikolai Miklin  International Centre for Theory of Quantum Technologies
(ICTQT), University of Gdansk, 80-308 Gdańsk, Poland  Marcin Pawłowski
International Centre for Theory of Quantum Technologies (ICTQT), University of
Gdansk, 80-308 Gdańsk, Poland

###### Abstract

Information Causality is a physical principle which states that the amount of
randomly accessible data over a classical communication channel cannot exceed
its capacity, even if the sender and the receiver have access to a source of
nonlocal correlations. This principle can be used to bound the nonlocality of
quantum mechanics without resorting to its full formalism, with a notable
example of reproducing the Tsirelson’s bound of the Clauser-Horne-Shimony-Holt
inequality. Despite being promising, the latter result found little
generalization to other Bell inequalities because of the limitations imposed
by the process of concatenation, in which several nonsignaling resources are
put together to produce tighter bounds. In this work, we show that
concatenation can be successfully replaced by limits on the communication
channel capacity. It allows us to re-derive and, in some cases, significantly
improve all the previously known results in a simpler manner and apply the
Information Causality principle to previously unapproachable Bell scenarios.

##  I Introduction

Information Causality (IC) is a physical principle proposed to bound
nonlocality of correlations without resorting to the full formalism of quantum
mechanics Pawłowski et al. (2009); Pawłowski and Scarani (2016). Instead, the
bounds are derived only from the axioms of information theory. In a nutshell,
the principle states that if one party has a single use of a communication
channel with a capacity $C$ to send the other party some information, then the
amount of information potentially available to the receiver cannot exceed $C$
even if the parties share some nonlocal resources. In Ref. Pawłowski et al.
(2009) it was shown that both classical and quantum information theories, and
every generalization of them adhering to some of their intuitive properties,
obey IC. At the same time, the principle of IC is strong enough to partially
recover the boundary of the set of quantum nonlocal correlations Allcock et
al. (2009a). Most notably, as shown in Ref. Pawłowski et al. (2009), IC can be
used to re-derive the Tsirelson’s bound of the Clauser-Horne-Shimony-Holt
(CHSH) inequality, answering the long-standing question of Popescu and
Rorhlich on the reason for its value Cirel’son (1980); Clauser et al. (1969);
Popescu and Rohrlich (1994). Moreover, IC was shown to rule out stronger-than-
quantum correlations, which could not be detected by other bipartite
principles, such as Macroscopic Locality Navascués and Wunderlich (2010);
Brassard et al. (2006); Cavalcanti et al. (2010). Finally, the principle of IC
is the only known candidate with the potential to recover the exact boundary
of the set of bipartite nonlocal quantum correlations Navascués et al. (2015).

The problem that one faces when deriving bound on the strength of nonlocal
correlations from IC is that one has to find a suitable communication protocol
that makes use of those correlations. Until now, it was believed that to
obtain the strongest results, one must use protocols that allow for
concatenation, the process of combining the outcomes of several copies of the
nonlocal source in a way that increases the amount of potentially available
information in stronger-than-quantum theories. This requirement significantly
limits the types of Bell scenarios for which the bounds from IC can be derived
Pawłowski et al. (2009); Pawłowski and Żukowski (2010); Cavalcanti et al.
(2010).

In this work, we argue that the procedure of concatenation is not required and
often suboptimal in proofs utilizing IC. More precisely, we show that by
considering a non-identity communication channel with suitably chosen capacity
and a single copy of a nonlocal resource, one can: (a) re-derive all the
results from Ref. Pawłowski et al. (2009); Pawłowski and Żukowski (2010); (b)
tighten the bounds found in Ref. Cavalcanti et al. (2010); (c) apply IC to
Bell scenarios for which no concatenation procedure is known. We expect that
with the modified construction, IC is likely to become a staple tool for
finding bounds on Bell nonlocality in situations when traditionally applied
numerical methods are computationally demanding Navascués et al. (2007).

##  II IC and concatenation

We start by restating the formulation of the IC principle from Ref. Pawłowski
and Scarani (2016). Consider a communication scenario in which the sender has
$N$ real-valued random variables $a_{0},...,a_{N-1}$ and a single use of a
channel with classical communication capacity $C$. Then

$\sum_{i=0}^{N-1}I(a_{i};b_{i})\leq C,$ (1)

where $b_{i}$ is a random variable denoting the receiver’s guess of the value
of $a_{i}$ if the receiver chooses to guess it, and $I(a_{i};b_{i})$ is the
Shannon’s mutual information.

In the original formulation of IC, as given by Ref. Pawłowski et al. (2009),
the bound on the right-hand side of Eq. (1) is defined as the size of a
message that the sender communicates in each round. The latter, somewhat vague
statement, was later clarified in authors’ subsequent work of Ref. Pawłowski
and Scarani (2016): “Notice that it does not matter how this information is
encoded: when we refer to ‘sending the $M$ bit message’, it should be
understood as a single use of a channel with classical communication capacity
$M$.”

In order to apply IC to quantum correlations, in Ref. Pawłowski et al. (2009)
the authors start by considering the following protocol van Dam (2013): The
parties share a pair of devices characterized by probability distribution
$P(a,b|x,y)$ with all variables $a,b,x,y$ binary, $x$ and $a$ being the input
and output of the sender, and $y$ and $b$ – the input and output of the
receiver (Here and later in the text, by $P(a,b|x,y)$ we mean all the
probabilities $P(a=i,b=j|x=k,y=l)$,$\forall i,j,k,l$). Let $N=2$ and
$P(a,b|x,y)$ be such that $b=a\oplus x\cdot y$ with probability $p$ for both
$y\in\\{0,1\\}$ ($\oplus$ denotes the summation modulo $2$). The sender
chooses $x=a_{0}\oplus a_{1}$ and transmits a message $m=a_{0}\oplus a$ to the
receiver. In order to learn about $i$-th bit $a_{i}$, the receiver chooses
$y=i$ and computes $b_{i}=m\oplus b$. It is straightforward to confirm that
$a_{i}=b_{i}$ with probability $p$. If values of $a_{0}$ and $a_{1}$ are
distributed uniformly, this protocol yields $I(a_{i};b_{i})=1-h(p)$, where
$h(.)$ is the Shannon’s binary entropy. If $m$ is send over a perfect channel,
then $C=1$ since $m$ is just a bit. The bound resulting from Eq. (1) is
$2(1-h(p))\leq 1$, which implies $p\leq 0.890$. It is easy to see that the
CHSH expression for $P(a,b|x,y)$ is also equal to $p$ Clauser et al. (1969).
Hence, we have derived a nontrivial bound on CHSH inequality from IC. However,
the maximum quantum value of $p$, known as the Tsirelson bound, is
$p_{Q}=\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\approx 0.854$ Cirel’son
(1980), which is significantly lower than what we have just derived.

To obtain a tighter bound on $p_{Q}$ from IC, the authors of Ref. Pawłowski et
al. (2009) propose to increase $N$, the number of bits given to the sender at
each round, and use concatenation. Concatenation is essentially a process of
locally combining inputs and outputs of many copies of the same pair of
devices. It is, however, different from the simple “wiring” of devices Allcock
et al. (2009b), as in the case of concatenation the input of each device of
the sender also depends on the bits $a_{i}$.

Here, we do not give the explicit description of concatenation, and refer the
reader to Ref. Pawłowski et al. (2009). Instead, we only state some details of
it as facts. This procedure works for $N=2^{k}$ for some positive integer $k$.
At sender’s the devices are placed in layers with $k$-th layer having
$2^{k-1}$ devices. Each of the devices produces a “message” just like in the
protocol above and the resulting $2^{k-1}$ “messages” are taken as inputs for
the devices in layer $k-1$. Each layer of concatenation introduces an error
diluting the information about each of the bits $a_{i}$. If at some layer
$k-1$ the probability of successfully decoding $a_{i}$ is $p_{k-1}$, at the
next level $k$ it will be

$p_{k}=p_{k-1}p+(1-p_{k-1})(1-p)=\frac{1+e_{k-1}e}{2},$ (2)

where $e_{k-1}$ and $e$ are biases of $p_{k-1}$ and $p$, i.e.,
$p_{k-1}=\frac{1+e_{k-1}}{2}$ and $p=\frac{1+e}{2}$. The condition in Eq. (1)
for a protocol with $k$ levels of concatenation becomes

$2^{k}\left(1-h\left(\frac{1+e^{k}}{2}\right)\right)\leq 1.$ (3)

For every $k$ it puts a lower bound on $e$ and for $k\to\infty$ the bound on
$e$ reaches $\frac{1}{\sqrt{2}}$ while the bound on $p$ converges to our aim,
$p_{Q}$.

##  III Replacing concatenation with noisy channel

There is, however, an easier way to obtain the bound $p_{Q}$ from IC without
the need of concatenation. Let us start by considering the same protocol as
before for a single pair of devices, but change the communications channel
between the sender and the receiver to a binary symmetric one. Such a channel
transmits an unchanged input bit with probability $p_{c}$, and with
probability $1-p_{c}$ it returns the flipped bit. The capacity of this channel
is $1-h(p_{c})$. The probability for $b_{i}$ to be equal to $a_{i}$ is
obtained with the same formula in Eq. (2), where instead of $p_{k-1}$ we use
$p_{c}$, yielding $\frac{1+e_{c}e}{2}$. The condition in Eq. (1), expressed in
the terms of biases becomes

$2\left(1-h\left(\frac{1+e_{c}e}{2}\right)\right)\leq
1-h\left(\frac{1+e_{c}}{2}\right).$ (4)

The bound implied by the above condition becomes stronger as $p_{c}$ and the
channel capacity decrease as shown in Fig. 1. For $p_{c}$ approaching
$\frac{1}{2}$ the bound on $p$ approaches $p_{Q}$. As we show below, this is a
consequence of a more general result.

Figure 1: Bound on $p$ as a function of $p_{c}$ characterizing binary
symmetric channel. $p\to p_{Q}$ as $p_{c}\to\frac{1}{2}$.

######  Result 1.

Any bound on nonlocality in the case of unbiased errors obtained with
concatenation procedure can also be obtained by a protocol involving a single
pair of devices and a suitably chosen discrete memoryless channel.

Before we move to the proof, we need to clarify what we mean by “unbiased
errors case”. Let us consider a single pair of the devices producing
probability distribution $P(a,b|x,y)$, where $a,b\in\\{0,1,\dots,d-1\\}$, and
$y\in\\{0,1,\dots,n-1\\}$. Defining the range of $x$ is not necessary for this
argument. Let us assume a protocol involving the communication of one of $d$
possible messages over a classical identity channel such that with probability
$p$, the receiver makes the right guess $b_{i}=a_{i}$. The “unbiased errors”
assumption means that $p$ does not depend on $i\in\\{0,1,\dots,n-1\\}$, every
term in $p$ corresponding to each value of $a_{i}$ is equal, and that all the
other “error” cases $b_{i}\neq a_{i}$ are equally probable and also uniformly
distributed with respect to $a_{i}$. Arguably, the case of unbiased errors is
a special one, but it is general enough to encompass all currently known
results Pawłowski et al. (2009); Pawłowski and Żukowski (2010); Cavalcanti et
al. (2010).

Let us now clarify what we mean by finding “bound on nonlocality”. Given
$P(a,b|x,y)$, one may ask whether this nonlocal behavior complies with IC’s
statement, giving a “yes/no” answer. However, one may instead ask a
quantitative question of how much noise needs to be added to $P(a,b|x,y)$ in
order for it to satisfy IC. It is standard to consider the white noise, and
the guessing probability $p$ in IC is proportional to the amount of white
noise required. Hence, $p$ quantifies nonlocality of $P(a,b|x,y)$. In some
cases, as it is for CHSH, the bound on $p$ also implies the bound on Bell
inequality.

###### Proof.

Following Ref. Cavalcanti et al. (2010) we generalize the definition of the
bias $e$ of probability $p$ as

$p=\frac{1+(d-1)e}{d}.$ (5)

If we choose to concatenate the protocol $k$ times, as it is done in Refs.
Pawłowski and Żukowski (2010); Cavalcanti et al. (2010), the success
probability of $b_{i}=a_{i}$ will be $p_{k}=\frac{1+(d-1)e^{k}}{d}$. From the
symmetry of the protocol and unbiasedness of errors, we conclude that all the
mutual information terms in the IC expression are equal, and given by
$I(a_{i};b_{i})=I_{d}(e^{k})$, where

$\begin{split}I_{d}(e)&=\log d-h\left(\frac{1+(d-1)e}{d}\right)\\\
&-\frac{(d-1)(1-e)}{d}\log(d-1).\end{split}$ (6)

The above expression is known as Fano’s inequality Fano (1961), which is
equality in this case (See Appendix A for a short proof). Let us assume that
$k$ is such that $k$ levels of concatenation are not enough to demonstrate
that $p$ violates IC, but $k+1$ are. In other words:

$\begin{split}n^{k}I_{d}(e^{k})\leq\log d\\\ n^{k+1}I_{d}(e^{k+1})>\log
d,\end{split}$ (7)

which implies

$nI_{d}(e^{k+1})>I_{d}(e^{k}).$ (8)

Let us now take a single pair of the devices and let the parties communicate
over a discrete memoryless channel channel with uniform errors, i.e., with
probability $p_{c}$ the message is unchanged and with $1-p_{c}$ probability it
is changed to one of the other $d-1$ messages in accordance with the uniform
distribution. Let $p_{c}=\frac{1+(d-1)e^{k}}{d}$, then the capacity of the
channel is $C=I_{d}(e^{k})$ (See Appendix A for a short proof). The
probability for $a_{i}=b_{i}$ is
$p_{c}p+\frac{(1-p_{c})(1-p)}{d-1}=\frac{1+(d-1)e^{k+1}}{d}$ and
$I(a_{i};b_{i})=I_{d}(e^{k+1})$, for all $i\in\\{0,1,\dots,n-1\\}$. Therefore,
in this case the IC condition in Eq. (1) reads

$nI_{d}(e^{k+1})\leq C=I_{d}(e^{k}).$ (9)

If the probability $p$ is such that Eq. (8) holds, the above inequality will
be violated, which means that $p$ can also be detected by the IC with a single
pair of the devices and a discrete memoryless channel with capacity $C$. ∎

##  IV New and tighter bounds

In this section, we demonstrate that the bounds on nonlocality obtained with
the new approach are strictly better in some cases. We also discuss cases in
which this approach can provide bounds while the concatenation procedure is
not applicable.

In the proof of Result 1, instead of choosing $p_{c}$ to be the success
probability corresponding to $k$-th level of concatenation, we can take
$p_{c}=\frac{1+(d-1)e_{c}}{d}$ and optimize over $e_{c}$. The condition that
we use to bound nonlocality is the following

$nI_{d}(e_{c}e)\leq I_{d}(e_{c}).$ (10)

Even though it is cumbersome to write the solution for the optimal $e_{c}$
explicitly, the optimization over a single real parameter can be done up to an
arbitrary numerical precision. In table 1 we compare the bounds on the bias
$e$ of the success probability $p$ implied by Eq. (10) with the bounds from
Ref. Cavalcanti et al. (2010), calculated using concatenation procedure for
$n=2$.

|  Optimal $e_{c}$ |  $e$ |  $e^{\prime}$
---|---|---|---
d=3 |  0.295 |  0.702 |  0.708
d=4 |  0.389 |  0.696 |  0.705
d=5 |  0.436 |  0.690 |  0.700
d=20 |  0.531 |  0.648 |  0.659
Table 1: Optimal $e_{c}$ is the value for which Eq. (10) puts the tightest
bound on $e$. $e$ is the improved bound, while $e^{\prime}$ is the bound from
Ref. Cavalcanti et al. (2010).

The major challenge of finding bounds on nonlocality with the concatenation
procedure is calculating each level’s success probabilities. This calculation
is easy only if one assumes the unbiasedness of errors, as described in Result
1. This assumption puts a significant constraint on the choice of the protocol
and the correlations $P(a,b|x,y)$ that can be easily bounded using the IC.

On the contrary, the method suggested in this paper can be applied to any
protocol. Below we give an example of bounding nonlocality in a Bell scenario
with $3$ settings per party and with binary outcomes (often referred to as
$3322$ scenario). Let us consider a non-signaling distribution
$P_{NS}(a,b|x,y)$ ($x,y\in\\{0,1,2\\}$) given by the following relation

$a\oplus b=\begin{cases}1,&\text{if }(x,y)\in\\{(1,2),(2,1),(2,2)\\}\\}\\\
0,&\text{otherwise},\end{cases}$ (11)

with all marginal probability distribution being uniform, i.e.,
$\sum_{i}P(a=i,b=j|x,y)=\sum_{j}P(a=i,b=j|x,y)=\frac{1}{2},\forall i,j$. This
distribution gives the maximal violation equal to $1$ of the $I_{3322}$
inequality Collins and Gisin (2004). Let us now consider a pair of devices
producing correlations $P_{e}(a,b|x,y)=eP_{NS}(a,b|x,y)+(1-e)P_{L}(a,b|x,y)$,
where $P_{L}(a,b|x,y)$ is the white noise distribution for which
$P_{L}(a=i,b=j|x,y)=\frac{1}{4},\forall i,j$ and all values of $x,y$. The
value of $I_{3322}$ inequality for the considered mixing is $2e-1$. We ask a
question of the maximal degree of nonlocality, specified by $e$, that is
allowed by IC. We can find a protocol, which we specify in Appendix B, that is
optimal for the nonlocal correlations $P_{NS}(a,b|x,y)$ from Eq. (11). Using
the symmetric channel, parameterized by $e_{c}$, in the limit of
$e_{c}\rightarrow 0$ we obtain a bound $e\rightarrow\frac{2}{3}$, which is not
far from the quantum bound of $\frac{3}{5}$, which can be confirmed (up to a
numerical precision) by the hierarchy of semidefinite programming of Ref.
Navascués et al. (2007). To compare, the bound which can be derived from IC
with a channel capacity of 1 is about $0.7445$, and there is no clear way to
construct a concatenation procedure for this protocol or to calculate the
corresponding guessing probabilities.

##  V Discussion

We have shown that concatenation can be successfully replaced by considering
different classical communication channels in the protocols used to bound
nonlocality with IC. Apart from showing that all the results already obtained
can be re-derived with the new approach, we also showed that some could be
improved. Additionally, we gave an example of a scenario that would be very
challenging to approach with the concatenation procedure. Perhaps the most
important goal of this paper is to show that IC’s scope of applicability can
be significantly widened. Therefore, our paper opens many possible ways for
future work.

We list some of the open questions which we believe deserve a separate study.
In the current paper, we limited ourselves to a particular type of discrete
memoryless channels, namely symmetric channels. These channels seem very well
suited to analyze Bell inequalities, which correspond to unique games, that is
games in which for every combination of parties’ inputs and one party output,
there is a unique output for the other party that wins the game. However, for
other Bell inequalities, different channels can yield stronger results, and
find optimal ones is an open question. Additionally, considering random data
in the IC scenario with different alphabets could be used to bound nonlocality
in distributions with a bias towards one of the outcomes. Finally, the
statement of IC can be read in the opposite direction. Namely, given a value
of guessing probability, one can obtain a lower bound on the minimal
communication required for such correlations, which can be used for randomness
certification.

##  VI Acknowledgements

We acknowledge support by the Foundation for Polish Science (IRAP project,
ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth
Operational Programme), M.P. acknowledges the support of NCN through grant
SHENG (2018/30/Q/ST2/00625). This research was made possible by funding from
QuantERA, an ERA-Net cofund in Quantum Technologies (www.quantera.eu), under
the project eDICT.

##  Appendix A Technical details of the proof of Result 1

Fano’s inequality for two random variables $a$ and $b$ taking values in
$\\{0,1,\dots,d-1\\}$ can be written as follows

$I(a;b)\geq\log d-h\left(p\right)-(1-p)\log(d-1),$ (12)

where $p=\mathrm{P}(a=b)$. In Eq. (6) we took $p=\frac{1+(d-1)e}{d}$ and
denoted the function on the right-hand side of Eq. (12) as $I_{d}(e)$. Let
$P(a=i,b=j)=r(j|i)P(a=i)$ be the decomposition of the joint probability
distribution of $a$ and $b$ with a response function (conditional
distribution) $r(j|i)$. For the case of probabilities in the derivations of
Result 1, the form of $r(j|i)$ is the following: $r(i|i)=p,\forall i$, and
$r(j|i)=\frac{1-p}{d-1},\forall j\neq i$. The mutual information $I(a;b)$ by
definition is equal to

$I(a;b)=\sum_{i=0}^{d-1}\sum_{j=0}^{d-1}P(a=i)r(j|i)\log\frac{r(j|i)}{\sum_{i=0}^{d-1}r(j|i)P(a=i)},$
(13)

Since the input $a$ is always taken to be uniformly distributed, we have that
$P(a=i)=\frac{1}{d}$, and hence
$\sum_{i=0}^{d-1}r(j|i)P(a=i)=\frac{1}{d},\forall j$. From here we can deduce
that

$\begin{split}I(a;b)&=\log(d)+\frac{1}{d}\sum_{i=0}^{d-1}\sum_{j=0}^{d-1}r(j|i)\log(r(j|i))\\\
&=\log(d)+p\log(p)+(1-p)\log\left(\frac{1-p}{d-1}\right),\end{split}$ (14)

which is exactly equal to the right-hand side of Eq. (12).

Now, we give a short proof that capacity of discrete memoryless channel, which
transmits $d$-dimensional message unchanged with probability
$p_{c}=\frac{1+(d-1)e_{c}}{d}$ and with probability $1-p_{c}$ changes it to
one of the $d-1$ other values according to the uniform distribution, is equal
to $I_{d}(e)$ (given by the right-hand side of Eq. (12)).

By definition, the capacity of a discrete memoryless channel is

$C=\max_{P(a)}I(a;b),$ (15)

where $b$ is a guess of $a$, and $P(a)$ is the distribution of $a$. The proof
is similar to the one above, since the response function $r(j|i)$ is exactly
the same for the considered channel (where we substitute $p$ with $p_{c}$).
The only part which requires the proof is that the optimal distribution $P(a)$
is the uniform one. It can be easily seen from the fact that, first of all,
entropy is a concave function, and that the form of $r(j|i)$ is symmetric.

##  Appendix B Protocol for the 3322 scenario

Here, we give a specification of a protocol for the $3322$ scenario. In this
protocol, the sender has access to three bits $a_{0},a_{1},a_{2}$, depending
on which the choice of measurement $x\in\\{0,1,2\\}$ is determined. The bit
message $m$ is a function of $a_{0},a_{1},a_{2}$ and the sender’s outcome $a$.
The decoding function determines the guess $b_{i}$ of $a_{i}$, depending on
$i\in\\{0,1,2\\}$. The receiver chooses the measurement according to $i$, in
the most obvious way $y=i$. The decoding functions are $b_{0}=m\oplus b\oplus
1$, $b_{1}=m\oplus b$, and $b_{2}=m\oplus b\oplus 1$. The message
$m=a_{0}\oplus a\oplus 1$. Below we specify the function for $x$ by a truth
table.

$a_{0}$ | $a_{1}$ | $a_{2}$ | $x$
---|---|---|---
$0$ | $0$ | $0$ | $0$
$0$ | $0$ | $1$ | $2$
$0$ | $1$ | $0$ | $0$
$0$ | $1$ | $1$ | $1$
$1$ | $0$ | $0$ | $1$
$1$ | $0$ | $1$ | $0$
$1$ | $1$ | $0$ | $2$
$1$ | $1$ | $1$ |  $0$.
(16)

The above protocol was obtained using simulated annealing Khachaturyan et al.
(1979).

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