# Crossed Andreev reflection and elastic co-tunneling in a three-site Kitaev
chain nanowire device

Alberto Bordin  Xiang Li  David van Driel  Jan Cornelis Wolff  Qingzhen Wang
Sebastiaan L. D. ten Haaf  Guanzhong Wang  Nick van Loo  Leo P. Kouwenhoven
<EMAIL_ADDRESS>Tom Dvir<EMAIL_ADDRESS>QuTech and Kavli
Institute of NanoScience, Delft University of Technology, 2600 GA Delft, The
Netherlands

###### Abstract

The formation of a topological superconducting phase in a quantum-dot-based
Kitaev chain requires nearest neighbor crossed Andreev reflection and elastic
co-tunneling. Here we report on a hybrid InSb nanowire in a three-site Kitaev
chain geometry — the smallest system with well-defined bulk and edge — where
two superconductor-semiconductor hybrids separate three quantum dots. We
demonstrate pairwise crossed Andreev reflection and elastic co-tunneling
between both pairs of neighboring dots and show sequential tunneling processes
involving all three quantum dots. These results are the next step towards the
realization of topological superconductivity in long Kitaev chain devices with
many coupled quantum dots.

## Introduction

The Kitaev chain was proposed over two decades ago as a platform that supports
unique non-local excitations known as Majorana bound states [1]. Proposals [2,
3, 4] for the realization of such a Kitaev chain rely on creating an array of
spin-polarized quantum dots (QDs) where neighboring QDs are coupled via two
mechanisms: elastic co-tunneling (ECT) and crossed Andreev reflection (CAR).
ECT involves the hopping of a single electron between two QDs. In the process
of CAR, two electrons from neighboring QDs simultaneously enter a
superconductor to form a Cooper pair, or, in reversed order, two electrons
forming a Cooper pair are split into two QDs [5, 6, 7]. Experiments have so
far focused on chains consisting of two QDs, showing both CAR and ECT in such
systems [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and even strong
coupling the QDs to form a minimal Kitaev chain [22]. Longer QD chains,
necessary for the formation of a topological phase, have so far not been
realized and are the long-term goal of this research.

In this work, we report on the fabrication of a three-site Kitaev device and
its characterization at zero magnetic field. We show CAR and ECT between each
pair of neighboring QDs and show that transport across the entire device is
possible through sequential events of CAR and ECT. By measuring the currents
on all of the terminals of our device, we can identify all the possible CAR
and ECT combinations.

## Device structure

Fig. 1: a. False-colored scanning electron micrograph of the measured device.
An InSb nanowire (green) is deposited on top of 11 finger gates (red) and it
is contacted with two superconducting leads S1 and S2 (blue) and two normal
leads $\mathrm{N}_{\textrm{L}}$ and $\mathrm{N}_{\textrm{R}}$ (yellow). Every
contact is connected to a corresponding voltage bias source and current meter.
b. Illustrative energy diagram. Brown symbols represent QD energy levels when
occupied by an electron. c., d. Spectroscopy of the hybrid segments.
$g_{\mathrm{L}}\equiv\frac{dI_{\mathrm{L}}}{dV_{\mathrm{L}}}$ and
$g_{\mathrm{R}}\equiv\frac{dI_{\mathrm{R}}}{dV_{\mathrm{R}}}$ are obtained by
numerical differentiation of the DC currents. Gate settings are reported in
Supplementary Information (Fig. S1, S2 and S3). e., f., g. Coulomb diamond
characterization of $\mathrm{QD}_{1}$ (panel e), $\mathrm{QD}_{2}$ (panel f),
$\mathrm{QD}_{3}$ (panel g). Fitting to a constant interaction model [23]
yields charging energies of 4, 3.5, $3.3\text{\,}\mathrm{m}\mathrm{V}$ and
lever arms of 0.32, 0.33, 0.31 for $\mathrm{QD}_{1}$, $\mathrm{QD}_{2}$, and
$\mathrm{QD}_{3}$ respectively.

In Fig. 1a we show a scanning electron micrograph of device A. This device
consists of an InSb nanowire placed on top of an array of 11 finger gates
separated by a thin dielectric. Two superconducting Al contacts (marked
$\textrm{S}_{\textrm{1}}$ and $\textrm{S}_{\textrm{2}}$) are evaporated on top
of the nanowire using the shadow-wall lithography technique [24, 25]. Both
sides of the device are further contacted by two normal Cr/Au probes (marked
$\mathrm{N}_{\textrm{L}}$ and $\mathrm{N}_{\textrm{R}}$). Every contact is
connected to an independent voltage source ($V_{\mathrm{L}}$,
$V_{\mathrm{S1}}$, $V_{\mathrm{S2}}$, $V_{\mathrm{R}}$) and current meter
($I_{\mathrm{L}}$, $I_{\mathrm{S1}}$, $I_{\mathrm{S2}}$, $I_{\mathrm{R}}$).
The two finger gates underneath the semiconductor-superconductor hybrid
segments control their chemical potential, while the other 9 gates form QDs on
each of the three bare InSb sections. The QD chemical potentials $\mu_{1}$,
$\mu_{2}$ and $\mu_{3}$, are controlled by the gate voltages
$V_{\mathrm{QD1}}$, $V_{\mathrm{QD2}}$ and $V_{\mathrm{QD3}}$ respectively
(Fig. 1b). See Supplementary Information for further nanofabrication details
and gate settings.

## Results

###  .1 Device characterization

Discrete Andreev bound states (ABS) in a hybrid semiconductor-superconductor
segment, separating two QDs, can efficiently mediate CAR and ECT between them
[26, 27, 28]. We find such discrete states by controlling the two finger gates
underneath contacts S1 and S2. Fig. 1c,d show the spectra measured on the
first and the second hybrid respectively, by using the finger gates separating
the superconducting and normal contacts as tunneling barriers [29]. In the
absence of an external magnetic field, both hybrid segments show a hard
superconducting gap, i.e., lack of subgap transport. A closer inspection of
the gate dependence (see Fig. S2) shows that ABS are present at energies close
to the energy gap. At 150 mT magnetic field, the ABS are more visible in the
spectrum. The remainder of the experiment was conducted at zero applied
magnetic field and fixed value of the hybrid gates.

In Fig. 1e-f we characterize QDs 1-3 respectively. The observed Coulomb
diamond structure allows us to estimate the charging energy of all QDs to be
between 3 and $4\text{\,}\mathrm{m}\mathrm{V}$ and the lever arm of the
underlying gates to be $\approx 0.3$ (see also Fig. S3). We also note the
presence of a superconducting gap in the spectrum.

### Pairwise CAR and ECT between neighboring QDs

Fig. 2: a., b. Schematic diagrams of CAR and ECT processes between
$\mathrm{QD}_{1}$ and $\mathrm{QD}_{2}$. CAR is measured by applying
$V_{\mathrm{b}}$ on $\mathrm{N}_{\textrm{L}}$ and $V_{\mathrm{b}}+\Delta/e$ on
$\mathrm{S}_{\textrm{2}}$ (panel a). ECT is measured by applying
$V_{\mathrm{b}}$ on $\mathrm{N}_{\textrm{L}}$ and $-V_{\mathrm{b}}-\Delta/e$
on $\mathrm{S}_{\textrm{2}}$ (panel b). c. CAR and ECT between
$\mathrm{QD}_{1}$ and $\mathrm{QD}_{2}$. The currents $I_{\mathrm{L}}$,
$I_{\mathrm{S1}}$, $I_{\mathrm{S2}}$, and $I_{\mathrm{R}}$ are measured as a
function of $V_{\mathrm{QD1}}$ and $V_{\mathrm{QD2}}$.
$V_{\mathrm{L}}=V_{\mathrm{b}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$,
while
$V_{\mathrm{S2}}=V_{\mathrm{b}}+\Delta/e=$380\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
(top row) or
$V_{\mathrm{S2}}=-$380\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$ (bottom
row). d. CAR and ECT between $\mathrm{QD}_{2}$ and $\mathrm{QD}_{3}$. The
currents through the leads as a function of $V_{\mathrm{QD2}}$ and
$V_{\mathrm{QD3}}$ are measured with
$V_{\mathrm{S1}}=$380\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$ in the
top row and
$V_{\mathrm{S1}}=-$380\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$ in the
bottom row, while
$V_{\mathrm{R}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$.

We begin by demonstrating CAR and ECT processes between pairs of neighboring
QDs. Fig. 2a shows schematically how CAR between $\mathrm{QD}_{1}$ and
$\mathrm{QD}_{2}$ is measured while $\mathrm{QD}_{3}$ is kept off-resonance
and thus does not participate in the transport. CAR involves current flowing
from a superconductor into the neighboring leads (or vice-versa). In recent
works, CAR was measured setting symmetric voltage biases, $V_{\mathrm{b}}$, on
two normal leads on both sides of the hybrid segment [20, 28, 21]. Here, to
account for the presence of the superconducting gap in S2
($\Delta\approx$230\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$),
we apply a bias of $V_{\mathrm{b}}+\Delta/e$ to the superconducting leads. In
this configuration, CAR can be sustained as long as $\mu_{1}=-\mu_{2}$ and the
two chemical potentials are in the bias window
$-\absolutevalue{eV_{\mathrm{b}}}<\mu_{1},\mu_{2}<\absolutevalue{eV_{\mathrm{b}}}$
[20]. ECT can be measured in an anti-symmetric bias configuration. Due to the
presence of the superconducting gap, such configuration similarly requires
adding $\Delta$ to the bias on S2, as shown schematically in Fig. 2b.

Fig. 2c shows the currents through all leads measured in the bias
configuration that allows for CAR (top row) and ECT (bottom row) as a function
of $V_{\mathrm{QD1}}$ and $V_{\mathrm{QD2}}$. In the top row, we find that the
currents $I_{\mathrm{L}}$, $I_{\mathrm{S1}}$, and $I_{\mathrm{S2}}$ are
largest along a diagonal line consistent with $\mu_{1}=-\mu_{2}$. Moreover, we
note that $I_{\mathrm{L}}$ and $I_{\mathrm{S2}}$ are positive and nearly equal
to each other and drain to the ground only through the lead S1. These
observations signal the presence of CAR between $\mathrm{QD}_{1}$ and
$\mathrm{QD}_{2}$. The bottom row is measured in a bias configuration that
supports ECT. The measurements show finite $I_{\mathrm{L}}$ and
$I_{\mathrm{S2}}$ currents with maxima along a diagonal compatible with
$\mu_{1}=\mu_{2}$. In this case, $I_{\mathrm{L}}$ and $I_{\mathrm{S2}}$ have
opposite signs, and almost no current flows through S1, indicating ECT
processes between $\mathrm{QD}_{1}$ and $\mathrm{QD}_{2}$.

Analogously, we measure CAR (and ECT) signatures between $\mathrm{QD}_{2}$ and
$\mathrm{QD}_{3}$ by applying effectively symmetric (and antisymmetric) biases
$V_{\mathrm{S1}}$ and $V_{\mathrm{R}}$. Measured currents on all leads as a
function of $V_{\mathrm{QD2}}$ and $V_{\mathrm{QD3}}$ are shown in Fig. 2d
attesting the presence of CAR and ECT between $\mathrm{QD}_{2}$ and
$\mathrm{QD}_{3}$. We also notice finite currents
$<$10\text{\,}\mathrm{p}\mathrm{A}$$ that depend only on the value of
$V_{\mathrm{QD3}}$. We interpret this as a sign of local Andreev reflection
(LAR) not being completely suppressed by the charging energy of
$\mathrm{QD}_{3}$ (see Fig. S3).

The results shown in Fig. 2 demonstrate both CAR and ECT — the crucial
ingredients of a Kitaev chain — between every pair of QDs.

To separately detect CAR and ECT between pairs of dots, we exploit above the
freedom to apply independent voltage biases to each superconductor. This
freedom might not always be accessible, e.g., in a Kitaev chain design with
the superconductors connected in a loop. In the following, we discuss the
signatures of CAR and ECT when both superconducting leads are grounded.

###  .2 Two-terminal CAR and ECT processes

We set $V_{\mathrm{S1}}=V_{\mathrm{S2}}=0$, and begin by discussing CAR and
ECT processes between $\mathrm{QD}_{2}$ and $\mathrm{QD}_{3}$ while keeping
$\mathrm{QD}_{1}$ off-resonance (see schematics in Fig. 3a,b). We observe
three transport mechanisms involving only leads $\mathrm{S}_{\textrm{2}}$ and
$\mathrm{N}_{\textrm{R}}$.

Fig. 3: a. Schematic illustration of the resonant CAR and ECT tunneling. When
$\mu_{2}=\mu_{3}=0$ both CAR and ECT are allowed between $\mathrm{QD}_{2}$ and
$\mathrm{QD}_{3}$, allowing a complete transport cycle to transfer a Cooper
pair between $\mathrm{N}_{\textrm{R}}$ and $\mathrm{S}_{\textrm{2}}$. b.
Schematic illustration of the Shiba-assisted local pair tunneling (SPT). c.
Current through the device as a function of $V_{\mathrm{QD2}}$ and
$V_{\mathrm{QD3}}$, with
$V_{\mathrm{QD1}}=$414.1\text{\,}\mathrm{m}\mathrm{V}$$, equivalent to
$\mu_{1}\approx$230\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
d. Same data of panel c but with subtracted LAR and saturated colorscale. LAR
is extracted from the average of the top and bottom linecuts at fixed
$V_{\mathrm{QD2}}$, see the code in the linked repository for more details.

The first transport mechanism, already mentioned above, is LAR, giving rise to
a signal that depends only on the chemical potential of $\mathrm{QD}_{3}$.

The second transport mechanism takes place when
$\mu_{2}=\mu_{3}=eV_{\mathrm{S2}}=0$, as depicted in Fig. 3a. In this
alignment of the chemical potentials, both CAR and ECT are allowed. A Cooper
pair can be transmitted between S2 and $\mathrm{N}_{\textrm{R}}$ by sequential
CAR and ECT processes: first, a Cooper pair is split from
$\mathrm{S}_{\textrm{2}}$ to both QDs. Then the electron in $\mathrm{QD}_{3}$
is drained to $\mathrm{N}_{\textrm{R}}$, allowing ECT to shuttle the other
electron between $\mathrm{QD}_{2}$ and $\mathrm{QD}_{3}$, which is finally
drained as well. Because of the resonant condition on the chemical potentials,
this process appears as a single spot in the measurements shown in Fig. 3c
(marked by the black arrow).

When $\mu_{2}\neq 0$, the resonant CAR-ECT process is not allowed anymore;
however, a third transport mechanism might happen. A small but finite LAR in
$\mathrm{QD}_{2}$ leads to the formation of a Yu-Shiba-Rusinov state [30, 31,
32, 33, 34, 35], which has an electron and a hole part. Both parts can assist
either ECT followed by ECT or CAR followed by CAR to transfer a Cooper pair
from $\mathrm{S}_{\textrm{2}}$ to $\mathrm{N}_{\textrm{R}}$. This process was
previously named “Shiba-assisted local pair tunneling” [18, 36] and is
depicted schematically in Fig. 3b. The Shiba-assisted local pair tunneling
occurs when $\mathrm{QD}_{3}$ is resonant with either the electron or the hole
part of the Yu-Shiba-Rusinov state on $\mathrm{QD}_{2}$. This process appears
as two current features observing $|\mu_{2}|=|\mu_{3}|$, one with a positive
slope and the other with a negative slope in the measurement (marked with a
grey arrow). To better visualize it, we highlight it in Fig. 3d with a grey
dashed line after subtracting the LAR background.

### Three-dot sequential CAR and ECT

Fig. 4: a. Schematic illustration of sequential ECT processes (with
$\mu_{1}=\mu_{2}=\mu_{3}$, green) and sequential CAR processes (with
$\mu_{1}=-\mu_{2}=\mu_{3}$, pink). b. Schematic illustration of CAR followed
by ECT (with $-\mu_{1}=\mu_{2}=\mu_{3}$, orange) and ECT followed by CAR (with
$\mu_{1}=\mu_{2}=-\mu_{3}$, blue). c. Current through the device as a function
of $V_{\mathrm{QD2}}$ and $V_{\mathrm{QD3}}$, with
$V_{\mathrm{R}}=-V_{\mathrm{L}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
and $V_{\mathrm{QD1}}=$413.8\text{\,}\mathrm{m}\mathrm{V}$$, equivalent to
$\mu_{1}\approx$130\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
d. Current through the device as a function of $V_{\mathrm{QD2}}$ and
$V_{\mathrm{QD3}}$, with
$V_{\mathrm{R}}=V_{\mathrm{L}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
and $V_{\mathrm{QD1}}=$413\text{\,}\mathrm{m}\mathrm{V}$$, equivalent to
$\mu_{1}\approx$-100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$$.
Note that the arrow colors in panels c and d correspond to the process colors
in panels a and b, while the black arrow corresponds to the resonant CAR and
ECT process shown in Fig.3a,c.

Setting $\absolutevalue{\mu_{1}}<\absolutevalue{eV_{\mathrm{L}}}$ allows
$\mathrm{QD}_{1}$ to participate in transport, opening the route for
sequential CAR and ECT processes involving all three QDs and four leads. Fig.
4a shows schematically such processes with antisymmetric bias settings
($V_{\mathrm{L}}=-V_{\mathrm{R}}$). In this configuration, electrons incoming
from $\mathrm{N}_{\textrm{L}}$ can be transferred all the way to
$\mathrm{N}_{\textrm{R}}$. This can happen in two ways: either with two
sequential ECT processes (marked by green arrows) or with two sequential CAR
ones (marked by pink arrows). Sequential ECT events can first transfer an
electron from $\mathrm{QD}_{1}$ to $\mathrm{QD}_{2}$ and then from
$\mathrm{QD}_{2}$ to $\mathrm{QD}_{3}$, provided that the QD chemical
potentials are all aligned ($\mu_{1}=\mu_{2}=\mu_{3}$). Alternatively, if the
QD chemical potentials are anti-aligned ($\mu_{1}=-\mu_{2}=\mu_{3}$),
sequential CAR can begin by transmitting a Cooper pair into
$\mathrm{S}_{\textrm{1}}$ and then splitting a Cooper pair from
$\mathrm{S}_{\textrm{2}}$, resulting in a net transfer of one electron from
$\mathrm{QD}_{1}$ to $\mathrm{QD}_{3}$. Equivalently, this sequential CAR
process can be seen as an electron from $\mathrm{QD}_{1}$ being converted into
a hole in $\mathrm{QD}_{2}$ and converted back to an electron into
$\mathrm{QD}_{3}$.

Fig. 4c shows the measured currents through our device as a function of
$V_{\mathrm{QD2}}$ and $V_{\mathrm{QD3}}$ for fixed
$V_{\mathrm{R}}=-V_{\mathrm{L}}$. These measurements show both the two-QD
processes discussed above (see black arrows) and the three-QD processes
discussed here. The sequential ECT processes appear as a single spot in
$I_{\mathrm{L}}$ and $I_{\mathrm{R}}$, where $\mu_{2}$ and $\mu_{3}$ are
aligned with $\mu_{1}$ (marked by the green arrow). Since the superconducting
leads drain no current in ECT, $I_{\mathrm{S1}}$ and $I_{\mathrm{S2}}$ do not
show this sequence. The sequence involving two CAR processes (marked by the
pink arrow) appears as a spot in the currents measured on all leads when
$\mu_{3}=\mu_{1}$ and $\mu_{2}=-\mu_{1}$. The currents alter in sign at every
lead, going from negative $I_{\mathrm{L}}$, to positive at $I_{\mathrm{S1}}$,
to negative at $I_{\mathrm{S2}}$, and back to positive at $I_{\mathrm{R}}$,
corresponding to Cooper pair formation in $\mathrm{S}_{\textrm{1}}$ followed
by Cooper pair splitting in $\mathrm{S}_{\textrm{2}}$.

Under symmetric bias conditions, the current is sustained when both leads
$\mathrm{N}_{\textrm{L}}$ and $\mathrm{N}_{\textrm{R}}$ drain electrons (see
Fig. 4b). The two sequences involving all QDs in agreement with this condition
are CAR followed by ECT and the opposite, ECT followed by CAR (Fig. 4b). The
first sequence of CAR followed by ECT can be seen in Fig. 4d in the features
marked by the orange arrow. It is seen as a current feature appearing when
$\mu_{3}\approx\mu_{2}\approx-\mu_{1}$ in $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$. This feature further appears in $I_{\mathrm{S1}}$ but not in
$I_{\mathrm{S2}}$, since CAR between $\mathrm{QD}_{1}$ and $\mathrm{QD}_{2}$
drains current to the ground through S1, whereas ECT between $\mathrm{QD}_{2}$
and $\mathrm{QD}_{3}$ drains no such current to ground via S2. The opposite
sequence, marked by the blue arrow, takes place with
$\mu_{3}\approx-\mu_{2}\approx-\mu_{1}$ and shows similar behavior. We
emphasize that this coupling between all three sites gives rise to a non-local
transport feature. For example, we observe in Fig. 4c,d that $I_{\mathrm{L}}$
is strongly modulated by $\mathrm{QD}_{3}$, two sites away.

Fig. 5: a. Current through the device as a function of $\mu_{2}$ and jointly
$\mu_{1}$ and $\mu_{3}$, which are set to equal values, measured with
antisymmetric bias configuration. b. Current through the device as a function
of $\mu_{2}$ and jointly $\mu_{1}$ and $\mu_{3}$, which are set to opposite
values, measured with symmetric bias configuration. Note that the color of the
arrows corresponds to the color of the processes in Fig. 4 (the grey arrow
highlights the faint SPT process discussed in Fig. 3).

Sequential CAR and ECT processes across the entire device require the
simultaneous tuning of the chemical potential of all three QDs. To better
visualize sequential CAR and ECT processes, we note that both sequential ECT
processes and sequential CAR processes always require $\mu_{1}=\mu_{3}$,
whereas CAR followed by ECT and ECT followed by CAR require
$\mu_{1}=-\mu_{3}$. In Fig. 5, we measure the currents through the devices
while tuning the QDs to follow these constraints. Fig. 5a was measured by
setting
$V_{\mathrm{R}}=-V_{\mathrm{L}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$.
$V_{\mathrm{QD1}}$ and $V_{\mathrm{QD3}}$ were swept together, imposing
$\mu_{1}=\mu_{3}$ for the full measurement (see Fig. S4 in Supplementary
Information for details regarding the tuning of chemical potential). Fig. 5a
features two diagonal lines. The positive-slope diagonal, compatible with
$\mu_{1}=\mu_{2}=\mu_{3}$, is prominent in panels $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ only, allowing us to attribute it to sequential ECT
processes. The negative-slope diagonal, compatible with
$\mu_{1}=-\mu_{2}=\mu_{3}$, appears in all panels and is associated with
sequential CAR processes. The measurements in Fig. 5b were conducted with
$V_{\mathrm{R}}=V_{\mathrm{L}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$.
Here, $V_{\mathrm{QD1}}$ and $V_{\mathrm{QD3}}$ are varied together, while
imposing $\mu_{1}=-\mu_{3}$. Similarly to the previous scenario, measured
currents feature a positive-slope diagonal alongside a negative-slope one.
Here, the positive-slope diagonal involves $I_{\mathrm{S2}}$ as expected for
ECT followed by CAR. The negative-slope diagonal involves $I_{\mathrm{S1}}$
instead, as required by CAR followed by ECT. In summary, the results of Fig. 4
and 5 show how all four possible compositions of CAR and ECT mediate transport
through the entire Kitaev chain device.

## Conclusion

In summary, we have fabricated and measured an InSb-Al device in a three-site
Kitaev geometry that includes QDs separated by semiconductor-superconductor
hybrids. We have shown the signatures of CAR and ECT between the two pairs of
neighboring QDs exhibiting the expected charge and energy conservation for
each respective process. We have further demonstrated control over sequential
CAR and ECT processes involving all QDs by tuning the biases applied to the
normal leads and the chemical potential of the QDs. Our measurements
demonstrate the possibility of extending the known Kitaev chain physics to
longer multi-site chains, leaving the fine-tuning of interdot couplings at
finite magnetic field to future work. Finally, we note that this experiment
demonstrates a general platform enabling long-range entanglement in condensed
matter systems [37, 5]. For instance, we note that the extension to three
sequential CAR events involving four QDs realizes a simple entanglement
swapping scheme [38, 39].

## Acknowledgements

This work has been supported by the Dutch Organization for Scientific Research
(NWO) and Microsoft Corporation Station Q. We wish to acknowledge Srijit
Goswami, Francesco Zatelli and Greg Mazur for useful discussions and Ghada
Badawy, Sasa Gazibegovic and Erik P. A. M. Bakkers for the nanowire growth.

## Author contributions

AB, XL, JCW, and DvD fabricated the devices. AB and XL performed the
electrical measurements with help from QW and SLDtH. AB and TD designed the
experiment and analyzed the data. TD and LPK supervised the project. AB, TD,
and LPK prepared the manuscript with input from all authors.

## Data availability

All raw data in the publication and the analysis code used to generate figures
are available at https://doi.org/10.5281/zenodo.8021184. In the same
repository, we share in addition a complete dataset of three-dimensional
current measurements as a function of $V_{\mathrm{QD1}},V_{\mathrm{QD2}}$ and
$V_{\mathrm{QD3}}$ for all combinations of symmetric and anti-symmetric biases
($V_{\mathrm{b}}=\pm V_{\mathrm{L}}=\pm V_{\mathrm{R}}$). We share a similar
dataset for a second device as well (see also Fig. S5 in Supplementary
Information). We include GIF animations for better visualization.

## References

  * Kitaev [2001] A. Y. Kitaev, Physics-uspekhi 44, 131 (2001).
  * Sau and Sarma [2012] J. D. Sau and S. D. Sarma, Nature Communications 3, 964 (2012), 1111.6600 .
  * Leijnse and Flensberg [2012] M. Leijnse and K. Flensberg, Physical Review B 86, 134528 (2012), 1207.4299 .
  * Fulga  _et al._ [2013] I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg, New journal of physics 15, 045020 (2013).
  * Recher  _et al._ [2001] P. Recher, E. V. Sukhorukov, and D. Loss, Physical Review B 63, 165314 (2001), cond-mat/0009452 .
  * Lesovik  _et al._ [2001] G. B. Lesovik, T. Martin, and G. Blatter, The European Physical Journal B-Condensed Matter and Complex Systems 24, 287 (2001).
  * Sauret  _et al._ [2004] O. Sauret, D. Feinberg, and T. Martin, Physical Review B 70, 245313 (2004).
  * Hofstetter  _et al._ [2009] L. Hofstetter, S. Csonka, J. Nygård, and C. Schönenberger, Nature 461, 960 (2009).
  * Herrmann  _et al._ [2010] L. G. Herrmann, F. Portier, P. Roche, A. L. Yeyati, T. Kontos, and C. Strunk, Phys. Rev. Lett. 104, 026801 (2010).
  * Wei and Chandrasekhar [2010] J. Wei and V. Chandrasekhar, Nature Physics 6, 494 (2010).
  * Hofstetter  _et al._ [2011] L. Hofstetter, S. Csonka, A. Baumgartner, G. Fülöp, S. d’Hollosy, J. Nygård, and C. Schönenberger, Physical review letters 107, 136801 (2011).
  * Das  _et al._ [2012] A. Das, Y. Ronen, M. Heiblum, D. Mahalu, A. V. Kretinin, and H. Shtrikman, Nature communications 3, 1165 (2012).
  * Schindele  _et al._ [2012] J. Schindele, A. Baumgartner, and C. Schönenberger, Physical review letters 109, 157002 (2012).
  * Schindele  _et al._ [2014] J. Schindele, A. Baumgartner, R. Maurand, M. Weiss, and C. Schönenberger, Physical Review B 89, 045422 (2014), 1311.0659 .
  * Tan  _et al._ [2015] Z. B. Tan, D. Cox, T. Nieminen, P. Lähteenmäki, D. Golubev, G. B. Lesovik, and P. J. Hakonen, Physical Review Letters 114, 096602 (2015), 1412.8451 .
  * Gramich  _et al._ [2017] J. Gramich, A. Baumgartner, and C. Schönenberger, Physical Review B 96, 195418 (2017), 1612.01201 .
  * Baba  _et al._ [2018] S. Baba, C. Jünger, S. Matsuo, A. Baumgartner, Y. Sato, H. Kamata, K. Li, S. Jeppesen, L. Samuelson, H. Xu, _et al._ , New Journal of Physics 20, 063021 (2018).
  * Scherübl  _et al._ [2020] Z. Scherübl, G. Fülöp, C. P. Moca, J. Gramich, A. Baumgartner, P. Makk, T. Elalaily, C. Schönenberger, J. Nygård, G. Zaránd, _et al._ , Nature communications 11, 1834 (2020).
  * Ranni  _et al._ [2021] A. Ranni, F. Brange, E. T. Mannila, C. Flindt, and V. F. Maisi, Nature communications 12, 6358 (2021).
  * Wang  _et al._ [2022a] G. Wang, T. Dvir, G. P. Mazur, C.-X. Liu, N. van Loo, S. L. ten Haaf, A. Bordin, S. Gazibegovic, G. Badawy, E. P. Bakkers, _et al._ , Nature 612, 448 (2022a).
  * Wang  _et al._ [2022b] Q. Wang, S. L. ten Haaf, I. Kulesh, D. Xiao, C. Thomas, M. J. Manfra, and S. Goswami, arXiv preprint arXiv:2211.05763 https://doi.org/10.48550/arXiv.2211.05763 (2022b).
  * Dvir  _et al._ [2023] T. Dvir, G. Wang, N. v. Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. t. Haaf, J.-Y. Wang, D. v. Driel, F. Zatelli, X. Li, F. K. Malinowski, S. Gazibegovic, G. Badawy, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Nature 614, 445 (2023).
  * Hanson  _et al._ [2004] R. Hanson, L. M. K. Vandersypen, L. H. W. v. Beveren, J. M. Elzerman, I. T. Vink, and L. P. Kouwenhoven, Physical Review B 70, 241304 (2004), cond-mat/0311414 .
  * Heedt  _et al._ [2021] S. Heedt, M. Quintero-Pérez, F. Borsoi, A. Fursina, N. v. Loo, G. P. Mazur, M. P. Nowak, M. Ammerlaan, K. Li, S. Korneychuk, J. Shen, M. A. Y. v. d. Poll, G. Badawy, S. Gazibegovic, N. d. Jong, P. Aseev, K. v. Hoogdalem, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature Communications 12, 4914 (2021), 2007.14383 .
  * Mazur  _et al._ [2022] G. P. Mazur, N. v. Loo, J. Wang, T. Dvir, G. Wang, A. Khindanov, S. Korneychuk, F. Borsoi, R. C. Dekker, G. Badawy, P. Vinke, S. Gazibegovic, E. P. A. M. Bakkers, M. Q. Pérez, S. Heedt, and L. P. Kouwenhoven, Advanced Materials , 2202034 (2022), 2202.10230 .
  * Fülöp  _et al._ [2015] G. Fülöp, F. Domínguez, S. d’Hollosy, A. Baumgartner, P. Makk, M. Madsen, V. Guzenko, J. Nygård, C. Schönenberger, A. L. Yeyati, _et al._ , Physical review letters 115, 227003 (2015).
  * Liu  _et al._ [2022] C.-X. Liu, G. Wang, T. Dvir, and M. Wimmer, Physical review letters 129, 267701 (2022).
  * Bordin  _et al._ [2022] A. Bordin, G. Wang, C.-X. Liu, S. L. ten Haaf, G. P. Mazur, N. van Loo, D. Xu, D. van Driel, F. Zatelli, S. Gazibegovic, _et al._ , arXiv preprint arXiv:2212.02274 https://doi.org/10.48550/arXiv.2212.02274 (2022).
  * van Driel  _et al._ [2022] D. van Driel, G. Wang, A. Bordin, N. van Loo, F. Zatelli, G. P. Mazur, D. Xu, S. Gazibegovic, G. Badawy, E. P. Bakkers, _et al._ , arXiv preprint arXiv:2212.10241 https://doi.org/10.48550/arXiv.2212.10241 (2022).
  * Luh [1965] Y. Luh, Acta Phys. Sin. 21, 75 (1965).
  * Shiba [1968] H. Shiba, Progress of theoretical Physics 40, 435 (1968).
  * Rusinov [1969] A. Rusinov, JETP Lett.(USSR)(Engl. Transl.);(United States) 9 (1969).
  * Bauer  _et al._ [2007] J. Bauer, A. Oguri, and A. C. Hewson, Journal of Physics: Condensed Matter 19, 486211 (2007), 0707.4368 .
  * Meng  _et al._ [2009] T. Meng, S. Florens, and P. Simon, Physical Review B 79, 224521 (2009), 0902.1111 .
  * Lee  _et al._ [2014] E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. D. Franceschi, Nature Nanotechnology 9, 79 (2014), 1302.2611 .
  * Scherübl  _et al._ [2022] Z. Scherübl, G. Fülöp, J. Gramich, A. Pályi, C. Schönenberger, J. Nygård, and S. Csonka, Physical Review Research 4, 023143 (2022).
  * Choi  _et al._ [2000] M.-S. Choi, C. Bruder, and D. Loss, Physical Review B 62, 13569 (2000), cond-mat/0001011 .
  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Physical review letters 70, 1895 (1993).
  * Zukowski  _et al._ [1993] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Physical review letters 71, https://doi.org/10.1103/PhysRevLett.71.4287 (1993).


11institutetext: Institute for Information Systems, TU Braunschweig,
Braunschweig, Germany
11email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
# On Dimensions of Plausibility for Narrative Information Access to Digital
Libraries

Hermann Kroll 11 0000-0001-9887-9276 Niklas Mainzer 11 Wolf-Tilo Balke 11
0000-0002-5443-1215

###### Abstract

Designing keyword-based access paths is a common practice in digital
libraries. They are easy to use and accepted by users and come with moderate
costs for content providers. However, users usually have to break down the
search into pieces if they search for stories of interest that are more
complex than searching for a few keywords. After searching for every piece one
by one, information must then be reassembled manually. In previous work we
recommended narrative information access, i.e., users can precisely state
their information needs as graph patterns called narratives. Then a system
takes a narrative and searches for evidence for each of its parts. If the
whole query, i.e., every part, can be bound against data, the narrative is
considered plausible and, thus, the query is answered. But is it as easy as
that? In this work we perform case studies to analyze the process of making a
given narrative plausible. Therefore, we summarize conceptual problems and
challenges to face. Moreover, we contribute a set of dimensions that must be
considered when realizing narrative information access in digital libraries.

###### Keywords:

Narrative Information Access Plausibility Dimensions

##  1 Introduction

Digital libraries are large-scale repositories containing a plethora of
heterogeneous but well-curated data: texts, images, videos, research data, and
many more. Today, keyword-based search (on texts or respective metadata) forms
the primary access path for collections: It is easy to use and thus a commonly
accepted interface, and implementing a respective search index comes with
acceptable costs for content providers. In addition, keyword queries can be
evaluated against different data representations; See keyword-based searches
in literature and books, in knowledge bases [10, 15], and data set
repositories [8]. Although the advantage of keyword-based search is apparent,
moving towards alternative access paths is heavily investigated, e.g., the
biomedical database SemMedDB [21] on top of Medline, the Open Knowledge
Research Graph [18], or the Europeana as Linked Open Data [16] and DBpedia
[2]. A good example is the current trend towards structured knowledge graphs:
Here users can formulate their information needs as structured queries (e.g.,
via SPARQL interfaces) asking for a combination of knowledge particles (i.e.,
automatically joining pieces of information).

However, integrating different and heterogeneous sources into a single
knowledge representation comes with new problems, such as the validity of
fused information [22], identifying and canonicalizing pieces of information
(entity linkage), and practical extraction problems when mining knowledge from
different sources; See [33] for a good overview. A remedy might be the use of
narrative information access to bypass the necessary integration of different
sources. In [23], narratives are defined by directed edge-labeled graph
patterns involving entities (things and concepts), events (state or state
changes), literals (values), and their respective relationships (properties,
temporal, and causal predicates, etc.).

Since the individual information particles within a narrative form proper
subgraphs, narratives can be understood as forming a logical overlay on top of
different knowledge repositories. A narrative query is then successfully
answered if evidence for every single relationship of the narrative can be
found in some collection’s data. This evidence forms so-called narrative
bindings connecting each relationship against a concrete piece of information
stated in some knowledge repository, i.e., any form of data storage (e.g.,
relational databases, knowledge graphs, data sets, etc.). In previous works we
formulated a conceptual model [23] for narrative information access, discussed
the binding process from a technical perspective [24], showed how queries can
be processed over collections of scientific publications [25], and discussed
conceptual problems like context compatibility when answering queries over
independent knowledge bases [26].

In contrast to fact-checking, where the task refers to deciding whether some
statement is indeed a fact (and thus true) or is rejected because it has no
evidence [32, 14], narrative structures may be composed out of several
statements. Moreover, we do not require a universally valid decision, i.e.,
based on a given ground truth (a theory in physics for example), we can make a
narrative plausible, whereas we cannot make the narrative plausible on a
different ground truth. While question answering and its methods [19, 20, 35,
35] are related to narrative information access, we see significant
differences in the evaluation: Narrative information access considers contexts
[26, 22], i.e., we do not fuse information from different contexts. For a good
example, consider claims from the biomedical domain: Combining claims made
when treating mice with claims when treating humans can be a serious threat to
the overall validity because no one guarantees that these claims should also
be valid for humans then.

In summary, the key advantage of narrative information access is to make user-
formulated graph patterns plausible in valid contexts. But beyond apparent
technical problems and the problem of context compatibility in narrative
information access, the notion of plausibility has yet to be discussed in
detail. Our current model considers a narrative as plausible if evidence for
every of its claims can be found. But is it as easy as that? In this paper we
summarize the conceptual problems of our narrative query model. Therefore, we
perform case studies in different domains to find the basic building blocks
for an actionable notion of plausibility. Moreover, we open up a design space
and discuss our notion of plausibility in detail by arguing on different
evaluation semantics, the trustworthiness of sources, and the quality of
bindings.

##  2 Case Studying Narrative Plausibility

Figure 1: Making narratives plausible by computing bindings against knowledge
repositories. Here our example Narrative 1 is bound against the digital
library JSTOR.

In brief, a user models a narrative of interest as a graph structure. Then, a
process searches for evidence for each relationship in a set of given
knowledge repositories. If we can find evidence for the whole narrative, i.e.,
every narrative’s relationship, we consider it plausible [23]. The process is
shown in Fig. 1. Note that we may build different plausibility validation
processes, which can vary in their included and excluded knowledge
repositories for the binding computation, e.g., peer-reviewed scientific
papers, fact-checked sources, or arbitrary websites. Due to the lack of
existing implementations for such a process, we performed manual case studies,
i.e., we searched for evidence through keyword queries in knowledge
repositories and evaluated the results manually.

###### Narrative 1

Of the approximately 275 million metric tons of plastic waste produced
annually, up to 12 million tons leak into oceans, wreaking havoc on
livelihoods and ecosystems (CIEL, 2020). The result is an estimated $13
billion in annual environmental damage to marine ecosystems._[1] (See Fig.
1)_.

The central entity of our example is plastic waste, around which the rest of
the narrative revolves. The narrative states two events directly relating to
plastic waste, its annual production, and that plastic waste is leaking into
the ocean. The havoc on livelihoods and ecosystems is a direct consequence of
ocean plastic and represents the third event of our example. Each event that
we described is further connected to a literal: There are approximately 275
million tons of annual plastic waste production, up to twelve million tons
leak into the ocean, and the havoc on livelihoods and ecosystems costs around
13 billion dollars per year. Note that we may extract different graph
representations of the same textual narrative but we discuss this
transformation ambiguity in the next section.

We next present a plausibility validation process based on the digital library
JSTOR (https://www.jstor.org/). One of the problems we faced in the beginning
was that some relationships require more context from the narrative itself,
e.g., Producing annually an amount of 275 million metric tons does not include
that it is about plastic waste. Either the model must include this detail in
the event Producing annually, or we must verify that the context of the
production must match plastic waste. Therefore, we started by evaluating that
around 275 million metric tons of plastic waste are annually produced. We
formulated different keyword queries to search for the given relationship. We
retrieved a source that claimed the 275 million metric ton estimation [17]. We
found two more sources, one of which reported an annual plastic production of
381 million metric tons in 2015, of which 79 percent are considered waste
[34]. We performed the calculation ourselves, and the result was around 300
million tonnes of plastic waste per annum [34] or respectively 321 million
metric tonnes per year [30].

We then evaluated that up to 12 million tons of plastic waste are annually
leaking into the ocean. [30] reported that in 2010, approximately nine million
tons of plastic were added to the ocean. [34] estimated that around 4.8 to
12.7 million tons of plastic entered the ocean in 2010. The plausibility
validation seemed to confirm the narrative claim in this way. We moved to the
next relationship: The leaking of plastic into the ocean leads to havoc on
livelihoods and ecosystems. Even in 1987, [3] already reported an endangerment
of marine environments caused by plastic in the ocean. A more recent article
described dangers to the marine animal and plant life due to microplastic as
well as the economic dangers [9]. For the last relation we searched evidence
that the impact of plastic waste in oceans costs 13 billion dollars annually.
[30] reported 75 billion Dollar annual environmental costs originating from
plastics. However, the source did not state how much of the total costs are
connected to plastic waste in the ocean. Apart from [30], we could not find
any other sources stating the actual costs.

###  2.1 Summary of Other Case Studies

We performed the plausibility validation for three additional narratives: 1\.
Deforestation of the Amazonian Rainforest and forests in general, 2\. Claims
about the connection between government measures during the Covid-19 pandemic
and the number of Covid-related deaths, and 3. Global warming and the effects
of carbon dioxide emissions on the climate. For 3. we only considered fact-
checked articles, whereas we allowed every website on the first result page of
our Google Search for the 1. and 2. narrative.

An interesting finding of the deforestation narrative occurred when we
evaluated the percentage of the Earth’s landmass covered by forests. The
narrative stated a value of 30 percent. One of our retrieved sources confirmed
this narrative claim by stating a value of 30.7 percent [4]. Another source,
however, stated that forests cover 38 percent of the habitable land area [31].
These values greatly differ and could be classified as counter-evidence
because 30 percent could not be verified precisely. However, we noticed that
[4] base their value on the total land area, while [31] makes this calculation
based on the habitable land area, explaining the difference in these values.

We performed a selection of our approaches multiple times over a period of
several weeks to observe how the results would change over time. We retrieved
the global warming narrative from [28]. When we initially accessed this
source, the narrative reported an average surface temperature of 1.18 degrees
Celsius on Earth. However, over time this value has been changed to one degree
Celsius, which is less specific than the initial estimate. While performing
the initial plausibility validation, we noticed various factors determining
Earth’s average surface temperature changes, such as the data collection
methods and the point in time used as a reference. Various sources described
warming of around one degree, but we could not precisely confirm the 1.18
degrees Celsius. The fact that NASA changed that value confirms our assumption
that there is too much uncertainty about the data to state such a specific
temperature change.

##  3 Concluding Discussion on Narrative Plausibility

In the following we summarize the main findings from our previous case
studies.

Confirmation Bias. Confirmation bias describes ”the seeking or interpreting of
evidence in ways that are partial to existing beliefs, expectations, or a
hypothesis at hand” [29]. As this definition suggests, bias can already occur
in the search for information, and its effects highly depend on how the
plausibility validation process searches for evidence. For instance,
evaluating plastic waste causes 13 billion dollars annually in environmental
costs by leaking into the ocean could lead to other results than searching
without a concrete cost value.

Narrative’s Context. The narrative itself spans a context that must be
considered when binding its relationships individually. For instance,
searching for an amount of leakage without plastic waste may result invalid
results. Similarly, evaluating the average surface temperature without a
concrete year may also yield results that do not match the user’s indented
context.

Finding the Narrative’s Source. Imagine a narrative extracted from a study
reports scientifically disputed findings. We would assume such a narrative to
be evaluated as implausible in a plausibility validation process. However, the
process will likely find the study’s sources again – even if they are
disputed.

Ambiguity of Narrative Extraction. Narratives can be interpreted differently
since they make various claims, usually without explaining exactly how
everything is meant. Without an objective understanding of what the narrative
is exactly stating, this will affect the narrative model extracted from the
actual textual narrative. Consequently, the plausibility validation process
computes narrative bindings for a selected instance of the possible
interpretations, making the conclusion about narrative plausibility dependent
on the chosen narrative model and, therefore, on the chosen interpretation of
the narrative. A plausibility validation process based on another
interpretation of the same narrative might result in a different conclusion
about the narrative’s plausibility.

Evidence. We observed three issues when searching for evidence: 1\. Indirect
Evidence: The process may not find direct but rather indirect evidence (e.g.,
a certain percentage of the annual plastic production), which requires further
computations. 2\. Counter-Evidence: We may find counter-evidence, i.e.,
evidence that contradicts a narrative’s relationship. 3\. Absence of Evidence:
Not finding evidence is always connected to uncertainty since we cannot be
sure if evidence does not exist or if we simply did not find it. All issues
are especially relevant if we utilize error-prone NLP and retrieval methods
[12, 6].

Trustworthiness of Sources. When computing bindings against different types of
sources, the credibility of sources must be considered. On the one hand, we
may utilize Provenance information that has already been curated [27, 13, 7].
Provenance allows knowledge curators to precisely describe how a certain piece
of information was created or obtained. If we bind against that certain piece,
we can then utilize its Provenance. On the other hand, we may integrate tests
like the CRAAP test [5] or automated heuristics, e.g., scientific publications
before websites, highly influential over low cited publications, etc. Note
that we are aware of highly influential papers that have been retracted but
handling such cases in digital libraries is a research field on its own [11].
But if we do not have curated information available, we must think about
alternatives here.

###  3.1 Dimensions of Narrative Plausibility

The contribution of this work is the introduction of dimensions influencing
narrative plausibility. When implementing a validation process in a digital
library, the designers must decide how the following criteria should be
handled.

#### Narrative Structure.

We have seen that transforming a story of interest (a user’s query) into a
graph representation can be challenging and may yield several different
representations. Moreover, each representation might end up in a different
result (plausible/not plausible) which could confuse users. Do we consider the
exact narrative, or do we allow a few derivations (e.g., relaxing the query a
bit)? What can users then expect if a narrative is considered plausible?

#### Validation Approach.

Which knowledge repositories are considered for the binding computation? Which
methods are used to compute the actual bindings? What do these methods
guarantee? And thus, what could a user then expect?

#### Types of Evidence.

Beyond suitable evidence we may face the indirect, missing, and counter-
evidence in practice. What should we do if we cannot find evidence for all of
a narrative’s relationships? Is the narrative simply considered as not
plausible? Or should we instead show the user how much of the narrative is
plausible and which parts lack evidence? Do we consider counter-evidence? And
if we do, what does the existence of counter-evidence then mean, e.g., that
the narrative is not plausible? As we have argued before, there is a fine line
between evidence not exactly confirming a relationship and counter-evidence.

#### Confidence of Bindings.

How trustworthy do the sources have to be? How should this trustworthiness be
determined or measured? How is the quality of the actual bindings from a
technical perspective, e.g., the confidence of a retrieval process or
similarity measures for values? Should we use thresholds to consider
trustworthy sources and high-quality bindings only? Which evaluation strategy
do we select, e.g., the trustworthiness of sources over binding quality? Do we
average both values into an overall confidence?

## References

  * [1] Plastics and the Environment. https://www.genevaenvironmentnetwork.org/resources/updates/plastics-and-the-environment/ (last access: 2022-05)
  * [2] Auer, S., Bizer, C., Kobilarov, G., Lehmann, J., Cyganiak, R., Ives, Z.G.: Dbpedia: A nucleus for a web of open data. In: Aberer, K., Choi, K., Noy, N.F., Allemang, D., Lee, K., Nixon, L.J.B., Golbeck, J., Mika, P., Maynard, D., Mizoguchi, R., Schreiber, G., Cudré-Mauroux, P. (eds.) The Semantic Web, 6th International Semantic Web Conference, 2nd Asian Semantic Web Conference, ISWC 2007 + ASWC 2007, Busan, Korea, November 11-15, 2007\. Lecture Notes in Computer Science, vol. 4825, pp. 722–735. Springer (2007). https://doi.org/10.1007/978-3-540-76298-0_52
  * [3] Azzarello, M.Y., Vleet, E.S.V.: Marine birds and plastic pollution. Marine Ecology Progress Series 37(2/3), 295–303 (1987), http://www.jstor.org/stable/24824704
  * [4] Bank, W.: Forest area (% of land area). https://data.worldbank.org/indicator/AG.LND.FRST.ZS, online; accessed 25 May 2022
  * [5] Blakeskee, S.: The craap test. LOEX Quarterly 31 (2004), https://commons.emich.edu/loexquarterly/vol31/iss3/4
  * [6] Blatz, J., Fitzgerald, E., Foster, G., Gandrabur, S., Goutte, C., Kulesza, A., Sanchis, A., Ueffing, N.: Confidence estimation for machine translation. In: COLING 2004: Proceedings of the 20th International Conference on Computational Linguistics. pp. 315–321. COLING, Geneva, Switzerland (aug 23–aug 27 2004), https://aclanthology.org/C04-1046
  * [7] Carroll, J.J., Bizer, C., Hayes, P.J., Stickler, P.: Named graphs. J. Web Semant. 3(4), 247–267 (2005). https://doi.org/10.1016/j.websem.2005.09.001
  * [8] Chapman, A., Simperl, E., Koesten, L., Konstantinidis, G., Ibáñez, L., Kacprzak, E., Groth, P.: Dataset search: a survey. VLDB J. 29(1), 251–272 (2020). https://doi.org/10.1007/s00778-019-00564-x
  * [9] Clark, J.R., Cole, M., Lindeque, P., Fileman, E., Blackford, J., Lewis, C., Lenton, T.M., Galloway, T.S.: Marine microplastic debris: a targeted plan for understanding and quantifying interactions with marine life. Frontiers in Ecology and the Environment 14(6), 317–324 (2016), http://www.jstor.org/stable/44001167
  * [10] Elbassuoni, S., Blanco, R.: Keyword search over rdf graphs. In: Proceedings of the 20th ACM International Conference on Information and Knowledge Management. p. 237–242. CIKM ’11, Association for Computing Machinery, New York, NY, USA (2011). https://doi.org/10.1145/2063576.2063615
  * [11] Fu, Y., Schneider, J.: Towards knowledge maintenance in scientific digital libraries with the keystone framework. In: Huang, R., Wu, D., Marchionini, G., He, D., Cunningham, S.J., Hansen, P. (eds.) JCDL ’20: Proceedings of the ACM/IEEE Joint Conference on Digital Libraries in 2020, Virtual Event, China, August 1-5, 2020. pp. 217–226. ACM (2020). https://doi.org/10.1145/3383583.3398514
  * [12] Gandrabur, S., Foster, G.F., Lapalme, G.: Confidence estimation for NLP applications. ACM Trans. Speech Lang. Process. 3(3), 1–29 (2006). https://doi.org/10.1145/1177055.1177057
  * [13] Group, W.W.: Prov-overview. an overview of the prov family of documents. https://www.w3.org/TR/prov-overview/ (2013)
  * [14] Guo, Z., Schlichtkrull, M., Vlachos, A.: A Survey on Automated Fact-Checking. Transactions of the Assoc. for Comp. Linguistics (TACL) 10 (2022)
  * [15] Han, S., Zou, L., Yu, J.X., Zhao, D.: Keyword search on rdf graphs - a query graph assembly approach. In: Proceedings of the 2017 ACM on Conference on Information and Knowledge Management. p. 227–236. CIKM ’17, Association for Computing Machinery, New York, NY, USA (2017). https://doi.org/10.1145/3132847.3132957
  * [16] Haslhofer, B., Isaac, A.: data.europeana.eu: The europeana linked open data pilot. In: Baker, T., Hillmann, D.I., Isaac, A. (eds.) Proceedings of the 2011 International Conference on Dublin Core and Metadata Applications, DC 2011, The Hague, The Netherlands, September 21-23, 2011. pp. 94–104. Dublin Core Metadata Initiative (2011), http://dcpapers.dublincore.org/pubs/article/view/3625
  * [17] Jambeck, J.R., Geyer, R., Wilcox, C., Siegler, T.R., Perryman, M., Andrady, A., Narayan, R., Law, K.L.: Plastic waste inputs from land into the ocean. Science 347(6223), 768–771 (2015). https://doi.org/10.1126/science.1260352
  * [18] Jaradeh, M.Y., Oelen, A., Farfar, K.E., Prinz, M., D’Souza, J., Kismihók, G., Stocker, M., Auer, S.: Open research knowledge graph: Next generation infrastructure for semantic scholarly knowledge. In: Kejriwal, M., Szekely, P.A., Troncy, R. (eds.) Proceedings of the 10th International Conference on Knowledge Capture, K-CAP 2019, Marina Del Rey, CA, USA, November 19-21, 2019\. pp. 243–246. ACM (2019). https://doi.org/10.1145/3360901.3364435
  * [19] Jin, Q., Yuan, Z., Xiong, G., Yu, Q., Ying, H., Tan, C., Chen, M., Huang, S., Liu, X., Yu, S.: Biomedical question answering: A survey of approaches and challenges. ACM Comput. Surv. 55(2) (jan 2022). https://doi.org/10.1145/3490238
  * [20] Khot, T., Sabharwal, A., Clark, P.: Answering complex questions using open information extraction. In: Barzilay, R., Kan, M. (eds.) Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, ACL 2017, Vancouver, Canada, July 30 - August 4, Volume 2: Short Papers. pp. 311–316. Association for Computational Linguistics (2017). https://doi.org/10.18653/v1/P17-2049
  * [21] Kilicoglu, H., Shin, D., Fiszman, M., Rosemblat, G., Rindflesch, T.C.: Semmeddb: a pubmed-scale repository of biomedical semantic predications. Bioinform. 28(23), 3158–3160 (2012). https://doi.org/10.1093/bioinformatics/bts591
  * [22] Kroll, H., Kalo, J., Nagel, D., Mennicke, S., Balke, W.: Context-compatible information fusion for scientific knowledge graphs. In: Hall, M.M., Mercun, T., Risse, T., Duchateau, F. (eds.) Digital Libraries for Open Knowledge - 24th International Conference on Theory and Practice of Digital Libraries, TPDL 2020, Lyon, France, August 25-27, 2020, Proceedings. Lecture Notes in Computer Science, vol. 12246, pp. 33–47. Springer (2020). https://doi.org/10.1007/978-3-030-54956-5_3
  * [23] Kroll, H., Nagel, D., Balke, W.: Modeling narrative structures in logical overlays on top of knowledge repositories. In: Dobbie, G., Frank, U., Kappel, G., Liddle, S.W., Mayr, H.C. (eds.) Conceptual Modeling - 39th International Conference, ER 2020, Vienna, Austria, November 3-6, 2020, Proceedings. Lecture Notes in Computer Science, vol. 12400, pp. 250–260. Springer (2020). https://doi.org/10.1007/978-3-030-62522-1_18
  * [24] Kroll, H., Nagel, D., Kunz, M., Balke, W.: Demonstrating narrative bindings: Linking discourses to knowledge repositories. In: Campos, R., Jorge, A.M., Jatowt, A., Bhatia, S., Finlayson, M.A. (eds.) Proceedings of Text2Story - Fourth Workshop on Narrative Extraction From Texts held in conjunction with the 43rd European Conference on Information Retrieval (ECIR 2021), Lucca, Italy, April 1, 2021 (online event due to Covid-19 outbreak). CEUR Workshop Proceedings, vol. 2860, pp. 57–63. CEUR-WS.org (2021), http://ceur-ws.org/Vol-2860/paper7.pdf
  * [25] Kroll, H., Pirklbauer, J., Kalo, J., Kunz, M., Ruthmann, J., Balke, W.: Narrative query graphs for entity-interaction-aware document retrieval. In: Ke, H., Lee, C.S., Sugiyama, K. (eds.) Towards Open and Trustworthy Digital Societies - 23rd International Conference on Asia-Pacific Digital Libraries, ICADL 2021, Virtual Event, December 1-3, 2021, Proceedings. Lecture Notes in Computer Science, vol. 13133, pp. 80–95. Springer (2021). https://doi.org/10.1007/978-3-030-91669-5_7
  * [26] Kroll, H., Plötzky, F., Pirklbauer, J., Balke, W.: What a publication tells you - benefits of narrative information access in digital libraries. CoRR (Accepted to JCDL2022) abs/2205.00718 (2022). https://doi.org/10.48550/arXiv.2205.00718
  * [27] Lebo, T., Sahoo, S., McGuinness, D.: PROV-O: The PROV Ontology. https://www.w3.org/TR/prov-o/ (2013)
  * [28] NASA: Global Temperature — Vital Signs - NASA Climate Change. https://climate.nasa.gov/evidence/, online; accessed 25 May 2022
  * [29] Nickerson, R.S.: Confirmation bias: A ubiquitous phenomenon in many guises. Review of General Psychology 2(2), 175–220 (1998). https://doi.org/10.1037/1089-2680.2.2.175
  * [30] Rhodes, C.J.: Plastic pollution and potential solutions. Science Progress 101, 207–260 (2018)
  * [31] Ritchie, H., Roser, M.: Forests and deforestation. Our World in Data (2021), https://ourworldindata.org/forests-and-deforestation
  * [32] Uscinski, J.E., Butler, R.W.: The Epistemology of Fact Checking. Critical Review 25(2) (2013)
  * [33] Weikum, G., Dong, X.L., Razniewski, S., Suchanek, F.M.: Machine knowledge: Creation and curation of comprehensive knowledge bases. Found. Trends Databases 10(2-4), 108–490 (2021). https://doi.org/10.1561/1900000064
  * [34] Williams, A., Rangel-Buitrago, N.: Marine Litter: Solutions for a Major Environmental Problem. Journal of Coastal Research 35(3), 648–663 (01 2019). https://doi.org/10.2112/JCOASTRES-D-18-00096.1
  * [35] Zhao, C., Xiong, C., Qian, X., Boyd-Graber, J.L.: Complex factoid question answering with a free-text knowledge graph. In: Huang, Y., King, I., Liu, T., van Steen, M. (eds.) WWW ’20: The Web Conference 2020, Taipei, Taiwan, April 20-24, 2020. pp. 1205–1216. ACM / IW3C2 (2020). https://doi.org/10.1145/3366423.3380197


For each pixel in a tile, we calculate the height $z$ with respect to $b$ $=$
0∘ by computing

$z=d\arctan(b).$ (12)

This approach assumes a single representative distance for each tile, that is,
it does not account for the variation between $b$ $=$ 0∘ and $b$ $=$ $\pm
10$∘. The maximum differences in $R_{\rm gal}$ and $d$ between the tile top
and center are around 0.074 kpc and 0.076 kpc, respectively. These are found
in tiles at large $R_{\rm gal}$ and $d$, where the variations in the two
quantities due to the tile height are negligible compared to the uncertainties
introduced by the resolution in radial velocity.

Our de-projection approach is different from the PPV grid interpolation and
mapping into a fixed PPP grid followed in, for example, Levine et al. (2006b),
Kalberla & Dedes (2008), and Koo et al. (2017). However, it does not yield
significant differences in the results obtained for the midplane displacement
(warp) and dispersion around the midplane (flaring), as shown in Sec. 4.2, and
it is computationally more efficient. Our analysis does not require the
continuous sampling of the PPP grid necessary for the Fourier mode analysis in
Levine et al. (2006b) or the three-dimensional mass reconstruction presented
in (Nakanishi & Sofue 2016), but rather aims to illustrate the variation in
filament orientation with increasing $R_{\rm gal}$. Given the one-to-one
mapping from $v_{\rm LSR}$ into $R_{\rm gal}$ in the outer galaxy under the
assumption of circular motions, our approach is close to a discrete sampling
of the PPP space.

###  B.3 Warp and flaring

For each one of the tiles de-projected into the Galactocentric cylindrical
coordinates, we calculated the mid-plane displacement with respect to the $b$
$=$0∘ position, or warp, by calculating the first-moment of the height $z$
weight by the intensity,

$\left<z\right>=\frac{\sum I_{ij}z_{ij}}{\sum I_{ij}},$ (13)

where the indices $i$ and $j$ run over the tile pixels.

We also calculated the dispersion around the mid-plane height, or flaring, by
computing the second-moment of the height $z$ weight by the intensity,

$\sigma_{z}=\left[\frac{\sum I_{ij}(z_{ij}-\left<z\right>)^{2}}{\sum
I_{ij}}\right]^{1/2}$ (14)

where $\left<z\right>$ is calculated using Eq. 13.

##  Appendix C Comparison between HI4PI and GALFA-HI

The only survey with enough coverage of the Galactic plane to expand the
results of the HI4PI analysis to higher resolutions is GALFA-HI. With 4′ FWHM
resolution, GALFA-HI provides a factor of four improvement with respect to
HI4PI in the ranges 36 .∘5 $<$ $l$ $<$ 70 .∘5 and 175 .∘3 $<$ $l$ $<$ 209 .∘4
We applied the Hessian matrix analysis to the GALFA-HI data using the
parameters presented in Table. 2. The analysis was applied to the data in its
native velocity resolution, 0.184 km s-1, and to the data smoothed in the
spectral axis to match the HI4PI velocity resolution. As with the HI4PI data,
the curvature threshold ($\lambda^{C}_{-}$) was estimated empirically using
noise-dominated velocity channels.

We present an example of the Hessian analysis applied to one 3∘ $\times$ 20∘
$\times$ 1.29 km s-1 tile in the HI4PI and GALFA-HI observations in the top
panels of Fig. 35. This example tile illustrates the connection in the
emission structures traced by the two surveys at different angular resolutions
and projected into the same radial velocity resolution.

We also present an example of the Hessian analysis applied to the GALFA-HI
observations in its native 0.184 km s-1 velocity resolution in the bottom
panels of Fig. 35. These tiles, which correspond to the same line of sight of
those in the top panels, illustrate the similarity of the intensity structures
in contiguous narrow velocity channels. This similarity, caused by the
linewidth of the Hi emission, is evident for the whole velocity range in Fig.
15.

Figure 35: Hessian matrix analysis applied to the HI4PI and the GALFA-HI
observations. Top. Results obtained on a 3∘ $\times$ 20∘ $\times$ 1.29 km s-1
tile in the HI4PI and the GALFA-HI observations, shown in the left and right
panels, respectively. Bottom. Results obtained on two 3∘ $\times$ 20∘ $\times$
0.184 km s-1 tiles in the GALFA-HI observations. In each of the panels, the
left-hand side map shows the Hi emission convolved with a 2D Gaussian with the
same diameter of the derivative kernel. The map in the center shows the second
eigenvalues of the Hessian matrix, $\lambda_{-}$, which highlights the
filamentary structure. The right-hand side map shows the orientation angle of
the filamentary structure.

We also tested the results of a larger derivative kernel and a corresponding
$\lambda^{C}_{-}$, as detailed in Table 3. We present the equivalent of Fig.
14 for the 8′ FWHM kernel in Fig. 36. As expected from the lower number of
gradients derived with a large kernel, the values of $V$ are reduced. The
equivalent of Fig. 15 for the 8′ FWHM kernel, shown in Fig. 37, indicates a
significant increase in the values of $V$ with respect to those obtain for the
HI4PI data.

Table 3: Hessian analysis parameters for the GALFA-HI data. Parameter |  | Value
---|---|---
Tile size |  | 3∘ $\times$ 20∘ $\times$ 0.184 km s-1
|  | 3∘ $\times$ 20∘ $\times$ 1.290 km s-1
Kernel size |  | 8′ FWHM
Intensity threshold |  | 0.3 K
Curvature threshold ($\lambda^{C}_{-}$) |  |  $-20.0$ K/deg2

Figure 36: Same as Fig. 14, but for an 8′ FWHM kernel.

Figure 37: Same as Fig. 15, but for an 8′ FWHM kernel.

##  Appendix D Filamentary structures in high velocity clouds

We present the result of the Hessian matrix analysis toward two high velocity
cloud (HVC) complexes in Fig. 38 and Fig. 39.

Figure 38: Atomic hydrogen (Hi) emission (top) and filamentary structures
identified using the Hessian matrix method (bottom) toward the HVC complex H.
Figure 39: Same as Fig. 38, but for the anticenter HVC complex.