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# High-Dimensional Pixel Entanglement: Efficient Generation and Certification

Natalia Herrera Valencia  [ Institute of Photonics and Quantum Sciences,
Heriot-Watt University, Edinburgh, UK  Vatshal Srivastav  Institute of
Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, UK  Matej
Pivoluska  Institute of Physics, Slovak Academy of Sciences, Bratislava,
Slovakia  Institute of Computer Science, Masaryk University, Brno, Czech
Republic  Marcus Huber  Institute for Quantum Optics and Quantum Information -
IQOQI Vienna, Austrian Academy of Sciences, Vienna, Austria  Vienna Center for
Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
Nicolai Friis  Institute for Quantum Optics and Quantum Information - IQOQI
Vienna, Austrian Academy of Sciences, Vienna, Austria  Will McCutcheon
Institute of Photonics and Quantum Sciences, Heriot-Watt University,
Edinburgh, UK  Mehul Malik  [ Institute of Photonics and Quantum Sciences,
Heriot-Watt University, Edinburgh, UK  Institute for Quantum Optics and
Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Vienna,
Austria

###### Abstract

Photons offer the potential to carry large amounts of information in their
spectral, spatial, and polarisation degrees of freedom. While state-of-the-art
classical communication systems routinely aim to maximize this information-
carrying capacity via wavelength and spatial-mode division multiplexing,
quantum systems based on multi-mode entanglement usually suffer from low state
quality, long measurement times, and limited encoding capacity. At the same
time, entanglement certification methods often rely on assumptions that
compromise security. Here we show the certification of photonic high-
dimensional entanglement in the transverse position-momentum degree-of-freedom
with a record quality, measurement speed, and entanglement dimensionality,
without making any assumptions about the state or channels. Using a tailored
macro-pixel basis, precise spatial-mode measurements, and a modified
entanglement witness, we demonstrate state fidelities of up to 94.4% in a
19-dimensional state-space, entanglement in up to 55 local dimensions, and an
entanglement-of-formation of up to 4 ebits. Furthermore, our measurement times
show an improvement of more than two orders of magnitude over previous state-
of-the-art demonstrations. Our results pave the way for noise-robust quantum
networks that saturate the information-carrying capacity of single photons.

##  1 Introduction

Quantum entanglement plays a pivotal role in the development of quantum
technologies, resulting in revolutionary concepts in quantum communication
such as superdense coding [1] and device-independent security [2, 3], as well
as enabling fundamental tests of the very nature of reality [4, 5, 6]. While
many initial demonstrations have relied on entanglement between qubits, recent
advances in technology and theory now allow us to fully exploit high-
dimensional quantum systems. In particular, the large dimensionality offered
by photonic quantum systems has provided the means for quantum communication
with record capacities [7, 8], noise-resistant entanglement distribution [9,
10], robust loophole-free tests of local realism [11, 12], and scalable
methods for quantum computation [13].

In order to make full use of the potential of high-dimensional entanglement,
it is of key importance to achieve the certification of entanglement with as
few measurements as possible. The characterisation of a bipartite state with
local dimension $d$ through full state tomography requires
$\mathcal{O}(d^{4})$ single-outcome projective measurements [14, 15], making
this task extremely impractical in high dimensions. More efficient tools for
quantifying high-dimensional entanglement involve entanglement witnesses that
use semi-definite programming [16], matrix completion techniques [17], or
compressed sensing [18]. In this context, it is crucial to certify
entanglement without compromising the security and validity of the
applications by introducing assumptions on the state, e.g., purity of the
generated state [19] or conservation of quantities [20]. Recent work has shown
how measurements in mutually unbiased bases (MUBs) allow the efficient
certification of high-dimensional entanglement [21, 22], without any such
assumptions on the underlying quantum state. Here we employ and extend this
method to improve upon the quality and speed of high-dimensional entanglement
certification.

High-dimensional entanglement has been demonstrated in multiple photonic
platforms, with encodings in the orbital angular momentum (OAM) [23, 22],
time-frequency [16], path [24], and transverse position-momentum degrees of
freedom (DoF) [18]. While time-frequency encoding offers the potential of
accessing spaces with very large dimensions, the difficulty of measuring
coherent superpositions of multiple time-bins hinders the scalability of the
technique, and in turn necessitates certification methods that require
unwanted assumptions on the reconstruction of the state in question to reach
their full potential [16]. Path-encoding in integrated photonic circuits
offers yet another promising avenue for realising high-dimensional
entanglement [24]. However, the precise fabrication and control of
$\frac{d(d-1)}{2}$ Mach-Zehnder interferometers required for universal
operations in $d$ dimensions poses significant practical challenges as the
dimension is increased [25].

Meanwhile, techniques for the creation, manipulation and detection of
entanglement in photonic OAM bases have seen rapid progress in recent years
[26, 27], where devices such as spatial light modulators (SLMs) enable
generalized measurements of complex amplitude modes. However, such
measurements necessarily suffer from loss [28] and have limited quality [29,
30], resulting in long measurement times and reduced entanglement quality in
large dimensions [22]. An alternative choice of basis is the discretised
transverse position-momentum, or “pixel” basis [31], where the lack of
efficient single-photon detector arrays necessitates scanning through
localised position or momentum modes, or subtracting a large noise background
[32]. Such measurements are also subject to extreme loss, and as a result
require long measurement times and strong assumptions on detector noise, such
as background or accidental count subtraction.

In this work, we report on significant progress towards overcoming the
challenges of scalability, speed, and quality in the characterisation of high-
dimensional entanglement with a strategy that combines three distinct
improvements in the generation and measurement of spatially entangled modes.
Working in the discretized transverse position-momentum DoF, we first tailor
our spatial-mode basis by adapting it to the characteristics of the two-photon
state generated and measured in our experimental setup. Second, we implement a
recently developed spatial-mode measurement technique [29] that ensures
precise projective measurements in any mode basis of our choice. Third, we
generalise a recently developed entanglement dimensionality witness [22] to
certify high-dimensional entanglement using any two high-dimensional MUBs.
Crucially, this allows us to bypass lossy localised mode measurements in the
transverse position or momentum bases.

The combination of these improvements in basis optimisation, spatial-mode
measurement, and certification tools allow us to certify high-dimensional
entanglement with a record quality, speed, and dimension, reaching state
fidelities up to 98%, certified entanglement dimensionality up to 55 (in local
dimension 97), and an entanglement-of-formation of up to 4 ebits. Below we
introduce our theoretical framework and elaborate on our improved techniques.

##  2 Theory

A two-photon state entangled in transverse position-momentum and produced via
the process of spontaneous parametric down-conversion (SPDC) is characterised
by its _joint-transverse-momentum-amplitude_ or JTMA (Fig. 1 a), which is well
approximated by the function [33, 34]

$\displaystyle
F(\vec{k}_{s},\vec{k}_{i})=\exp\Bigl{\\{}-\mbox{$\textstyle\frac{1}{2}$}\frac{|\vec{k}_{s}+\vec{k}_{i}|^{2}}{\sigma_{P}^{2}}\Bigr{\\}}\operatorname{sinc}\Bigl{\\{}\frac{1}{\sigma_{S}^{2}}|\vec{k}_{s}-\vec{k}_{i}|^{2}\Bigr{\\}},$
(1)

where $\vec{k}_{s}$ and $\vec{k}_{i}$ are the transverse momentum components
of the signal and idler photons. The parameters $\sigma_{P}$ and $\sigma_{S}$
are the widths of the minor and major axis of the JTMA, which are dependent on
the pump transverse momentum bandwidth and the crystal characteristics,
respectively. In effect, $\sigma_{P}$ indicates the degree of momentum
correlation, while $\sigma_{S}$ is a measure of the number of correlated modes
in the state.

Experimental studies of spatial-mode entanglement usually involve projective
measurements made with a combination of a hologram and single-mode fiber
(SMF), which together act as spatial-mode filter. The addition of collection
optics for coupling to an SMF results in the collected bi-photon JTMA of

$G(\vec{k}_{s},\vec{k}_{i})=\exp\Bigl{\\{}-\mbox{$\textstyle\frac{1}{2}$}\frac{|\vec{k}_{s}|^{2}}{\sigma_{C}^{2}}\Bigr{\\}}\exp\Bigl{\\{}-\mbox{$\textstyle\frac{1}{2}$}\frac{|\vec{k}_{i}|^{2}}{\sigma_{C}^{2}}\Bigr{\\}}F(\vec{k}_{s},\vec{k}_{i}),$
(2)

where $\sigma_{C}$ denotes the collection bandwidth, that is, the size of the
back-projected detection mode. The effect of the collection on the
correlations is illustrated in Fig. 1, where the Gaussian functions attributed
to the SMF modes decrease the probability of detecting higher-order modes
associated with the edges of the JTMA. Finally, we include the hologram
functions $\Phi_{s}(\vec{k}_{s})$ and $\Phi_{i}(\vec{k}_{i})$ used for
performing projective measurements on the signal and idler photons,
respectively, which leads to the two-photon coincidence probability

$\Pr(\Phi_{s},\Phi_{i})=\biggl{|}\int d^{2}\vec{k}_{s}\int
d^{2}\vec{k}_{i}\Phi_{s}(\vec{k}_{s})\Phi_{i}(\vec{k}_{i})G(\vec{k}_{s},\vec{k}_{i})\biggr{|}^{2}.$
(3)

Maximizing the quality and dimensionality of an experimentally generated
entangled state involves a complex interplay of optimizing the generation and
measurement parameters introduced above. Recent work in this direction has
focused on pump shaping [35, 36] and entanglement witnesses that adapt to the
non-maximally entangled nature of the state [22]. Below, we introduce a new
approach to engineering high-quality spatial-mode entanglement based on three
distinct improvements: tailoring the spatial-mode measurement basis, precise
two-photon measurements via intensity-flattening, and a modified witness for
high-dimensional entanglement.

Figure 1: Joint-transverse-momentum-amplitude (JTMA) and pixel basis
tailoring. a) A contour plot (green/yellow) depicting the absolute value of a
2-d slice $F\left((0,k_{sy}),(0,k_{iy})\right)$ of the JTMA corresponding to
$k_{sx}=k_{ix}=0$. The Gaussian distributions (purple) on the $k_{sy}$ and
$k_{iy}$ axes are the collection mode envelopes
$\exp\\{-\mbox{$\textstyle\frac{1}{2}$}\left(\frac{k_{sy}}{\sigma_{C}}\right)^{2}\\}$.
The square regions indicate values of $k_{sy}$ and $k_{iy}$ used for
generating an optimised pixel basis mask. As such, integrals over the square
regions are closely related to the coincidence rate for projections on to the
given pair of pixels (up to the additional 2-d integral over $k_{sx}$ and
$k_{ix}$), demonstrating the necessity to increase the pixel radii for those
positioned over the less intense regions of the JTMA. b) An example
31-dimensional tailored pixel mask. The color function displays the marginal
JTMA intensity of the signal photon corresponding to $\int
d^{2}\vec{k}_{i}F(\vec{k}_{s},\vec{k}_{i})$. The line at $x_{s}=0$ intersects
5 pixels having 3 unique radii, and its intersections with their boundaries
are marked by red dots. These points are mapped through the optical system
onto the corresponding boundaries of the regions in momentum space at the
crystal shown in the preceding figure.

###  2.1 Pixel basis design

In our experiment, we choose to work in the discretized transverse-momentum,
or macro-pixel basis. This basis provides several advantages over other
spatial mode bases. First, projective measurements in bases that are unbiased
with respect to the pixel basis require phase-only measurements. In contrast
to the lossy amplitude and phase measurements required for Laguerre-Gauss
modes [30, 29] and their superpositions, measurements with phase-only
modulation are lossless in theory and produce a count rate that is independent
of the specifics of the chosen basis. As a result, measurements in such bases
maximize photon flux, are resistant to detector noise, and allow us to
minimize the total number of measurements required. Second, as the
distribution of macro-pixels is determined by circle packing formulas, this
basis is compatible with state-of-the-art quantum communication technologies
based on multi-core fibres [37, 38, 39, 40] and was recently employed for
high-dimensional entanglement transport through a commercial multi-mode fibre
[41] as well as a test of genuine high-dimensional steering [42]. Third and
most significantly, informed by knowledge of the JTMA, we can optimise this
basis to approach a maximally entangled state of the form
$\left|\right.\\!{\Phi}\\!\left.\right\rangle=\frac{1}{\sqrt{d}}\sum_{m}\left|\right.\\!{mm}\\!\left.\right\rangle$,
where $\left|\right.\\!{m}\\!\left.\right\rangle$ stands for one photon in
mode m and none in the others. This in turn enables us to use powerful
theoretical techniques that rely on mutually unbiased bases for entanglement
certification. In contrast with Procrustean filtering techniques that achieve
a similar goal by adding mode-dependent loss [43, 23], this optimisation can
be done in a relatively lossless manner by tailoring the size and spacing of
individual pixel modes.

As illustrated in Fig. 1, the JTMA indicates that outer macro-pixel modes
exhibit the strongest correlation albeit with lower amplitudes, while the
inner pixel modes show weaker correlation with higher amplitudes. In order to
obtain the highest fidelity to the maximally entangled state, the spacing and
size of pixels can be optimised to minimise cross-talk arising from non-
vanishing pump transverse bandwidth $\sigma_{P}$, while simultaneously
equalizing pixel probability amplitudes and maximizing photon count rates.
This proceeds as follows: the diameter of the circular region containing the
macro-pixel basis is determined by the width $\sigma_{S}$ of the JTMA major
axis, such that the outermost pixel modes for a chosen dimension have
sufficient amplitude. The maximum coincidence rate for the outermost pixels is
obtained by giving them the maximum radius allowed for a given dimension and
the chosen spacing. Taking into account the decreasing correlation strength,
we then proceed to choose the radius for the inner pixels such that the photon
count rate is equal for all pixels, thereby approximating a state with equal
Schmidt coefficients extremely well.

Figure 2: Experimental Setup. a) An example computer-generated hologram used
for projective measurements in a 19-dimensional Wootters-Fields (WF) basis
state (Eq. (4)). b) A grating-stabilized UV laser (405 nm) is shaped by a
telescope system of lenses and used for generating a pair of infrared photons
(810 nm) entangled in their transverse position-momentum via Type-II
spontaneous-parametric-down-conversion (SPDC) in a non-linear ppKTP crystal.
After removing the UV pump with a dichroic mirror (DM), the photons are
separated with a polarising-beam-splitter (PBS) and made incident on two
phase-only spatial light modulators (SLM) at an angle of $5^{\circ}$ (the
$45^{\circ}$ angle shown is solely for the purpose of the illustration).
Precise projective measurements in the pixel basis and any of its mutually
unbiased bases are performed with the combination of SLMs, intensity-
flattening telescopes (IFT), and single-mode-fibers. The filtered photons are
detected by single-photon-avalanche-detectors (SPAD) connected to a
coincidence counting logic (CC) that records time coincident events with a
coincidence window of 0.2 ns.

###  2.2 Intensity-flattening telescopes

As shown in Fig. 2, the detection of our entangled state depends on a spatial-
mode filtering scheme composed of holographic mode projectors implemented on
spatial light modulators (SLMs) and single-mode fibers (SMFs). As Eq. (2)
indicates, the collection bandwidth of the entangled state is limited by the
SMF Gaussian mode width $\sigma_{C}$ that depends on the specific
characteristics of the fiber and coupling optics used. As a result, higher
order modes are relatively suppressed, which has an especially adverse effect
when measuring coherent high-dimensional superpositions of spatial modes. In
recent work, we demonstrated an “intensity-flattening” technique that
dramatically improves the quality of general mode-projective measurements at
the expense of adding loss [29].

This technique can be extended to the two-photon case, as depicted by the
illustration of the collection modes superposed on the JTMA (represented by
the purple curves in Fig. 1 a). If the Gaussian back-propagated modes are made
wider, higher order modes associated with the edges of the JTMA will couple
more efficiently to the SMF, while lower order modes are relatively
suppressed. This effectively allows us to make the approximation
$G(\vec{k}_{s},\vec{k}_{i})\approx F(\vec{k}_{s},\vec{k}_{i})$ in Eq. (2),
equating the collected JTMA with the generated one. We implement this
technique by using intensity-flattening telescopes (IFTs) that afocally
magnify the back-propagated collection modes to a size that optimally
increases the two-photon collection bandwidth while keeping loss at a
tolerable level. While our previous work demonstrated the IFT technique with
classical light from a laser, this work extends it to an entangled state for
the first time, demonstrating a marked improvement in reconstructed two-photon
state fidelity [22]. For additional details on the experimental setup we refer
to Appendix A.I.

###  2.3 High-dimensional entanglement witness

Our entanglement certification technique builds on an entanglement-
dimensionality witness that uses measurements in the chosen standard basis,
complemented by measurements in a basis that is unbiased with respect to it,
in order to efficiently certify high-dimensional entanglement [22]. Here, we
briefly introduce this witness, identify its limitations when applied to
experiment, and discuss our modifications that significantly improve its
utility.

The local Schmidt basis of a position-momentum entangled state, Eq. (1), which
we designate as the _standard basis_ , normally consists of localised
transverse spatial (or momentum) modes. When performing single-outcome
projective measurements (as described above) in such a basis, the detector
count rates are restricted by the limited projection onto the collection mode
(a pixel). Therefore, measurements in such a spatially localised basis result
in significantly fewer counts (per unit time) than measurements in bases that
are unbiased111For brevity, we will refer to such bases simply as ‘unbiased’
from here on, which is to be understood as being in reference to a chosen
standard basis. with respect to the spatially localized basis. This is the
case because the vectors in the unbiased basis correspond to equally weighted
superpositions of the spatially localised modes. Measurements in a spatially
delocalised unbiased basis can thus collect photons incident on any of the
pixels.

As a result, ‘unbiased’ measurements take $\mathcal{O}(d)$ less time
individually, or $\mathcal{O}(d^{2})$ less time in the bipartite case, than
localised standard basis single-outcome projective measurements for a given
photon flux. As we demonstrate here, one may construct an efficient witness
for certifying high-dimensional entanglement purely from measurements in two
(or more) spatially delocalised bases that are mutually unbiased with respect
to each other (and with respect to the original standard basis), without
resorting to measurements in the standard basis at all. This construction is
extremely beneficial and permits us to lower the measurement times for
certifying high-dimensional entanglement significantly.

The method described in Ref. [22] allows one to certify high-dimensional
entanglement by estimating a lower bound on the fidelity $F(\rho,\Phi)$ of a
measured state $\rho$ with a chosen target state
$\left|\right.\\!{\Phi}\\!\left.\right\rangle=\frac{1}{\sqrt{d}}\sum_{m}\left|\right.\\!{mm}\\!\left.\right\rangle$.
By performing two-photon measurements in the standard basis
$\\{\left|\right.\\!{mn}\\!\left.\right\rangle\\}$ and a second basis
$\\{\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle\\}$
that is unbiased with respect to the standard basis, one can obtain a lower
bound to the fidelity $F(\rho,\Phi)$. As explained in Ref. [22], the fidelity
can in turn be used to bound the Schmidt number of $\rho$ from below. That is,
the certified entanglement dimensionality is the maximal $d_{\mathrm{ent}}$
such that $(d_{\mathrm{ent}}-1)/d\geq F(\rho,\Phi)$. The unbiased bases can be
constructed in a standard manner by following the prescription by Wootters and
Fields [44], i.e.,

$\displaystyle\left|\right.\\!{\tilde{j}_{k}}\\!\left.\right\rangle$
$\displaystyle=\tfrac{1}{\sqrt{d}}\sum_{m=0}^{d-1}\omega^{jm+km^{2}}\left|\right.\\!{m}\\!\left.\right\rangle,$
(4)

where $\omega=\exp(\tfrac{2\pi\hskip 0.5pti}{d})$ is the principal complex
d$-th$ root of unity, $k\in\\{0,\dots,d-1\\}$ labels the chosen basis, and
$j\in\\{0,\dots,d-1\\}$ labels the basis elements. We refer to these bases as
Wootters-Fields (WF) bases. Notice that when $d$ is an odd prime, which we
will assume from now on (see Appendix A.II), the set of $d$ bases
$\\{\left|\right.\\!{\tilde{j}_{k}}\\!\left.\right\rangle\\}_{j}$ together
with the standard basis $\\{\left|\right.\\!{m}\\!\left.\right\rangle\\}_{m}$
forms a complete set of $d+1$ mutually unbiased bases (MUBs) with the property
that the overlap between any two basis states chosen from any two different
bases in the set have the same magnitude.

In order to certify entanglement without using the standard basis, we use a
property of the maximally entangled state
$\left|\right.\\!{\Phi}\\!\left.\right\rangle=\frac{1}{\sqrt{d}}\sum_{m}\left|\right.\\!{mm}\\!\left.\right\rangle$.
Such states are invariant under transformations $(U\otimes U^{*})$ for any
unitary operator $U$, and thus have the same form in any WF basis, allowing us
to express our target state as
$\left|\right.\\!{\Phi}\\!\left.\right\rangle=\frac{1}{\sqrt{d}}\sum_{m}\left|\right.\\!{\tilde{m}_{k}\tilde{m}^{*}_{k}}\\!\left.\right\rangle$.
The fidelity of our experimental state to this target state $F(\rho,\Phi)$ can
then be expressed in terms of $k$-th WF basis and be split into two
contributions, $F(\rho,\Phi)=F_{1}(\rho,\Phi)+F_{2}(\rho,\Phi)$, where

$\displaystyle F_{1}(\rho,\Phi)$
$\displaystyle:=\tfrac{1}{d}\sum_{m}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}^{*}_{k}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{m}^{*}_{k}}\\!\left.\right\rangle,$
(5a) $\displaystyle F_{2}(\rho,\Phi)$ $\displaystyle:=\tfrac{1}{d}\sum_{m\neq
n}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{n}_{k}\tilde{n}^{*}_{k}}\\!\left.\right\rangle.$
(5b)

While the first term $F_{1}(\rho,\Phi)$ can be calculated directly from
measurements in $k$-th WF basis
$\\{\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle\\}$,
we show in Appendix A.III how one can determine a lower bound for
$F_{2}(\rho,\Phi)$ with measurements in a second WF basis (labeled by
$k^{\prime}\neq k$ in the construction above)
$\\{\left|\right.\\!{\tilde{m}_{k^{\prime}}\tilde{n}_{k^{\prime}}^{*}}\\!\left.\right\rangle\\}$,
allowing us to calculate a lower bound for the fidelity to the maximally
entangled state given by

$\displaystyle
F(\rho,\Phi)\geq\,\tilde{F}(\rho,\Phi)=\,\tfrac{1}{d}\sum_{m}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right\rangle\,$
$\displaystyle\hskip
11.38109pt+\sum\limits_{m}\left\langle\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}^{*}}\\!\left.\right\rangle-\,\tfrac{1}{d}$
(6) $\displaystyle-\sum\limits_{\begin{subarray}{c}m\neq m^{\prime}\\!\\\
m\neq n\\\ n\neq n^{\prime}\\!\\\ n^{\prime}\neq m^{\prime}\\\
\end{subarray}}\tilde{\gamma}_{mnm^{\prime}n^{\prime}}\,\sqrt{\left\langle\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right\rangle\left\langle\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle}\,,$

where the term $\tilde{\gamma}_{mnm^{\prime}n^{\prime}}$ vanishes whenever
$(m-m^{\prime}-n+n^{\prime})\\!\\!\mod(d)\neq 0$, and is equal to
$\frac{1}{d}$ otherwise.

This result shows that measurements in any two WF bases can be used to lower
bound the fidelity to a target state, and hence to certify the entanglement
dimensionality. In particular, this allows us to bypass the use of
measurements in the standard basis used in [22], while providing significant
flexibility in the measurement settings. By construction, the witness works
best if the chosen target state is close to the state $\rho$ whose
entanglement is being certified. Nevertheless, the witness is valid for any
choice of target state, and thus requires no assumptions about the state
$\rho$. We do make certain assumptions about our local measurement devices,
which means that our witness is not fully device-independent. Below, we
discuss these assumptions on the state and devices in the context of our
experiment.

While full device-independence is challenging to achieve, almost beyond
practicability with current technology, it is usually also more than what is
required for practical security. Assumptions on states and devices can enter
in different form, including assumptions on certain properties of the state
(purity, conservation of energy or momentum), to assuming certain device
properties (perfect measurements, background/dark count subtraction). Our only
assumption is that our measurements work in a $d$-dimensional Hilbert space
and that the eigenstates corresponding to measurement outcomes in different
bases are mutually unbiased with a specific phase relation. This assumption
can and has been tested by preparing eigenstates of one basis and measuring in
another basis, to estimate if all possible results in the latter basis are
equally distributed [29]. No further assumptions on the states are needed in
our setting, and we do not perform any background subtraction. All assumptions
that we make can thus be tested experimentally. We believe that this presents
a reasonable compromise between fully device-independent security and
practicability of implementation. Moreover, while entanglement witnesses based
on MUBs can in principle be constructed in a device-independent way [45],
device-independent certification of the Schmidt number is generally not
possible [46].

##  3 Results

We implement the above improvements in basis optimisation, spatial-mode
measurements, and entanglement certification in a two-photon entanglement
experiment. Figure 2 a) depicts a tailored diffractive hologram used for
performing spatial-mode projective measurements in a 19-dimensional WF basis.
Notice that contrary to the holograms used for measuring in the standard
basis, where projections on each state are made by “switching on” a single
macro-pixel at a time, the WF basis states require us to display all macro-
pixels at once, each with an appropriate phase selected according to Eq. (4).

The pixel orientations are determined by circle packing formulas and the pixel
sizes and spacings are optimised according to the JTMA as described above in
Sec. 2.1. The experimental setup is depicted in Fig. 2 b). While not discussed
in the previous section, an additional consideration involves the use of
Gaussian beam simulations to ensure that mode waists and crystal conjugate
planes are optimally located in the setup (see Appendix A.I). A grating-
stabilised CW laser at 405 nm is loosely focused onto a 5 mm ppKTP crystal to
produce photon pairs at 810 nm entangled in their transverse position-momentum
via type-II spontaneous parametric down-conversion (SPDC). Single-outcome
projective measurements in the optimized macro-pixel basis (or any of its
MUBs) are performed on the photon pairs using diffractive holograms
implemented on spatial light modulators (SLMs), where each macro-pixel mode is
defined by displaying a grating over the corresponding circular area. To
perform transformations over the modes of interest, different phases for each
macro-pixel are encoded as a relative shift in the grating with respect to the
other macro-pixels. As shown in Figure 2a), while the black area around the
macro-pixel modes is effectively “switched off” because no grating is
displayed, the grating inside each macro-pixel effectively “switches on” the
selected modes so they efficiently couple to single-mode fibers. After the
SLMs, two intensity-flattening telescopes (IFT) are used for ensuring precise,
generalized spatial-mode measurements across the entire modal bandwidth of
interest (see Appendix A.I).

Figure 3: Experimental data in 19 dimensions. Normalised two-photon
coincidence counts showing correlations in the standard 19-dimensional pixel
basis $\\{\left|\right.\\!{mn}\\!\left.\right\rangle\\}_{mn}$ and its 19
mutually unbiased bases
$\\{\left|\right.\\!{\tilde{i}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle\\}_{i,j}$.
With these measurements, we obtain a fidelity of $F(\rho,\Phi^{+})=94.4\pm
0.4$% to a 19-dimensional maximally entangled state, certifying 18-dimensional
entanglement.

[tb]  : Fidelities $F(\rho,\Phi^{+})$ to the maximally entangled state,
obtained via measurements in all mutually unbiased bases in dimension $d$ $d$
$d_{\mathrm{ent}}$ $F(\rho,\Phi^{+})$ $E_{\mathrm{oF}}$ (ebits) 3 3 $98.2\pm
0.9$ % $1.4\pm 0.1$ 5 5 $97.5\pm 0.5$ % $2.2\pm 0.1$ 7 7 $96.4\pm 0.7$ %
$2.6\pm 0.1$ 11 11 $93.9\pm 0.7$ % $3.1\pm 0.1$ 13 13 $94.1\pm 0.6$ % $3.3\pm
0.1$ 17 17 $94.3\pm 0.3$ % $3.6\pm 0.1$ 19 18 $94.4\pm 0.4$ % $3.8\pm 0.1$

Our modified entanglement witness allows characterizing the generated state by
lower bounding its fidelity to the maximally entangled state through
measurements in any two mutually unbiased bases. However, if one uses the
standard basis and all of its MUBs, the fidelity bound becomes tight and one
can estimate the exact fidelity to the maximally entangled state [22]. To
showcase the effect of our improvements on the resulting pixel entanglement
quality, we measure correlations in all MUBs for subspaces of prime dimension
up to $d=19$. From these measurements, we obtain record fidelities to the
maximally entangled state of $\approx 94\%$ or greater in all subspaces,
certifying entanglement dimensions of $d_{\mathrm{ent}}=d$ for $d=3,...,17$
(see Table 3). In $d=19$, we obtain an estimate of exact fidelity of $94.4\pm
0.4$%, certifying an entanglement dimensionality of $d_{\mathrm{ent}}=18$ (and
just below the fidelity bound of $94.7\%$ for $d_{\mathrm{ent}}=19$). We also
calculate the entanglement-of-formation ($E_{\mathrm{oF}}$), which is an
entropic measure of the amount of entanglement needed to create our state [47,
48] (please see Appendix A.IV for more details).

Our $E_{\mathrm{oF}}$ reaches a value of $3.8\pm 0.1$ ebits in $d=19$, which
is already higher than previously reported values without subtracting
accidental/background counts or making any assumptions on the state [18, 16]
Measured correlation data in all 20 mutually unbiased bases of the $d=19$
pixel basis are shown in Fig. 3. The uncertainty in fidelity is calculated
assuming Poisson counting statistics and error propagation via a Monte-Carlo
simulation of the experiment.

We now demonstrate the speed of our measurement technique, which is enabled by
the use of phase-only pixel-basis holograms and our entanglement witness that
allows the use of any two pixel MUBs for certifying entanglement. We perform
measurements in two MUBs for prime dimensions ranging from $d=19$ to $97$. In
$d=19$, we are able to certify a fidelity lower bound of ($93\pm 2$)% using a
total data acquisition time of 3.6 minutes (corresponding to 722 single-
outcome measurements), certifying an entanglement dimensionality of
$d_{\mathrm{ent}}=18\pm 1$. In comparison, prior experiments on OAM
entanglement required a measurement time two orders of magnitude larger for
certifying just $d_{\mathrm{ent}}=9$ in 11 dimensions [22]. A summary of
results for dimensions 19 and higher is shown in Table 3. As can be seen, the
entanglement quality starts dropping above $d\approx 30$, which is a result of
the limited resolution of our devices and the SPDC generation bandwidth
itself. Nevertheless, we are able to certify an entanglement dimensionality of
$d_{\mathrm{ent}}=55\pm 1$ (See Fig. 4) in a 97-dimensional space and an
entanglement-of-formation of $E_{\mathrm{oF}}=4.0\pm 0.1$ ebits in a
31-dimensional space, which both constitute a record-breaking entanglement
dimensionality and entanglement-of-formation certified without any assumptions
on the state. A very recent experiment demonstrated comparable state
fidelities and $E_{\mathrm{oF}}$ values in $d=32$ using multi-path down-
conversion [49]. The required measurement time in the 97-dimensional pixel
space was on the order of a day, while using standard basis measurements would
require $\sim$14 years of measurement time for data of similar quality.

[b!]  : Fidelity bounds $\tilde{F}(\rho,\Phi^{+})$ obtained via measurements
in two mutually unbiased bases in dimension $d$ $d$
$d_{\mathrm{ent}}^{\,\dagger\dagger}$ $F(\rho,\Phi^{+})$ $E_{\mathrm{oF}}$
(ebits) 19 18(+1) $93\pm 2$ % $3.6\pm 0.2$ 23 22(-1) $92\pm 2$ % $3.6\pm 0.2$
29 27(-1) $90\pm 2$ % $3.8\pm 0.2$ 31 29(-1) $92\pm 2$ % $4.0\pm 0.1$ 37
32(-1) $84\pm 1$ % $3.4\pm 0.1$ 51† 38(-1) $73\pm 1$ % $2.8\pm 0.1$ 97 55($\pm
1$) $56\pm 1$ % $1.9\pm 0.1$ † Our two-MUB witness remains valid even for this
non-prime dimension, as the two measurement bases used ($k=0,1$) are still
mutually unbiased, and their phase relationship allows the bound to hold.
†† Here we present the entanglement dimensionalities certified by the mean
values of the corresponding fidelity bounds, while other Schmidt-number
thresholds within the confidence interval are shown in parenthesis.

Figure 4: Experimental data in 97 dimensions. a) An example of the diffractive
hologram used for projective measurements in a 97-dimensional WF basis.
Normalised two-photon coincidence counts showing correlations in the b) first
and c) second mutually unbiased bases ($k=0,1$) to the 97-dimensional pixel
basis
$\\{\left|\right.\\!{\tilde{i}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle\\}_{i,j}$.
Using these measurements, we obtain a fidelity bound of
$\tilde{F}(\rho,\Phi^{+})=56\pm 1$% that is above the bound
$B_{54}=54/97=0.5567$, thus certifying an entanglement dimensionality of
$d_{\mathrm{ent}}=55\pm 1$.

##  4 Conclusion

We have demonstrated the certification of photonic high-dimensional
entanglement in the transverse position-momentum degree-of-freedom with a
record quality, measurement time, and entanglement dimensionality. These
results are made possible through the combination of three new methods:
tailored design of the spatial-mode basis, precise two-photon spatial-mode
measurements, and a versatile entanglement witness based on measurements in
mutually unbiased bases (MUBs). As a demonstration of the quality of our
entanglement, we achieve state fidelities of $\approx 94\%$ or above in local
dimensions of 19 or below, certifying up to 18-dimensional entanglement. In
addition, the use of any two MUBs enables the measurement of 18-dimensional
entanglement in 3.6 minutes, a reduction in measurement time by more than two
orders of magnitude over previous demonstrations [22]. Finally, we are able to
certify an entanglement dimensionality of at least 55 local dimensions and an
entanglement-of-formation of 4 ebits, which to our knowledge is the highest
amount certified without any assumptions on the state. While we have used
projective single-outcome measurements in our experiment, recent progress on
generalized multi-outcome measurement devices [50, 51, 52] and superconducting
detector arrays [53] promises to increase measurement speeds even further. Our
results show that high-dimensional entanglement can indeed break out of the
confines of an experimental laboratory and enable noise-resistant
entanglement-based quantum networks that saturate the information-carrying
potential of a photon.

## Acknowledgements

This work was made possible by financial support from the QuantERA ERA-NET Co-
fund (FWF Project I3773-N36) and the UK Engineering and Physical Sciences
Research Council (EPSRC) (EP/P024114/1). NF acknowledges support from the
Austrian Science Fund (FWF) through the project P 31339-N27. MH acknowledges
funding from the Austrian Science Fund (FWF) through the START project
Y879-N27. MP acknowledges funding from VEGA project 2/0136/19 and GAMU project
MUNI/G/1596/2019.

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## Appendix

We have demonstrated photonic high-dimensional entanglement in the discretized
transverse position-momentum degree of freedom that allows the assumption-free
certification of entanglement with a record quality, measurement time, and
entanglement dimensionality. In this appendix we provide additional
information on the experimental setup used for engineering our entangled
state, give a detailed proof of the fidelity bound that we use for certifying
high-dimensional entanglement, and provide a description of the entanglement
of formation bound we have used in the main text for quantifying the amount of
entanglement of the produced state.

###  A.I Experimental Setup

A nonlinear ppKTP crystal ($1~{}mm\times 2~{}mm\times 5~{}mm$) is pumped with
a continuous wave grating-stabilized 405 nm laser in order to generate a pair
of photons at 810 nm entangled in their transverse position-momentum via the
process of type-II spontaneous parametric down conversion (SPDC). The crystal
is temperature-tuned and phase-matching conditions are met by housing it in a
custom-built oven that keeps it at 30∘C. With the purpose of increasing the
dimensionality of the generated state, we consider Gaussian beam propagation
and the transformation by two lenses to determine how to shape our beam and
loosely focus it on the nonlinear crystal. As shown in Fig. A.1, the pump
laser goes through a telescope system where the focal lengths and position of
the lenses are chosen such that the final beam waist is located at the crystal
with a large enough size to increase the number of generated modes, but
without clipping the beam by the crystal aperture.

Figure A.1: Gaussian beam propagation for the pump. The collimated UV pump
(beam waist $\omega_{0}=950\mu$m) propagates through a telescope system
composed of two lenses with $f_{1}=250$ mm and $f_{2}=50$ mm. After the
telescope, the pump beam waist is located at a distance of 50 mm away from the
second lens, with a size of $w_{\textrm{p}}=188\mu$m. We place the crystal
such that this final beam waist is at its longitudinal center.

After the pump is removed with a dichroic mirror (DM), the entangled pair of
photons is separated with a polarising beam-splitter (PBS). Each of the
photons is made incident on a phase-only spatial light modulator (SLM,
Hamamatsu X10468-02) that is placed in the Fourier plane of the crystal using
a lens (see Fig. A.2). For the reflected photon to be manipulated by the SLM,
we use a half-wave-plate (HWP) to rotate its polarisation from vertical to
horizontal. Computer-generated holograms displayed on each SLM allow us to
select particular spatial modes of the incident light and convert them into a
Gaussian mode, which effectively couples (using a 10X objective) into a
single-mode fiber (SMF) that carries these filtered photons to a single photon
avalanche detector (SPAD). In effect, this allows us to perform projective
measurements of any complex spatial mode. Time-coincident events between the
two detectors are registered by a coincidence counting logic (CC). Thus, the
states considered in our experiment are post-selected on detecting two
photons.

Figure A.2: Gaussian beam propagation from Crystal to SLM. For the design of
the experimental setup, we model the propagation of the generated photons to
the spatial light modulators assuming a Gaussian mode that has an initial beam
waist of $\omega_{\textrm{SPDC}}=188$ $\mu$m, which corresponds to the pump
beam waist at the longitudinal center of the crystal. Using a lens of $f=250$
mm, we place the Fourier plane of the crystal at the SLM (represented in the
figure by the blue line at the right). A Gaussian mode is this plane would
have a beam waist of $\omega_{\textrm{SLM}}=343$ $\mu$m on the SLM after
propagation from the crystal. Figure A.3: Intensity Flattening Telescope.
Consider a virtually backward-propagating collimated beam that is launched by
a 10X microscope objective with an initial beam waist of
$\omega_{\textrm{BP}}=1.117$ mm. The intensity flattening telescope magnifies
the collection mode by a factor of 3.3 using lenses with $f_{1}=150$ mm and
$f_{2}=500$ mm. The resulting collection mode beam waist in the plane of the
SLM is $\omega_{\textrm{c}}=3723$ $\mu$m, which determines the maximum mode
size that we can use for the pixel holograms without introducing large amounts
of loss or measurement imprecision.

The accuracy of the projective measurement performed by the combination of an
SLM and SMF is ensured through the use of intensity flattening telescopes
(IFT) [29]. As represented by the purple curves in Fig. 1a), the use of
single-mode fibres introduces a Gaussian component into the collected JTMA and
restricts the detection of higher-order modes. In order to reduce this
negative effect, we use the IFT to afocally decrease the size of the mode that
is propagating from the SLM to the objective lens, thus removing the Gaussian
component introduced by the use of a SMF and recovering the orthogonality
between spatial modes of a given basis. This technique can be better
understood when one considers the overlap between the generated and collected
modes on the plane of the SLM, where the telescopes magnify the back-
propagated collection mode and effectively “flatten” the intensity
distribution of the Gaussian component on the collected state. This is
especially important when measuring coherent superpositions of spatial modes
or pixel MUBs. In our experiment, the back-propagated mode is magnified by a
factor of 3 (see Fig. A.3), which allows us to increase the bandwidth of the
measured modes while minimizing losses that would hinder our efficiency.

The quality of our measurements also relies on the design of our bases. We
choose to work with the Pixel basis, a discrete position-momentum basis
composed of circular macro-pixels arranged inside a circle. The mutually
unbiased bases to the pixel basis are a coherent superposition of the pixel
states with a specific phase relationship between them [44]. As described in
the main text, in order to obtain the highest fidelities to a maximally
entangled state, we try and equalise the probabilities of measuring each pixel
mode by adapting the size of the macro-pixels, the spacing between them, and
the size of the circle they are in, according to the characteristics of the
joint-transverse-momentum-amplitude (JTMA) of the state we are producing.
Doing so, we improve the visibility of the correlations in mutually unbiased
pixel bases, which increases the fidelity of our state to the maximally
entangled state.

###  A.II Derivation of the dimensionality witness

In this section, we provide a derivation of the fidelity lower bound based on
two measurement bases that was originally introduced in Ref. [22]. In Sec.
A.III, we then present an adaption of these fidelity witnesses that permits us
to increase the speed with which the required measurements can be performed in
our experiment.

Consider a bipartite quantum system with Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$AB$}}}=\mathcal{H}_{\raisebox{0.0pt}{\tiny{$A$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$B$}}}$,
with equal local dimensions
$\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$A$}}})=\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$B$}}})=d$,
and an a priori unknown state $\rho$ of this system. For the certification of
the Schmidt number w.r.t. the bipartition into subsystems $A$ and $B$, we want
to use only a few measurement settings (local product bases) to give a lower
bound on the fidelity $F(\rho,\Phi)$ to the target state
$\left|\right.\\!{\Phi}\\!\left.\right\rangle=\sum_{n}\lambda_{n}\left|\right.\\!{nn}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$AB$}}}$,
given by:

$\displaystyle
F(\rho,\Phi)=\textnormal{Tr}(|{\Phi}\rangle\langle{\Phi}|\rho)=\sum_{m,n=0}^{d-1}\langle{mm}|{\rho}|{nn}\rangle.$
(A.1)

The entanglement dimensionality can be deduced from the fidelity taking into
account that for any state $\rho$ of Schmidt number $r\leq d$, the fidelity of
Eq. (A.1) is bounded by:

$\displaystyle F(\rho,\Phi)\leq
B_{r}(\Phi):=\sum_{m=0}^{r-1}\lambda_{i_{m}}^{2},$ (A.2)

where the sum runs over the $i_{m}$ with $m\in\\{0,...,d-1\\}$ such that
$\lambda_{i_{m}}\geq\lambda_{i_{m}^{\prime}}\forall m\geq m^{\prime}$. Hence,
any state with $F(\rho,\Phi)>B_{r}(\Phi)$ must have an entanglement
dimensionality of at least $r+1$.

Our goal is to obtain a (lower bound on the) fidelity that is as large as
possible for the target state whose Schmidt rank (the number of non-vanishing
coefficients $\lambda_{n}$) is as close as possible to the local dimension
$d$. Here, we want to focus on the case where the target state is the
maximally entangled state $\left|\right.\\!{\Phi^{+}}\\!\left.\right\rangle$,
i.e., where $\lambda_{n}=1/\sqrt{d}\,\forall\,n$.

The method for certifying high-dimensional entanglement (in particular, the
Schmidt number) described in Ref. [22] works in the following way. First, one
designates a standard basis
$\\{\left|\right.\\!{mn}\\!\left.\right\rangle\\}_{m,n=0,\ldots,d-1}$,
measures locally w.r.t. this basis, obtaining estimates for the matrix
elements
$\\{\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle\\}_{m,n}$
from the coincidence counts $\\{N_{ij}\\}_{i,j}$ in the chosen basis via

$\displaystyle\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle$
$\displaystyle=\,\frac{N_{mn}}{\sum_{i,j}N_{ij}}.$ (A.3)

Then, one measures in $M$ of $d$ possible mutually unbiased bases
$\\{\left|\right.\\!{\tilde{i}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle\\}_{i,j}$,
where the label $k\in\\{0,1,\ldots,d-1\\}$ labels the chosen basis, and the
asterisk denotes complex conjugation w.r.t. the standard basis. The local
basis vectors are constructed according to the prescription by Wootters and
Fields [44], that is,

$\displaystyle\left|\right.\\!{\tilde{j}_{k}}\\!\left.\right\rangle$
$\displaystyle=\tfrac{1}{\sqrt{d}}\sum_{m=0}^{d-1}\omega^{jm+km^{2}}\left|\right.\\!{m}\\!\left.\right\rangle,$
(A.4)

where $\omega=\exp(\tfrac{2\pi\hskip 0.5pti}{d})$ are the complex $d$-th roots
of unity, and we hence refer to these bases as Wootters-Fields (WF) bases.
When $d$ is an odd prime, which we will assume from now on, the set of all
bases $\\{\left|\right.\\!{\tilde{j}_{k}}\\!\left.\right\rangle\\}_{j}$
together with the standard basis
$\\{\left|\right.\\!{m}\\!\left.\right\rangle\\}_{m}$ forms a complete set of
$d+1$ mutually unbiased bases with the property that the overlaps between any
two basis states of any two different bases from the set have the same
magnitude. Measurements in any of these $d$ bases then provide the matrix
elements
$\\{\left\langle\right.\\!{\tilde{i}_{k}\tilde{j}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{i}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle\\}_{i,j}$
for any of the $M$ chosen values of $k$.

For simplicity, let us now first concentrate on the case where one measures
only in a single of these Wootters-Fields bases, i.e., the case $M=1$. We use
the corresponding matrix elements to bound the target state fidelity
$F(\rho,\Phi)$ by splitting it in two contributions,
$F(\rho,\Phi)=F_{1}(\rho,\Phi)+F_{2}(\rho,\Phi)$, where

$\displaystyle F_{1}(\rho,\Phi)$
$\displaystyle:=\tfrac{1}{d}\sum_{m}\left\langle\right.\\!{mm}\\!\left.\right|\rho\left|\right.\\!{mm}\\!\left.\right\rangle,$
(A.5a) $\displaystyle F_{2}(\rho,\Phi)$
$\displaystyle:=\tfrac{1}{d}\sum_{m\neq
n}\left\langle\right.\\!{mm}\\!\left.\right|\rho\left|\right.\\!{nn}\\!\left.\right\rangle.$
(A.5b)

The first term $F_{1}(\rho,\Phi)$ can be calculated directly from the
measurements in the standard basis, while the term $F_{2}(\rho,\Phi)$ can be
bounded from below by the quantity $\tilde{F}_{2}(\rho,\Phi)\leq
F_{2}(\rho,\Phi)$, given by

$\displaystyle\tilde{F}_{2}$
$\displaystyle:=\sum\limits_{j=0}^{d-1}\left\langle\right.\\!{\tilde{j}_{k}\tilde{j}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{j}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle-\tfrac{1}{d}-\sum\limits_{\begin{subarray}{c}m\neq
m^{\prime}\\!,m\neq n\\\ n\neq n^{\prime}\\!,n^{\prime}\neq m^{\prime}\\\
\end{subarray}}\tilde{\gamma}_{mm^{\prime}nn^{\prime}}\,\sqrt{\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right\rangle\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle},$
(A.6)

where we have noted that
$\sum_{m,n=0}^{d-1}\\!\\!\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle=1$
by construction due to Eq. (A.3), and the prefactor
$\tilde{\gamma}_{mm^{\prime}nn^{\prime}}$ is given by

$\displaystyle\tilde{\gamma}_{mm^{\prime}nn^{\prime}}$
$\displaystyle=\begin{cases}0\ \ \mbox{if}\ \
(m-m^{\prime}-n+n^{\prime})\\!\\!\\!\\!\mod(d)\neq 0\\\\[2.84526pt]
\displaystyle\tfrac{1}{d}\ \ \mbox{otherwise}.\end{cases}$ (A.7)

With the fidelity bound for this particular target state at hand, one can then
certify a Schmidt number of $r$ whenever one finds a fidelity (bound) larger
than $\tfrac{r}{d}$, see [22].

To prove this, we focus on the information given by the elements of the
density matrix that we obtain from measuring in the WF basis. In particular,
the sum over the correlated WF matrix elements can be split into three terms,
i.e.,

$\displaystyle\sum\limits_{j=0}^{d-1}\left\langle\right.\\!{\tilde{j}_{k}\tilde{j}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{j}_{k}\tilde{j}_{k}^{*}}\\!\left.\right\rangle$
$\displaystyle=:\Sigma\,=\,\Sigma_{1}\,+\,\Sigma_{2}\,+\,\Sigma_{3},$ (A.8)

which are given by

$\displaystyle\Sigma_{1}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m,n}\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle=\tfrac{1}{d},$
(A.9a) $\displaystyle\Sigma_{2}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m\neq
n}\left\langle\right.\\!{mm}\\!\left.\right|\rho\left|\right.\\!{nn}\\!\left.\right\rangle,$
(A.9b) $\displaystyle\Sigma_{3}$
$\displaystyle:=\tfrac{1}{d^{2}}\sum\limits_{\begin{subarray}{c}m\neq
m^{\prime},m\neq n\\\ n\neq n^{\prime},n^{\prime}\neq
m^{\prime}\end{subarray}}\operatorname{Re}\Bigl{(}c_{mnm^{\prime}n^{\prime}}\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle\Bigr{)},$
(A.9c)

where we have introduced the quantity

$\displaystyle c_{mnm^{\prime}n^{\prime}}$
$\displaystyle:=\sum_{j}\omega^{j(m-m^{\prime}-n+n^{\prime})+k(m^{2}-m^{\prime
2}-n^{2}+n^{\prime 2})}.$ (A.10)

We can then bound the real part appearing in the summands of $\Sigma_{3}$ by
their modulus, i.e.,

$\displaystyle\operatorname{Re}\Bigl{(}c_{mnm^{\prime}n^{\prime}}\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle\Bigr{)}\,\leq\,|c_{mnm^{\prime}n^{\prime}}\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle|\,=\,|c_{mnm^{\prime}n^{\prime}}|\cdot|\\!\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle\\!|\,$
(A.11)

and use the Cauchy-Schwarz inequality and the spectral decomposition
$\rho=\sum_{i}p_{i}\left|\right.\\!{\psi_{i}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{i}}\\!\left.\right|$
such that

$\displaystyle|\\!\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle\\!|$
$\displaystyle=|\sum\limits_{i}\sqrt{p_{i}}\left\langle\right.m^{\prime}n^{\prime}\left|\right.\psi_{i}\left.\right\rangle\sqrt{p_{i}}\left\langle\right.\psi_{i}\left|\right.mn\left.\right\rangle|$
(A.12)
$\displaystyle\leq\sqrt{\sum\limits_{i}p_{i}\left\langle\right.m^{\prime}n^{\prime}\left|\right.\psi_{i}\left.\right\rangle\left\langle\right.\psi_{i}\left|\right.m^{\prime}n^{\prime}\left.\right\rangle}\sqrt{\sum\limits_{i}p_{i}\left\langle\right.mn\left|\right.\psi_{i}\left.\right\rangle\left\langle\right.\psi_{i}\left|\right.mn\left.\right\rangle}=\sqrt{\left\langle\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right|\rho\left|\right.\\!{m^{\prime}n^{\prime}}\\!\left.\right\rangle\left\langle\right.\\!{mn}\\!\left.\right|\rho\left|\right.\\!{mn}\\!\left.\right\rangle}.$

Finally, one notes that for any single basis choice (labelled by $k$), one has

$\displaystyle|c_{mnm^{\prime}n^{\prime}}|$
$\displaystyle=\,|\sum_{j}\omega^{j(m-m^{\prime}-n+n^{\prime})}|,$ (A.13)

which vanishes whenever $(m-m^{\prime}-n+n^{\prime})\\!\\!\mod(d)\neq 0$, and
is equal to $d$ otherwise, resulting in
$\tilde{\gamma}_{mm^{\prime}nn^{\prime}}$ as in Eq. (A.7).

###  A.III Changing the designation of ‘standard basis’

The fidelity bound we have derived is based on designating one of the measured
bases as the ‘standard basis’, while the other is a WF basis constructed
w.r.t. the standard basis. However, the bases are mutually unbiased and
because the target state is maximally entangled, we can also reverse their
roles to obtain another fidelity bound that can differ in its value from the
original bound, depending on the estimated matrix elements (i.e., depending on
the experimental data). Moreover, if one has measured in several of the WF
bases, any of them can be designated the ‘standard basis’ and one may combine
it either with the original standard basis, or with any of the other WF bases
for which one has taken data.

####  A.III.1 Exchanging standard basis and WF basis

To see which modifications this entails, let us first spell out the specific
relationship between these bases. Suppose we designate the WF basis
$\\{\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle\\}_{n}$ labelled by
$k$ as the new ‘standard basis’. Then we can express the vectors of the
original standard basis $\\{\left|\right.\\!{m}\\!\left.\right\rangle\\}_{m}$
as

$\displaystyle\left|\right.\\!{m}\\!\left.\right\rangle$
$\displaystyle=\,\sum\limits_{n}\left\langle\right.\tilde{n}_{k}\left|\right.m\left.\right\rangle\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle\,=\,\tfrac{1}{\sqrt{d}}\sum\limits_{n}\omega^{-n\hskip
0.5ptm-k\hskip
0.5ptm^{2}}\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle\,=\,\tfrac{1}{\sqrt{d}}\sum\limits_{n}\,c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle,$
(A.14)

where $c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}=\omega^{-n\hskip
0.5ptm-k\hskip 0.5ptm^{2}}$. Similarly, we can choose a decomposition into the
conjugated basis vectors, i.e.,

$\displaystyle\left|\right.\\!{m}\\!\left.\right\rangle$
$\displaystyle=\,\sum\limits_{n}\left\langle\right.\tilde{n}_{k}^{*}\left|\right.m\left.\right\rangle\,\left|\right.\\!{\tilde{n}_{k}^{*}}\\!\left.\right\rangle\,=\,\tfrac{1}{\sqrt{d}}\sum\limits_{n}\,c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})\hskip
0.5pt*$}}}\,\left|\right.\\!{\tilde{n}_{k}^{*}}\\!\left.\right\rangle.$ (A.15)

We can then write the analogous expression to $\Sigma$ in Eq. (A.8) as

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}\,=\,\sum\limits_{m}\left\langle\right.\\!{mm}\\!\left.\right|\rho\left|\right.\\!{mm}\\!\left.\right\rangle\,=\,\tfrac{1}{d^{2}}\sum\limits_{\begin{subarray}{c}n,n^{\prime}\\\
q,q^{\prime}\end{subarray}}\sum\limits_{m}c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})\hskip
0.5pt*$}}}c_{mn^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}c_{mq}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}c_{mq^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})\hskip
0.5pt*$}}}\left\langle\right.\\!{\tilde{n}_{k}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{q}_{k}\tilde{q}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right\rangle,$ (A.16)

which we can also split up into three terms, i.e.,

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}$
$\displaystyle=\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{1}\,+\,\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{2}\,+\,\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{3},$
(A.17)

where

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{1}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m,n}\left\langle\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle\,=\,\tfrac{1}{d},$
(A.18a)
$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{2}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m\neq
n}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{n}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle,$
(A.18b)
$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{3}$
$\displaystyle:=\tfrac{1}{d^{2}}\sum\limits_{\begin{subarray}{c}m\neq
m^{\prime},m\neq n\\\ n\neq n^{\prime},n^{\prime}\neq
m^{\prime}\end{subarray}}c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}\left\langle\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle,$
(A.18c)

while all other contributions to the sum in Eq. (A.16) vanish. The coefficient
$c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}$
appearing in $\Sigma^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}_{3}$ is
given by

$\displaystyle
c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}$
$\displaystyle=\,\sum\limits_{j}c_{jm^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})\hskip
0.5pt*$}}}c_{jn^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}c_{jm}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}c_{jn}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})\hskip
0.5pt*$}}}\,=\,\sum\limits_{j}\omega^{-j(m-m^{\prime}-n+n^{\prime})},$ (A.19)

and since
$|c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,\mathrm{st.})$}}}|=|c_{mnm^{\prime}n^{\prime}}|$,
the coefficient $\tilde{\gamma}_{mm^{\prime}nn^{\prime}}$ in Eq. (A.7) is
unaffected. We can thus simply exchange the role of the standard basis and any
one of the WF bases to obtain a fidelity bound of the form

$\displaystyle F(\rho,\Phi)$
$\displaystyle\geq\,\tfrac{1}{d}\sum_{m}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right\rangle\,+\,\sum\limits_{m}\left\langle\right.\\!{mm}\\!\left.\right|\rho\left|\right.\\!{mm}\\!\left.\right\rangle-\tfrac{1}{d}$
$\displaystyle\ \ -\sum\limits_{\begin{subarray}{c}m\neq m^{\prime}\\!,m\neq
n\\\ n\neq n^{\prime}\\!,n^{\prime}\neq m^{\prime}\\\
\end{subarray}}\tilde{\gamma}_{mm^{\prime}nn^{\prime}}\,\sqrt{\left\langle\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right\rangle\left\langle\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle}\,.$
(A.20)

####  A.III.2 Using only two WF bases

Now, let us consider what happens when we designate one of the WF bases
(labelled by $k$) as the new ‘standard basis’ and use a second WF basis,
labelled by $k^{\prime}$ as a mutually unbiased basis to construct our
fidelity bound. As before, we express these bases w.r.t. to each other as

$\displaystyle\left|\right.\\!{\tilde{m}_{k^{\prime}}}\\!\left.\right\rangle$
$\displaystyle=\,\sum\limits_{n}\left\langle\right.\tilde{n}_{k}\left|\right.\tilde{m}_{k^{\prime}}\left.\right\rangle\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle\,=\,\tfrac{1}{d}\sum\limits_{n,p}\omega^{p(m-n)+p^{2}(k^{\prime}-k)}\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle\,=\,\tfrac{1}{\sqrt{d}}\sum\limits_{n}\,c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}\,\left|\right.\\!{\tilde{n}_{k}}\\!\left.\right\rangle,$
(A.21)

where we have defined

$\displaystyle c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}$
$\displaystyle:=\tfrac{1}{\sqrt{d}}\sum\limits_{p}\omega^{p(m-n)+p^{2}(k^{\prime}-k)}.$
(A.22)

In order to continue, let us recall the formula for quadratic Gauss sums for
odd roots of unity.

$\displaystyle
g(a:d)=\sum_{i=0}^{d-1}\omega^{ai^{2}}=\left(\frac{a}{d}\right)\varepsilon_{d}\sqrt{d},$
(A.23)

where $d$ is odd, $\left(\frac{a}{d}\right)\in\\{-1,0,1\\}$ is the Jacobi
symbol and

$\displaystyle\varepsilon_{d}=\begin{cases}1&\text{if }d\equiv 1\mod 4\\\
i&\text{if }d\equiv 3\mod 4.\end{cases}$ (A.24)

This itself is not of the same form as (A.22), but is a main ingredient in its
evaluation.

In fact (A.22) is of a more general form, which is often denoted as
generalized quadratic Gauss sum:

$\displaystyle G(a,b,c)$ $\displaystyle=\sum\limits_{n=0}^{c-1}e^{2\pi
i\frac{an^{2}+bn}{c}}.$ (A.25)

In our special case, evaluation of such sums is given by the following lemma.

######  Lemma 1.

Let $c$ be odd and $\gcd(a,c)=1$ the solution takes this form:

$\displaystyle
G(a,b,c)=\varepsilon_{c}\sqrt{c}\left(\frac{a}{c}\right)e^{-2\pi
i\frac{\psi(a)b^{2}}{c}},$ (A.26)

where

$\displaystyle\varepsilon_{c}=\begin{cases}1&\text{if }c\equiv 1\mod 4\\\
i&\text{if }c\equiv 3\mod 4.\end{cases}$ (A.27)

$\left(\frac{a}{c}\right)\in\\{1,0,-1\\}$ is the Jacobi symbol (in our case
$1$ or $-1$, because $a$ and $c$ are coprime) and $\psi(a)$ is a number such
that $4\psi(a)a\equiv 1\mod c$.

###### Proof.

Since $c$ is odd and $\gcd(a,c)=1$, numbers $2,4$ and $a$ are all invertible
modulo $c$. Therefore we can “complete the square” by rewriting $an^{2}+bn$ as
$a(n-h)^{2}+k$, where $h=-\frac{b}{2a}$ and $k=-\frac{b^{2}}{4a}$ to get

$\displaystyle G(a,b,c)$ $\displaystyle=\sum\limits_{n=0}^{c-1}e^{2\pi
i\frac{a(n-h)^{2}+k}{c}}\,=\,e^{2\pi
i\frac{k}{c}}\sum\limits_{n=0}^{c-1}e^{2\pi
i\frac{a(n-h)^{2}}{c}}\,=\,e^{-2\pi
i\frac{\psi(a)b^{2}}{c}}\sum\limits_{n=0}^{c-1}e^{2\pi
i\frac{an^{2}}{c}}\,=\,e^{-2\pi i\frac{\psi(a)b^{2}}{c}}g(a:c),$ (A.28)

where the second to last equality follows from the fact that both sums iterate
over the same set $\mathbb{Z}_{c}:=\\{0,\dots,c-1\\}$. ∎

In our case we can set $a=k^{\prime}-k$, $b=(m-n)$, and $c=d$, where $d$ is an
odd prime and $\gcd(k^{\prime}-k,d)=1$ (as both $k\neq k^{\prime}$ and
$k,k^{\prime}\in\\{0,\dots,d-1\\}$). Consequently, all conditions are met and
we have

$\displaystyle c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}$
$\displaystyle:=\tfrac{1}{\sqrt{d}}\sum\limits_{p}\omega^{p(m-n)+p^{2}(k^{\prime}-k)}\,=\,\varepsilon_{d}\left(\frac{k^{\prime}-k}{d}\right)\omega^{-\psi(k^{\prime}-k)(m-n)^{2}}.$
(A.29)

We also need the relation between the two conjugated bases, i.e.,

$\displaystyle\left|\right.\\!{\tilde{m}_{k^{\prime}}^{*}}\\!\left.\right\rangle$
$\displaystyle=\,\sum\limits_{n}\left\langle\right.\tilde{n}_{k}^{*}\left|\right.\tilde{m}_{k^{\prime}}^{*}\left.\right\rangle\,\left|\right.\\!{\tilde{n}_{k}^{*}}\\!\left.\right\rangle\,=\,\tfrac{1}{\sqrt{d}}\sum\limits_{n}\,c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})\hskip
0.5pt*$}}}\,\left|\right.\\!{\tilde{n}_{k}^{*}}\\!\left.\right\rangle.$ (A.30)

We can then follow the exact same steps as in Sec. A.III.1: We first form the
sum over correlated matrix elements in the second basis,

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}\,=\,\sum\limits_{m}\left\langle\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}}\\!\left.\right\rangle\,=\,\tfrac{1}{d^{2}}\\!\\!\sum\limits_{\begin{subarray}{c}n,n^{\prime}\\\
q,q^{\prime}\end{subarray}}\sum\limits_{m}c_{mn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})\hskip
0.5pt*$}}}c_{mn^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}c_{mq}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}c_{mq^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})\hskip
0.5pt*$}}}\left\langle\right.\\!{\tilde{n}_{k}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{q}_{k}\tilde{q}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right\rangle,$ (A.31)

which can again be split up into three terms, i.e.,

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}$
$\displaystyle=\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{1}\,+\,\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{2}\,+\,\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{3},$
(A.32)

where

$\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{1}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m,n}\left\langle\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle\,=\,\tfrac{1}{d},$
(A.33a) $\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{2}$
$\displaystyle:=\,\tfrac{1}{d}\sum\limits_{m\neq
n}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{n}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle,$
(A.33b) $\displaystyle\Sigma^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}_{3}$
$\displaystyle:=\tfrac{1}{d^{2}}\sum\limits_{\begin{subarray}{c}m\neq
m^{\prime},m\neq n\\\ n\neq n^{\prime},n^{\prime}\neq
m^{\prime}\end{subarray}}c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}\left\langle\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle,$
(A.33c)

and one can once more confirm that all other terms vanish. The main difference
now lies in the form of the coefficient
$c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}$,
which, with help of Eq. (A.29) can be written as

$\displaystyle
c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}$
$\displaystyle=\,\sum\limits_{j}c_{jm^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})\hskip
0.5pt*$}}}c_{jn^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}c_{jm}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}c_{jn}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})\hskip
0.5pt*$}}}\,=\,(\varepsilon_{d}^{*})^{2}\varepsilon_{d}^{2}\left(\frac{k^{\prime}-k}{d}\right)^{4}\sum\limits_{j}\omega^{\psi(k^{\prime}-k)[(j-m^{\prime})^{2}{-(j-n^{\prime})^{2}}{-(j-m)^{2}}+{(j-n)^{2}}]}$
$\displaystyle=\omega^{\psi(k^{\prime}-k)({m^{{}^{\prime}2}-m^{2}-n^{{}^{\prime}2}+n^{2}})}\sum\limits_{j}\omega^{\psi(k^{\prime}-k)2j(m-m^{\prime}-n+n^{\prime})}.$
(A.34)

It remains to be noted that
$|c_{mnm^{\prime}n^{\prime}}^{\raisebox{0.0pt}{\tiny{$(k,k^{\prime})$}}}|=|c_{mnm^{\prime}n^{\prime}}|$.
We can hence write the fidelity bound in the usual form

$\displaystyle F(\rho,\Phi)$
$\displaystyle\geq\,\tfrac{1}{d}\sum_{m}\left\langle\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{m}_{k}^{*}}\\!\left.\right\rangle\,+\,\sum\limits_{m}\left\langle\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k^{\prime}}\tilde{m}_{k^{\prime}}^{*}}\\!\left.\right\rangle-\,\tfrac{1}{d}$
$\displaystyle\ \ -\sum\limits_{\begin{subarray}{c}m\neq m^{\prime}\\!,m\neq
n\\\ n\neq n^{\prime}\\!,n^{\prime}\neq m^{\prime}\\\
\end{subarray}}\tilde{\gamma}_{mnm^{\prime}n^{\prime}}\,\sqrt{\left\langle\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}^{\prime}\tilde{n}_{k}^{\prime\hskip
0.5pt*}}\\!\left.\right\rangle\left\langle\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right|\rho\left|\right.\\!{\tilde{m}_{k}\tilde{n}_{k}^{*}}\\!\left.\right\rangle}\,.$
(A.35)

For any two chosen bases, the bound can hence be evaluated in a
straightforward way.

###  A.IV Entanglement of Formation

In this section, we discuss how we bound the entanglement of formation
$(E_{\mathrm{oF}})$ of our bipartite state from the measurement data for any
two MUBs. Here, the $E_{\mathrm{oF}}$ quantifies how many Bell states are
required to convert to a single copy of our high-dimensional entangled state
via local operations and classical communication (LOCC) [47].

Using the uncertainty relations reviewed in Ref. [48], the bound on the
$E_{\mathrm{oF}}$ is given by

$E_{\mathrm{oF}}\geq\log_{2}(d)-\mathcal{H}(A_{1}|B_{1})-\mathcal{H}(A_{2}|B_{2}),$
(A.36)

Figure A.4: Entanglement of formation $(E_{\mathrm{oF}})$ bounds. Using
measurements in two mutually unbiased bases, bounds for the entanglement of
formation are determined through Eq. (A.36) for dimensions up to $d=97$
(orange points). Error bars are calculated assuming Poissonian counting
statistics and error propagation via a Monte-Carlo simulation of the
experiment. The theoretical upper bound given by $log_{2}(d)$ is depicted by
blue crosses. The maximum value achieved in our experiment
($E_{\mathrm{oF}}=4.0\pm 0.1$ ebits for $d=31$) is indicated by dotted lines.

where $\mathcal{H}(A_{i}|B_{i})$ is the conditional Shannon entropy for the
$i$-th mutually unbiased basis (MUB), i.e.,

$\displaystyle\mathcal{H}(A_{i}|B_{i})$
$\displaystyle=\mathcal{H}(\\{\rho^{(i)}_{jk}\\})-\mathcal{H}(\\{\rho^{(i)}_{j}\\}),\quad\text{for
}i=1,2,$ (A.37)

with

$\displaystyle\rho^{(i)}_{jk}$
$\displaystyle=\langle{jk}|{\rho}|{jk}\rangle_{i},\quad\rho^{(i)}_{j}=\sum_{k}\langle{jk}|{\rho}|{jk}\rangle_{i}.$
(A.38)

Since we know the terms $\\{\rho^{(i)}_{jk}\\}$ are related to coincidences
measured in $i$-th MUB through Eq. (A.3), the expression in (A.36) can be
evaluated for any pair of MUB measurements. Figure A.4 shows the
$E_{\mathrm{oF}}$ achieved in our experiment as a function of dimension,
compared with the theoretical maximum of $log_{2}(d)$. As can be seen, the
$E_{\mathrm{oF}}$ increases until approximately $d=31$, where it reaches a
maximum value of $4.0\pm 0.1$ ebits. As the dimension is increased further,
experimental imperfections and the inherent dimensionality limits of the
source bring this number back down.