# Quantifying high dimensional entanglement with two mutually unbiased bases

Paul Erker  Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona,
Spain  Faculty of Informatics, Università della Svizzera italiana, Via G.
Buffi 13, 6900 Lugano, Switzerland  Facoltà indipendente di Gandria, Lunga
scala, 6978 Gandria, Switzerland  Mario Krenn  Vienna Center for Quantum
Science and Technology (VCQ), Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria.  Institute for Quantum Optics and
Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3,
A-1090 Vienna, Austria.  Marcus Huber  Group of Applied Physics, University of
Geneva, 1211 Geneva 4, Switzerland  Universitat Autonoma de Barcelona, 08193
Bellaterra, Barcelona, Spain  ICFO-Institut de Ciencies Fotoniques, 08860
Castelldefels, Barcelona, Spain  Institute for Quantum Optics and Quantum
Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090
Vienna, Austria.

###### Abstract

We derive a framework for quantifying entanglement in multipartite and high
dimensional systems using only correlations in two unbiased bases. We
furthermore develop such bounds in cases where the second basis is not
characterized beyond being unbiased, thus enabling entanglement quantification
with minimal assumptions. Furthermore, we show that it is feasible to
experimentally implement our method with readily available equipment and even
conservative estimates of physical parameters.

Entanglement has long been recognized as the key concept that takes quantum
communication beyond the classically possible. It finds applications in secure
key distribution [1], super-dense coding [2] and improves communication
capacities in a general sense [3].
So far, most photonic implementations rely on two-dimensional degrees of
freedom, limiting the capacity of each exchanged photon to one bit of
information. Recent progress in understanding high dimensional degrees of
freedom of photons, however, revealed the high capacity of entanglement in
photon pairs naturally emerging from down-conversion processes. Prominent
examples are path entangled photons in waveguides [4, 5, 6, 7], access to the
entangled angular momentum of photon pairs [8, 9, 10, 11, 12, 13, 14, 15] and
energy-time binned photons [16, 17, 18]. Such high dimensional entanglement
endows each photon pair with more shared information, greatly improving the
efficiency of known protocols and enabling secure quantum communication at
noise levels that would be prohibitive for qubit systems [19, 20, 21, 22, 23].
A lot of attention has been devoted to proving entanglement for such high
dimensional systems [24], revealing the underlying dimensionality of
entanglement [6, 11, 15] or generally characterizing the potential for
accommodating many dimensions [25, 26, 27, 28, 29]. These initial
investigations reveal the clear potential of the underlying systems, but in
order to properly quantify the advantage provided one needs to actually
quantify the number of entangled bits (e-bits, which is the number of two-
dimensional Bell states that are necessary to produce the state) shared by the
photons. The dimensionality of entanglement (i.e. the Schmidt rank) is given
by the rank of the reduced density matrix and denotes the minimal dimension
that is needed to reproduce the correlations of the state, whereas the e-bits
are given by the entropy of that reduction, giving a clear operational meaning
to the amount of information encoded in the entanglement. For example
$|\psi\rangle=\frac{1}{\sqrt{2+k\epsilon^{2}}}\left(|0,0\rangle+|1,1\rangle+\epsilon\sum_{i=1}^{k}|i,i\rangle\right)$
with $\epsilon$ being very small illustrates this difference. While this state
is $k+2$-dimensional entangled (i.e. Schmidt rank $k+2$), it takes only little
more than a single Bell state to create it (which shows up in its entanglement
entropy being close to one as long as $k\epsilon^{2}\ll 1$). On the other
hand, this state’s Schmidt number is very fragile with respect to noise, so
experimentally certified Schmidt-numbers need more entangled bits to be
robust. Calculating entangled bits for general states, however, is a
notoriously difficult task [30] and even at full access to a reconstructed
density matrix there is no known method for computing this number in an
efficient way (the best known algorithm just for deciding whether it is
nonzero is exponential in the system’s dimension [31]). While entanglement
witnesses in general only provide an answer to the question whether a given
state exhibits entanglement [32], their actual value can also be used to
quantify the amount of entanglement [33, 34, 35, 36, 37, 38, 30].
Unfortunately, generic entanglement witnesses in this context require a number
of local measurement bases that scales with the system size, thus increasing
the complexity rapidly with a growing number of degrees of freedom. Using
mutually unbiased bases on the other hand provides a means of revealing
entanglement with just two local measurements [39], a fact that has also been
exploited for high dimensional experiments [40, 41, 42].

In this work we combine the advantage of both approaches and quantify high
dimensional entanglement purely from correlations in two mutually unbiased
bases (MUBs), and show that the resulting quantification can readily be
implemented experimentally with modern cameras. In fact, for sufficiently pure
states the entire high dimensional entanglement can be certified with this
minimal access. Since with only two measurement settings the word mutually
seems superfluous we will sometimes only refer to unbiased bases. A natural
candidate for two such bases are discretised position and momentum
correlation. They are known to be readily accessible at a high quality and
have thus been used to ascertain entanglement before [43, 44, 45]. Therefore,
to showcase our theorem we derive how modern cameras and lenses (which perform
a Fourier transformation in the far field/focus) can be harnessed to quantify
the spatial correlations in down-conversion photons. After treating the
bipartite case we move on to multipartite systems and show how to quantify
multipartite entanglement using two local unbiased measurement settings.
Before we do, let us introduce the relevant concepts. Sets of basis vectors
$\\{|v_{i}^{k}\rangle\\}$ are called (mutually) unbiased, if they are both
orthonormal $\langle v_{i}^{k}|v_{j}^{k}\rangle=\delta_{ij}$ and their
overlaps are unbiased $|\langle
v_{i}^{k}|v_{j}^{k^{\prime}}\rangle|^{2}=\frac{1}{d}$. MUBs can be used for an
efficient tomography [46], cryptography protocols [47] and in prime power
dimensions there exist exactly $d+1$ such bases (see [48] for a review and
further applications). It is still an unsolved problem how many MUBs exist in
general (the smallest example being $d=6$). Our intention however is to make
use of only two such bases, which always exist for any dimension $d$.
To quantify entanglement we use entanglement of formation (EOF) [49], which
for pure states quantifies the asymptotic conversion rate between maximally
entangled states and the quantum state under investigation. For mixed states
its regularised asymptotic version quantifies precisely this entanglement cost
as a rate of ”‘target states per Bell state”’. In other words, given $N$
copies of qubit Bell states, how many copies $k$ of the target state can we
deterministically create using only local operations and classical
communication (LOCC)? This asymptotic conversion rate $\frac{N}{k}$ is then
found in the limit $N\to\infty$. For pure states it corresponds to the entropy
of entanglement
$E_{oF}(|\psi_{AB}\rangle):=S(\text{Tr}_{A/B}(|\psi_{AB}\rangle\langle\psi_{AB}|))$,
where $S(\rho)=-\text{Tr}(\rho\log(\rho))$. For general quantum states
Entanglement of Formation can be evaluated via a convex roof construction as
the minimal average entanglement across all possible decompositions
$E_{oF}(\rho)=\inf_{\mathcal{D}(\rho)}\sum_{i}p_{i}E_{oF}(|\psi_{i}\rangle)$.
Even if the whole state $\rho$ is known exactly, it is a hard problem even to
decide whether the measure is nonzero [50], but we now want to find a lower
bound from only two measurement outcomes.
The central figure of merit will be similar to the ones developed in [39],
such that we can easily apply our methods to already existing experimental
data (e.g. from [41, 40]). For bipartite systems the existing method makes use
of the sum over all diagonal correlations in $m$ different MUBs, i.e.
$C_{m}(\rho)=\sum_{k=1}^{m}\sum_{i=0}^{d-1}\langle
v_{i}^{k}(v_{i}^{k})^{*}|\rho|v_{i}^{k}(v_{i}^{k})^{*}\rangle\,.$ The simplest
example are polarisation encoded states, for which $m=3$ MUBs can be used to
compute $C_{3}=\frac{\langle H,H\,\rangle+\langle V,V\,\rangle+\langle
D,D\,\rangle+\langle A,A\,\rangle+\langle L,L\,\rangle+\langle
R,R\,\rangle}{\langle H,H\,\rangle+\langle H,V\,\rangle+\langle
V,H\,\rangle+\langle V,V\,\rangle}$, where $\langle X,Y\,\rangle$ denotes
coincidence counts in $X$ and $Y$ for the first and second photon,
respectively and $H,V,D,A,L,R$ denote the six different polarisation states.
This quantity is known to be bounded for separable states by
$C_{m}(\rho_{sep})\leq 1+\frac{m-1}{d}$, whereas maximally entangled states
can reach a value of $m$ [39]. We are interested in the case $m=2$ as it’s the
minimal number of MUBs that can be used to verify entanglement, and it is
experimentally the simplest case to access. Importantly, for large classes of
states (and all pure entangled states), $C_{2}$ is already sufficient to
detect entanglement. But how much can we say about the ability to give
quantitative bounds on entanglement measures, such as EOF using only two
bases?

A motivating example is provided in an idealised two qubit case: The
eigenstates of the Pauli matrices $\sigma_{i}$ form MUBs. If we now measured
correlations in such as e.g. $\langle\sigma_{x}\otimes\sigma_{x}\rangle$ and
$\langle\sigma_{y}\otimes\sigma_{y}\rangle$ and found both of these values to
be (close to) $-1$, positivity of the density matrix implies that
$\langle\sigma_{z}\otimes\sigma_{z}\rangle\approx-1$. So despite only having
measured two out of the three defining correlations of a qubit Bell state, we
can infer that the state is indeed close to a Bell state and thus it’s
entanglement is close to $1$. Indeed, such an example is given in Ref. [51]
along with a semi-definite programming characterisation (SDP) to evaluate the
convex roof extended linear entropy, even if the values are not close to $-1$.
This further motivates two questions that naturally follow from this rather
idealised setting:

  * •

Can we make these considerations analytical, noise robust and quantitative?

  * •

Are two measurements still sufficient for any dimension?

We affirmatively answer both questions by introducing the following quantity

$\begin{split}B(\rho)=N\Big{[}d\left(\sum_{i=0}^{d-1}\langle
v_{i}^{2}(v_{i}^{2})^{*}|\rho|v_{i}^{2}(v_{i}^{2})^{*}\rangle\right)-1-\sum_{\begin{subarray}{c}m\neq
n,m\neq l\\\ l\neq o,n\neq o\end{subarray}}\sqrt{\langle
v_{m}^{1}v_{n}^{1}|\rho|v_{m}^{1}v_{n}^{1}\rangle\langle
v_{l}^{1}v_{o}^{1}|\rho|v_{l}^{1}v_{o}^{1}\rangle}\\\ -\sum_{i\neq
j}\sqrt{\langle v_{i}^{1}v_{j}^{1}|\rho|v_{i}^{1}v_{j}^{1}\rangle\langle
v_{j}^{1}v_{i}^{1}|\rho|v_{j}^{1}v_{i}^{1}\rangle\,}\Big{]}\end{split}$ (1)

where we have used $N=\sqrt{\frac{2}{d(d-1)}}$, and the additional terms can
be recorded alongside with the original measurements (without the need of
adjusting the local measurement settings). Our main result is the fact that
$B(\rho)$ is indeed a direct lower bound to the generalised concurrence and
thus a lower bound on the Entanglement of Formation can be easily computed
whenever $B(\rho)\geq 0$, via

$\displaystyle E_{oF}(\rho)\geq-\log(1-\frac{B(\rho)^{2}}{2})\,.$ (2)

The details of this derivation can be found in the appendix (2). We want to
stress that this result holds for any choice of the two MUBs. One immediate
consequence is that with just two global measurement settings we can certify
that the maximally entangled state
$|\phi^{+}\rangle=\frac{1}{\sqrt{d}}(\sum_{i=0}^{d-1}|ii\rangle)$ has an
entanglement of formation of $E_{oF}=\log(d)$, i.e. all the entanglement can
be quantified exactly through
$B(|\phi^{+}\rangle\langle\phi^{+}|)=\sqrt{2(1-\frac{1}{d})}$.
While at first glance it seems surprising that an entanglement of $\log(d)$
can be certified using only two measurement settings, there is of course a
trade-off. The extra terms required make the bounds more sensitive to noise.
So in order to gauge the practical usefulness it will be essential to study
the performance of the bound in experimentally feasible settings with all
sources of noise taken into account. But before we continue our discussion on
experimental applicability and an analysis on the noise robustness we now give
a brief excursion into multipartite variants of this theorem. Quantifying
multipartite entanglement is a notoriously hard task, as there is no unique
”currency”, from which every state can be created via LOCC. Recent progress
has been made using maximally entangled sets [52, 53], however it remains an
open problem to evaluate measures based on that concept on many high
dimensional parties. We will make use of a more crude classification,
capturing necessary resources, but leaving open the sufficiency entirely. Here
we can make use of a multipartite generalization of entanglement of formation
proposed in [36, 54, 55]. The operational interpretation is simply the minimal
necessary average entanglement across every cut for creating this state via
LOCC. It does not reveal a deeper structure, but is nonzero for every
genuinely multipartite entangled state and zero if there exists a
decomposition into at least biseparable states. Formally it can be defined as

$\displaystyle
E_{GME}:=\inf_{\mathcal{D}(\rho)}\sum_{i}p_{i}\min_{A_{i}}S(\text{Tr}_{A_{i}}(|\psi_{i}\rangle\langle\psi_{i}|))\,.$
(3)

Figure 1: A: The proposed experimental setup. A laser produces pairs of
spatially entangled photons with wavelength of 800nm in a SPDC process in the
nonlinear ppKTP crystal. After the pump beam is removed by a dicroic mirror,
the photon-pair is deterministically separated with a calcit polarizer. The
two parts of the beam then go to different positions at a camera. Depending on
the lens configuration, the photons are detected in the position-basis (if the
two imaging lenses (IL) are in the beam path) or in the momentum-basis (if
only the Fourier lens (FL) is used). B: Here we showcase two different effects
that predominantly reduce the certifiable e-bits, white noise and cross-
correlations between different pixels. The purple and orange graphs show
different weighting of the individual terms. Purple depicts a beam with
$\sigma=100$ pixels, and orange $\sigma=25$ pixels. $p$ stands for the
probability of white noise or cross-correlations, respectively. C: The
considered areas of pixels at the screen of the ICCD camera as considered in
the calculations. Due to the significant impact of white noise, it is optimal
to use only small parts of the camera.

While, again, two mutually unbiased bases can suffice for some states to
quantify all multipartite entanglement exactly, the multitude of potential
correlation structures giving rise to Genuine Multipartite Entanglement (GME)
makes the construction of one universal criterion impossible. While the
criteria themselves constitute genuine lower bounds and need no assumption
about the underlying state, their capability of revealing multipartite
entanglement will be tailored to specific classes of states. In fact the same
is true for bipartite systems, however, with a change of local basis every
pure state can be written as
$|\psi\rangle=\sum_{i=0}^{r_{S}}\lambda_{i}|v_{i}v_{i}^{*}\rangle$. This
Schmidt decomposition is not possible for multipartite states, so our criteria
will have to be adapted to the target state in question. In the appendix we
therefore describe a generic recipe how this can be accomplished and
illustrate it with known states of a specific dimensionality structure [56],
that can be readily produced in optical setups [57].

In order to explore the feasibility of our criterion we come back to bipartite
systems and the issue of noise resistance. As mentioned before, the generality
of bounding entanglement through any arbitrary pair of MUBs, without any
information on the relative phases comes at the price of high noise
sensitivity. The amount of noise that can be tolerated, of course, depends on
the nature of the noise itself. To start, let us analyse two paradigmatic
noise models and how they theoretically impact the bound. The first is
dephasing noise, i.e. a noise model where the intended (maximally entangled)
target state is mixed with its dephased version:

$\rho_{dp}(p)=p|\phi^{+}\rangle\langle\phi^{+}|+\frac{(1-p)}{d}(\sum_{i=0}^{d-1}|ii\rangle\langle
ii|)\,.$ (4)

In this case $B(\rho_{dp})=N(d(p+\frac{(1-p)}{d^{2}}-1)$, which implies
entanglement detection up to a noise threshold of $p_{crit}=\frac{1}{d+1}$,
which actually increases noise resistance with the dimension $d$. The other
extreme case is of course white noise, which provides the toughest challenge
for entanglement detection.

$\rho_{wn}(p)=p|\phi^{+}\rangle\langle\phi^{+}|+\frac{(1-p)}{d^{2}}\mathbbm{1}\,.$
(5)

Here $B(\rho_{wn})=N(d(p+\frac{(1-p)}{d^{2}}-1-\frac{d-2}{d}(d-1)^{2}(1-p))$,
which leads to a decreasing noise resistance of
$p_{crit}=\frac{d^{2}-3d+2}{d^{2}-2d+2}$ which approaches $1$ for
$d\rightarrow\infty$. Both of these noise models are somewhat unrealistic,
however, and the true practicality can only be ascertained through an
experimental proposal.

We therefore consider a scenario where a pair of photons illuminate a single-
photon sensitive camera. With simple optics, we can access two mutual unbiased
bases (the position-basis and the momentum-basis), which we will use to
estimate the strength and noise-dependence of the criterion. This is the ideal
testbed, as it also plays on the strength of not needing to know the exact
phase relation between the mutually unbiased bases.

The photon pairs are created in a nonlinear spontaneous parametric down-
conversion (SPDC) crystal [58], which is the working horse in quantum optical
experiments. The two photons, which are deterministically separated (for
instance when they have opposite polarization in a type-II SPDC, or different
wavelengths), illuminate two different regions of an ICCD (intensified charge-
coupled device) camera. ICCDs have high detection efficiencies (20% for 800nm,
up to 50% for green) and have a very low dark count rate, which makes it
possible to identify single photons and photon pairs. (This is in contrast to
electron multiplying charge-coupled device (EMCCD) cameras, which have high
efficiencies but high dark count rates. There, quantum correlations can be
inferred via averaging techniques [59, 60]). In the first measurement, the
crystal is imaged onto the camera, which results in strong position
correlations. The second measurement is performed using a lens to go to the
Fourier-plane of the crystal, which results in strong momentum correlations -
thus in strong position anti-correlations. The strength of spatial
correlations can be quantified with the Federov-Ratio $F$, which is the ratio
between the marginal and conditional probability [61, 58]. Experimentally, one
can reach very large values - realistically $F$ could be $25$ [58], which is
the value we calculate with .

To estimate the detectable number of e-bits from (2), one has to consider
several effects that reduce the extractable e-bits: First, dark counts are
roughly evenly distributed over the camera pixels, resulting in white noise.
Secondly, cross-correlations between different states further reduce the
certifiable entanglement. And thirdly, non-equal weighting between different
modes are influencing the final result as well. The interplay of these effects
can be seen in fig.(1B).

Different camera configurations will lead to a differently sized influence of
these effects. Larger regions grouped together reduce cross-talk, but will
increase dark counts per region. However, with careful considerations of the
influencing parameters, fig.(1B), one can nonetheless find configurations that
can certify more than $3$ e-bits, with only two MUBs. These values are reached
entirely without phase characterization of the MUBs. If we restrict ourselves
to areas of $7\times 7$ pixels, we find that the optimal number of areas to be
considered is $10$, where we find a noise-level of $0.6\%$. If we use more
areas, the white noise probability increases (as more pixels are considered).
With this information, we find a good configuration which reduces as much as
possible the negative influence of the other two effects. Specifically, we
introduce large empty areas between the considered pixels in order to reduce
cross-talk. While this will reduce the number of photons detected per unit of
time proportional to the area not counted, it will strongly increases the
certified e-bits. This is because of the assumed large single-photon beam,
where the probability to find a photon in any of the different areas is very
similar. In the end, we find that this configuration leads to $2.4$ e-bits,
mainly reduced due to white noise. If we decrease the number of pixels further
to $3\times 3$ areas, we find $3.05$ e-bits. For the reader’s convenience we
present a detailed step-by-step calculation of the $d=3$ case (i.e. $3$ areas)
in the appendix.

Except for the careful design of the optimal areas for correlations (which can
be done after the data is collected), another experimental challenge is the
stabilization of the setup for longer times. The ICCD camera has a limit of
roughly $10$ images per second. As one wants to reduce multi-photon events (as
they would increase the white-noise fraction), the photon production rate has
to be set at a very low level to roughly $1$ Hz. Only one pair in $100$
arrives at the considered areas - together with the detection efficiency of
both photons, it results in roughly $10^{-4}$ Hz of detected photon pairs. In
order to collect a suitable amount of correlated photons, one could expect the
measurement time to be $100-200$ hours. This caveat could be addressed in
practice by using compressed sensing approaches, as e.g. in [45]. An
alternative way to improve the extractable e-bits is the shift from the
standard 800nm photon pairs (for which the camera has an efficiency of 20%) to
green entangled photons [62] which can be detected with an efficiency of 50%.
Also the useage of novel sCMOS cameras [63, 64], which are faster and have
lower noise compared to the ICCD camera considered here, will improve the
measurement time and extractable ebits. This concludes the analysis of the
feasibility of using only two MUBs to experimentally quantify entanglement.

While in principle any amount of bipartite entanglement can be certified for
pure states, realistic noise assessment shows that with this limited number of
local observables and current technology we can still certify more than three
times the entanglement of a perfect qubit entangled pair. We hope that this
number can still be increased through rapidly developing camera technology or
possibly even through a more clever detection design. While this method serves
as a technique to experimentally prove the potential of down conversion
sources for high dimensional quantum communication, there are still many open
questions. Especially quantum key distribution protocols display a strong
trade-off between security and implementability. In particular fully device
independent quantum key distribution requires (almost) loophole free Bell
inequality violation, a feat that is hard to achieve at sensible key rates.
Prepare and measure schemes can be easily attacked if the source or
measurement devices are hacked. It may be possible to use entanglement
quantifiers, such as the one presented here, to certify security of the source
for high dimensional QKD, interpolating between prepare and measure and fully
device independent schemes. Another path to pursue would of course be the
inclusion of further measurements to strengthen the resistance to noise, which
in our experimental proposal would also require a precise phase control in the
Fourier-plane pixels. Our method can also be implemented for other quasi-
continuous variable entangled systems. In particular it might be particular
useful for wavelength-entangled systems, where access to two MUBs (wavelength
and its complementary, time), is possible - but higher numbers of MUBs are not
easily accessible. There our method might be able to certify the great
potential of wavelength-entangled quantum systems[65, 66, 67].

_Acknowledgements_ We are grateful to Robert Boyd, Anton Zeilinger, Ebrahim
Karimi, Robert Fickler and Mehul Malik for discussions that started and shaped
this project. We thank Claude Klöckl for productive discussions at LIQUID and
Robert Fickler, Armin Hochrainer and Nicolai Friis for helpful feedback. MH
and PE were supported by the European Commission (STREP “RAQUEL”), by the
Spanish MINECO, projects FIS2008-01236 and FIS2013-40627-P, with the support
of FEDER funds, and by the Generalitat de Catalunya CIRIT, project
2014-SGR-966. PE furthermore acknowledges funding from the Swiss National
Science Foundation (SNF) and the National Centres of Competence in Research
”Quantum Science and Technology” (QSIT). MH furthermore acknowledges funding
from the Juan de la Cierva fellowship (JCI 2012-14155), the Swiss National
Science Foundation (AMBIZIONE PZ00P2$\\_$161351) and the Austrian Science Fund
(FWF) through the START project Y879-N27. MK acknowlegdes support from the
European Research Council (SIQS Grant No. 600645 EU-FP7-ICT) and the Austrian
Science Fund (FWF) with SFB F40 (FOQUS).

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##  1 Analysing the noise robustness

First we analyse the noise robustness of the criterion presented in the main
text in a realistic experimental scheme. Because of three different
experimental effects, the detected state differs from a maximally entangled
high-dimensional pure state: Unequal weighting due to gaussian shape of the
photons’ spatial distribution, cross correlations due to non-perfect
correlations, and white noise due to accidental dark counts of the camera.

###  1.1 Unequal weighting

The unequal weighting of the different pixel-areas is mainly due to the
Gaussian character and the discretisation of the grid. The state can can be
written as

$\displaystyle\left|\psi\right\rangle=N_{1}\Big{(}\sum_{x=0}^{x_{max}}w_{x}\overline{\left|x,x\right\rangle}\Big{)}$
(6)

where $w_{x}$ can be found numerically and $\overline{\left|x,x\right\rangle}$
is explained in the next section. The weighting could be flattened by
adjusting the size of the discretisation. However, as we will show, the white
noise contributions is significant for large number of pixel-areas.

###  1.2 Cross correlations

Due to physical constraints in the production of photon pairs (such as the
size of the pump beam or the length of the crystal), the spatial correlations
can not be infinitely strong. It results in cross-correlations between pixel
areas that reduce the entanglement.

We consider only first-order contributions from cross correlations (which
means, only direct neighboring pixel areas contribute). The cross-correlations
depend on the geometry of the considered pixels. In Fig. 1C in the main text,
one can see that pixel all pixels have two direct neighbors. In general, one
finds:

$\displaystyle\overline{\left|x,x\right\rangle}=N_{2}\Big{(}c_{x}\left|x,x\right\rangle+\sum_{y\in
NB(x)}c_{x,y}\big{(}\left|x,y\right\rangle+\left|y,x\right\rangle\big{)}\Big{)}$
(7)

where $c_{x}$ and $c_{x,y}$ are found numerically, and $NB(x)$ is the set of
neighbors of $x$. One could significantly reduce the off-correlations by
introducing unobserved pixel-lines between the $N\times N$-areas, as shown in
Fig. 1C in the main text.

###  1.3 White Noise

The dominating source of white noise comes from photon loss and from dark
counts introduced in the measurement of the ICCD camera. As the loss of the
photons and the dark counts are evenly distributed over the camera area, we
can model them as following:

$\displaystyle\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{d^{2}}\mathbbm{1}_{d^{2}}\,,$
(8)

where $|\psi\rangle$ is defined in eq. (6).

The white noise comes from dark counts, and from multi-pair detection from the
crystal. This is not negligible, because the integration time is 0.1sec. For a
given setting of dark count-rate, detector-efficiency and used area of the
ICCD, there is an optimal value of the photon-pair rate P (which is very low,
roughly $10^{-2}$ per seconds).

We only consider camera images with exactly two photons, one in region A and
one in region B. All other events are rejected. Such two photon events can
happen in the following ways:

  1. 1.

Two dark counts; one in Region 1 (where photon A usually appears), one in
Region 2

  2. 2.

One dark count, one real count

  3. 3.

Two real counts

Now a simple calculation shows:

all counts $\displaystyle=\bar{P}\left(D_{1}D_{2}\right)+$ (9)
$\displaystyle+P^{1}\left(D_{1}D_{2}\bar{\epsilon_{1}}\bar{\epsilon_{2}}+D_{1}\epsilon_{2}\bar{D_{2}}\bar{\epsilon_{1}}+\epsilon_{1}D_{2}\bar{D_{1}}\bar{\epsilon_{2}}+\epsilon_{1}\epsilon_{2}\bar{D_{1}}\bar{D_{2}}\right)+$
$\displaystyle+P^{2}\left(D_{1}D_{2}\bar{\epsilon_{1}}^{2}\bar{\epsilon_{2}}^{2}+2D_{1}\epsilon_{2}\bar{D_{2}}\bar{\epsilon_{1}}^{2}\bar{\epsilon_{2}}+2\epsilon_{1}D_{2}\bar{D_{1}}\bar{\epsilon_{2}}^{2}\bar{\epsilon_{1}}+4\epsilon_{1}\epsilon_{2}\bar{D_{1}}\bar{D_{2}}\bar{\epsilon_{1}}\bar{\epsilon_{2}}\right)+$
$\displaystyle+...$
$\displaystyle=\sum_{n=0}^{\infty}P^{n}\left(D_{1}D_{2}\bar{\epsilon_{1}}^{n}\bar{\epsilon_{2}}^{n}+nD_{1}\epsilon_{2}\bar{D_{2}}\bar{\epsilon_{1}}^{n}\bar{\epsilon_{2}}^{n-1}+n\epsilon_{1}D_{2}\bar{D_{1}}\bar{\epsilon_{2}}^{n}\bar{\epsilon_{1}}^{n-1}+n^{2}\epsilon_{1}\epsilon_{2}\bar{D_{1}}\bar{D_{2}}\bar{\epsilon_{1}}^{n-1}\bar{\epsilon_{2}}^{n-1}\right)$

where the last sum has a simple closed form and $\bar{P}$ is the probability
that no photon-pair is created. The counts we are interested in are two real
counts, without dark counts:

good counts
$\displaystyle=P^{1}\left(\epsilon_{1}\epsilon_{2}\bar{D_{1}}\bar{D_{2}}\right)$
(10)

$\displaystyle p_{good}=\frac{\textnormal{good counts}}{\textnormal{all
counts}}$ (11)

We use $D_{1}=D_{2}$ (dark counts in the two areas are the same) and
$\epsilon_{1}=\epsilon_{2}$ (the efficiency in the two areas are the same).
The white-noise probability is $p_{whitenoise}=1-p_{good}$.

##  2 Detailed derivation for bipartite systems

We now want to present a detailed derivation of the fact that two MUB
measurements can quantify entanglement, even without any knowledge about the
detailed structure of the second MUB. All one needs is to make sure that the
outcomes are indeed unbiased. We start the derivation by reminding the reader
of the lower bounds for the concurrence (i.e. the square root of the linear
entropy) developed in [54, 55] as

$\displaystyle I:=\sqrt{\frac{2}{d(d-1)}}\left(\sum_{m\neq
n}\underbrace{|\langle mm|\rho|nn\rangle|}_{I_{1}}-\underbrace{\sqrt{\langle
mn|\rho|mn\rangle\langle
nm|\rho|nm\rangle}}_{I_{2}}\right)\leq\inf_{\mathcal{D}(\rho)}\sum_{i}p_{i}\sqrt{(2(1-\text{Tr}({\rho_{A}^{i}}^{2}))}\,.$
(12)

To get a bound on entanglement of formation we can use the relation between
the linear entropy and the Renyi 2-entropy
$S_{2}(\rho)=-\log(\text{Tr}(\rho^{2}))$ and the fact that the family of Renyi
entropies is monotonically decreasing in $\alpha$, i.e. that for
$S_{\alpha}:=\frac{1}{1-\alpha}\text{Tr}(\rho^{\alpha})$ it holds that
$S_{\alpha}\geq S_{\beta}\,\forall\alpha\leq\beta$. This directly implies that
entanglement of formation, defined as

$\displaystyle E_{oF}:=\inf_{\mathcal{D}(\rho)}\sum_{i}p_{i}S(\rho_{A}^{i})$
(13)

is directly lower bounded by

$\displaystyle E_{oF}\geq-\log(1-\frac{I^{2}}{2})$ (14)

So the task at hand is to experimentally estimate the bound $I$ with only two
mutually unbiased measurements. The first thing to notice is that $I_{2}$ is
directly accessible from correlations in the first basis, i.e. choosing
$\\{|v_{1}^{i}\rangle\\}=\\{|i\rangle\\}$

$\displaystyle\langle mn|\rho|mn\rangle=\frac{N_{m,n}}{\sum_{i,j}N_{i,j}}\,,$
(15)

where $N_{m,n}$ is just the total number of correlated clicks recorded between
$m$ on Alice’s side and $n$ on Bob’s side. Since the term is strictly negative
it is desirable to have these ”wrong” correlations as suppressed as possible
and thus define the basis in terms of the most correlated elements between
Alice and Bob to be labeled by the same numbers $m$.
Now comes the more tricky part: Estimating the total number of coherences in
the first term of $I$. First we notice that to be mutually unbiased to the
computational basis every overlap $|\langle
v^{2}_{i}|j\rangle|^{2}=\frac{1}{d}$, that means we can write the second basis
as

$\displaystyle|v_{k}\rangle:=\frac{1}{\sqrt{d}}\sum_{m=0}^{d-1}e^{-i\phi_{m}^{k}}|m\rangle\,.$
(16)

Now we can evaluate a specific sum of correlations in the mutually unbiased
basis:

$\displaystyle\Sigma:=C_{2}-C_{1}=\sum_{k}\langle
v_{k}v_{k}^{*}|\rho|v_{k}v_{k}^{*}\rangle=\frac{\sum_{k}N_{k,k}}{\sum_{i,j}N_{i,j}}$
(17)

we first notice that
$|v_{k}v_{k}^{*}\rangle=\frac{1}{d}\sum_{m,n}e^{-i(\phi_{m}^{k}-\phi_{n}^{k})}|mn\rangle$,
such that

$\displaystyle\Sigma=\frac{1}{d^{2}}\sum_{k}\sum_{m,n,l,o}e^{i(\phi_{m}^{k}-\phi_{n}^{k}+\phi_{o}^{k}-\phi_{l}^{k})}\langle
mn|\rho|lo\rangle$ (18)

Now we can use that
$\sum_{m=0}^{d-1}e^{i(\phi_{m}^{k}-\phi_{m}^{k^{\prime}})}=0\,\forall\,k\neq
k^{\prime}$ due to the fact that $\langle
v_{k}^{2}|v_{k^{\prime}}^{2}\rangle=0\,\forall\,k\neq k^{\prime}$. The sum
$\Sigma$ can be split in three terms
$\Sigma=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}$:

  * •

$m=l$ and $n=o$: This amounts to
$\Sigma_{1}=\frac{1}{d}\underbrace{\sum_{m,n}\langle
mn|\rho|mn\rangle}_{=\text{Tr}(\rho)=1}=\frac{1}{d}$

  * •

$m=n$ and $l=o$: This is exactly the desired sum, i.e.
$\Sigma_{2}=\frac{1}{d}\sum_{m\neq l}\langle mm|\rho|ll\rangle$

  * •

For the term $m=l$ and $n\neq o$:Here we find a pre-factor of
$\sum_{k=0}^{d-1}e^{i(\phi_{o}^{k}-\phi_{n}^{k})}=0$. The same for $m=n$ and
$l\neq o$, $n=o$ and $m\neq l$ and $l=o$ and $m\neq n$.

  * •

The remaining terms that do not vanish fulfill with $m\neq n$, $m\neq l$,
$n\neq o$ and $l\neq o$. terms for which
$\sum_{k}e^{i(\phi_{m}^{k}-\phi_{n}^{k}+\phi_{o}^{k}-\phi_{l}^{k})}=c_{m,n,l,o}$
yield $\Sigma_{3}=\frac{1}{d^{2}}\sum_{\begin{subarray}{c}m\neq n,m\neq l\\\
l\neq o,n\neq o\end{subarray}}c_{m,n,l,o}\Re e[\langle mn|\rho|lo\rangle]$

Now we can make use of additional experimental data taken in the first basis
and use the Cauchy-Schwarz-inequality to show

$\displaystyle d\Sigma_{3}-\sum_{\begin{subarray}{c}m\neq n,m\neq l\\\ l\neq
o,n\neq o\end{subarray}}\sqrt{\langle mn|\rho|mn\rangle\langle
lo|\rho|lo\rangle}\leq 0$ (19)

Finally since $\Re e[z]\leq|z|$ we can now state the main result here being:

$\displaystyle I_{1}\geq
d\left(\Sigma-\frac{1}{d}-\frac{1}{d}\sum_{\begin{subarray}{c}m\neq n,m\neq
l\\\ l\neq o,n\neq o\end{subarray}}\sqrt{\langle mn|\rho|mn\rangle\langle
lo|\rho|lo\rangle}\right)\,.$ (20)

and thus

$\displaystyle I_{1}-I_{2}\geq
d\left(\Sigma-\frac{1}{d}-\frac{1}{d}\sum_{\begin{subarray}{c}m\neq n,m\neq
l\\\ l\neq o,n\neq o\end{subarray}}\sqrt{\langle mn|\rho|mn\rangle\langle
lo|\rho|lo\rangle}\right)-\sum_{m\neq n}\sqrt{\langle mn|\rho|mn\rangle\langle
nm|\rho|nm\rangle}\,,$ (21)

such that we finally arrive at

$\displaystyle E_{oF}\geq-\log(1-\frac{B(\rho)^{2}}{2})\,.$ (22)

##  3 The multipartite case

Note that in contrast to the bipartite case where our results hold for any
pair of MUBs in the following we choose a particular pair of MUBs to
facilitate the derivation:

$\displaystyle|\tilde{i}_{k}\rangle:=\sum_{m=0}^{d-1}\omega^{\tilde{i}m}|m\rangle\,,$
(23)

where $\omega:=e^{\frac{2\pi i}{d}}$. Using these we first introduce the
following linear combination of diagonal density matrix elements:

$\displaystyle
C_{n,d}:=\sum_{\alpha}f_{\alpha}\langle\tilde{k}_{\alpha}|\rho|\tilde{k}_{\alpha}\rangle$
(24)

where $\alpha={i_{1},\dots,i_{n}}$ is a multi-index with
$i_{1},\dots,i_{n}\in\\{0,\dots,d-1\\}$. Furthermore

$\displaystyle f_{\alpha}:=\begin{cases}1\ &\mbox{if }0\leq
s_{\alpha}\leq\left\lfloor\frac{d}{4}\right\rfloor\\\ -1\ &\mbox{if
}\left\lceil\frac{d}{4}\right\rceil\leq
s_{\alpha}\leq\left\lfloor\frac{3d}{4}\right\rfloor\\\ 1\ &\mbox{if
}\left\lceil\frac{3d}{4}\right\rceil\leq s_{\alpha}\leq d-1\end{cases}$ (25)

where

$\displaystyle s_{\alpha}:=\sum_{j=1}^{n}i_{j}\mod{d}.$ (26)

Now we have that

$\displaystyle C_{n,d}=-g+\frac{1}{\xi}\sum_{\gamma}\Re e\langle
k_{\alpha}|\rho|k_{\beta}\rangle$ (27)

where $\gamma:=\\{\alpha,\beta|i\in\alpha,k\in\beta:\langle
i_{j}|k_{j}\rangle=0\ \forall\ 1\leq j\leq n\\}$. Moreover if one defines
$p_{l}$ as the number of combinations for which $s_{\alpha}=l$ then

$\displaystyle\frac{1}{\xi}=\frac{1}{2d^{n}}\sum_{l=0}^{d-1}p_{l}|\Re
e(\omega^{l})|$ (28)

and

$\displaystyle g=1-\frac{2p_{0}}{d^{n}}.$ (29)

This quantity $C_{n,d}$ is reminiscent of the figure of merit in the bipartite
case (i.e. $\sum_{i=0}^{d-1}\langle
v_{i}^{2}(v_{i}^{2})^{*}|\rho|v_{i}^{2}(v_{i}^{2})^{*}\rangle$) as it features
off-diagonal elements potentially involved in multipartite entangled states.
In the following we will use a similar strategy, relating these elements
together with diagonal matrix elements to lower bounds on concurrences for
multipartite systems [36, 37, 38, 54, 55].

$\displaystyle C_{GME}\geq\sqrt{\frac{2}{d(d-1)}}(\sum_{j=0}^{d-1}\sum_{i\neq
j}|\langle i|^{\otimes n}\rho|j\rangle^{\otimes
n}|-\sum_{\kappa}P_{ij}^{\kappa})$ (30)

where

$\displaystyle P_{ij}^{\kappa}=\sqrt{\langle i|^{\otimes n}\langle j|^{\otimes
n}\Pi_{\kappa}\rho^{\otimes 2}\Pi_{\kappa}|i\rangle^{\otimes
n}|j\rangle^{\otimes n}}.$ (31)

We can now use the linear combination above such that

$\displaystyle C_{GME}\geq\sqrt{\frac{2}{d(d-1)}}(\xi
C_{n,d}-\overline{C}_{n,d}-\sum_{\kappa}P_{ij}^{\kappa})$ (32)

with

$\displaystyle\overline{C}_{n,d}:=\xi
C_{n,d}-\sum_{j=0}^{d-1}\sum_{i<j}|\langle i|^{\otimes
n}\rho|j\rangle^{\otimes n}|-g.$ (33)

Making use of the Cauchy-Schwarz inequality, we get our main result, i.e. for
$\gamma^{\prime}:=\gamma/\\{\alpha,\beta|i\in\alpha,k\in\beta:\langle
i_{j}|k_{j}\rangle=0\ ,\ i_{j}=i_{h}\ ,\ k_{j}=k_{h}\ \forall\ 1\leq j,h\leq
n\\}$ it holds that

$\displaystyle
C_{GME}\geq\sqrt{\frac{2}{d(d-1)}}\left[\xi\left(C_{n,d}-\sum_{\gamma^{\prime}}\sqrt{\langle
k_{\alpha}|\rho|k_{\alpha}\rangle\langle
k_{\beta}|\rho|k_{\beta}\rangle}\right)+g-\sum_{\kappa}P_{ij}^{\kappa}\right].$
(34)

As mentioned in the main text, this procedure will strongly depend on the off-
diagonal elements that are revealed through the MUBs and the tailoring of the
lower bounds themselves for specific target states. As a demonstration of
feasibility we focus on the tripartite case, but stress that in principle for
each target state one could try to develop a similar approach.

###  3.1 Explicit tripartite examples

Here we give explicit examples for the resulting inequalities for the case of
$n=3$, i.e. a tripartite scenario. As the examples become quite cumbersome
with increasing $d$ we will restrict to explicitly stating only the cases
$d=2$ and $d=3$.

####  3.1.1 Three Qubits

We start out with the known bound for the genuine multipartite concurrence
[54, 55]

$\displaystyle C_{GME}\geq B_{GME}(\rho):=$ $\displaystyle 2(\Re e\langle
111|\rho|000\rangle$ $\displaystyle-\sqrt{\langle 001|\rho|001\rangle\langle
110|\rho|110\rangle}-\sqrt{\langle 010|\rho|010\rangle\langle
101|\rho|101\rangle}-\sqrt{\langle 011|\rho|011\rangle\langle
100|\rho|100\rangle})$ (35)

and choose the following linear combination of diagonal density matrix
elements

$\displaystyle C_{3,2}$
$\displaystyle=\langle+++|\rho|+++\rangle+\langle+--|\rho|+--\rangle+\langle-+-|\rho|-+-\rangle$
(36)
$\displaystyle+\langle--+|\rho|--+\rangle-\langle++-|\rho|++-\rangle-\langle+-+|\rho|+-+\rangle$
$\displaystyle-\langle-++|\rho|-++\rangle-\langle---|\rho|---\rangle$
$\displaystyle=2(\Re e\langle 111|\rho|000\rangle+\Re e\langle
001|\rho|110\rangle+\Re e\langle 010|\rho|101\rangle+\Re e\langle
100|\rho|011\rangle)$ (37)

This leads to the following bound

$\displaystyle C_{GME}\geq C_{3,2}-4\left(\sqrt{\langle
001|\rho|001\rangle\langle 110|\rho|110\rangle}+\sqrt{\langle
010|\rho|010\rangle\langle 101|\rho|101\rangle}+\sqrt{\langle
011|\rho|011\rangle\langle 100|\rho|100\rangle}\right)\,.$ (38)

Just as in the bipartite case, this bound gives the correct value for a pure
GHZ state (i.e. $B_{GME}(|GHZ\rangle\langle GHZ|)=C_{GME}(|GHZ\rangle\langle
GHZ|)$, for both explicit examples. For $d=2$ and thus
$|GHZ\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$ this results in a
resistance to white noise (i.e. the critical value of $p$ such that
$\rho_{noise}:=p|GHZ\rangle\langle GHZ|+\frac{1-p}{d^{3}}\mathbbm{1}$ is still
detected to be genuinely multipartite entangled ) of $p_{crit}=\frac{3}{5}$.

####  3.1.2 Three Qutrits

In the tripartite qutrit case it was shown that [54, 55]

$\displaystyle C_{GME}$ $\displaystyle\geq\frac{2}{\sqrt{3}}(\Re e\langle
000|\rho|111\rangle+\Re e\langle 111|\rho|222\rangle+\Re e\langle
222|\rho|000\rangle$ $\displaystyle-\sqrt{\langle 001|\rho|001\rangle\langle
110|\rho|110\rangle}-\sqrt{\langle 010|\rho|010\rangle\langle
101|\rho|101\rangle}-\sqrt{\langle 100|\rho|100\rangle\langle
011|\rho|011\rangle}$ $\displaystyle-\sqrt{\langle 112|\rho|112\rangle\langle
221|\rho|221\rangle}-\sqrt{\langle 121|\rho|121\rangle\langle
212|\rho|212\rangle}-\sqrt{\langle 122|\rho|122\rangle\langle
211|\rho|211\rangle}$ $\displaystyle-\sqrt{\langle 002|\rho|002\rangle\langle
220|\rho|220\rangle}-\sqrt{\langle 020|\rho|020\rangle\langle
202|\rho|202\rangle}-\sqrt{\langle 022|\rho|022\rangle\langle
200|\rho|200\rangle})$ (39)

The linear combination we will make use of is defined as

$\displaystyle C_{3,3}$
$\displaystyle=\langle\tilde{0}\tilde{0}\tilde{0}|\rho|\tilde{0}\tilde{0}\tilde{0}\rangle+\langle\tilde{1}\tilde{1}\tilde{1}|\rho|\tilde{1}\tilde{1}\tilde{1}\rangle+\langle\tilde{2}\tilde{2}\tilde{2}|\rho|\tilde{2}\tilde{2}\tilde{2}\rangle$
(40)
$\displaystyle+\langle\tilde{0}\tilde{1}\tilde{2}|\rho|\tilde{0}\tilde{1}\tilde{2}\rangle+\langle\tilde{1}\tilde{2}\tilde{0}|\rho|\tilde{1}\tilde{2}\tilde{0}\rangle+\langle\tilde{2}\tilde{0}\tilde{1}|\rho|\tilde{2}\tilde{0}\tilde{1}\rangle$
$\displaystyle+\langle\tilde{1}\tilde{0}\tilde{2}|\rho|\tilde{1}\tilde{0}\tilde{2}\rangle+\langle\tilde{0}\tilde{2}\tilde{1}|\rho|\tilde{0}\tilde{2}\tilde{1}\rangle+\langle\tilde{2}\tilde{1}\tilde{0}|\rho|\tilde{2}\tilde{1}\tilde{0}\rangle$
$\displaystyle-\langle\tilde{0}\tilde{0}\tilde{1}|\rho|\tilde{0}\tilde{0}\tilde{1}\rangle-\langle\tilde{0}\tilde{1}\tilde{0}|\rho|\tilde{0}\tilde{1}\tilde{0}\rangle-\langle\tilde{1}\tilde{0}\tilde{0}|\rho|\tilde{1}\tilde{0}\tilde{0}\rangle$
$\displaystyle-\langle\tilde{2}\tilde{2}\tilde{0}|\rho|\tilde{2}\tilde{2}\tilde{0}\rangle-\langle\tilde{2}\tilde{0}\tilde{2}|\rho|\tilde{2}\tilde{0}\tilde{2}\rangle-\langle\tilde{0}\tilde{2}\tilde{2}|\rho|\tilde{0}\tilde{2}\tilde{2}\rangle$
$\displaystyle-\langle\tilde{1}\tilde{1}\tilde{2}|\rho|\tilde{1}\tilde{1}\tilde{2}\rangle-\langle\tilde{1}\tilde{2}\tilde{1}|\rho|\tilde{1}\tilde{2}\tilde{1}\rangle-\langle\tilde{2}\tilde{1}\tilde{1}|\rho|\tilde{2}\tilde{1}1\rangle$
$\displaystyle-\langle\tilde{0}\tilde{0}\tilde{2}|\rho|\tilde{0}\tilde{0}\tilde{2}\rangle-\langle\tilde{0}\tilde{2}\tilde{0}|\rho|\tilde{0}\tilde{2}\tilde{0}\rangle-\langle\tilde{2}\tilde{0}\tilde{0}|\rho|\tilde{2}\tilde{0}\tilde{0}\rangle$
$\displaystyle-\langle\tilde{1}\tilde{1}\tilde{0}|\rho|\tilde{1}\tilde{1}\tilde{0}\rangle-\langle\tilde{1}\tilde{0}\tilde{1}|\rho|\tilde{1}\tilde{0}\tilde{1}\rangle-\langle\tilde{0}\tilde{1}\tilde{1}|\rho|\tilde{0}\tilde{1}\tilde{1}\rangle$
$\displaystyle-\langle\tilde{2}\tilde{2}\tilde{1}|\rho|\tilde{2}\tilde{2}\tilde{1}\rangle-\langle\tilde{2}\tilde{1}\tilde{2}|\rho|\tilde{2}\tilde{1}\tilde{2}\rangle-\langle\tilde{1}\tilde{2}\tilde{2}|\rho|\tilde{1}\tilde{2}\tilde{2}\rangle$
$\displaystyle=\frac{2}{3}\\{$ $\displaystyle+\Re e\langle
000|\rho|111\rangle+\Re e\langle 000|\rho|222\rangle+\Re e\langle
001|\rho|220\rangle+\Re e\langle 002|\rho|110\rangle$ $\displaystyle+\Re
e\langle 002|\rho|221\rangle+\Re e\langle 010|\rho|121\rangle+\Re e\langle
010|\rho|202\rangle+\Re e\langle 011|\rho|122\rangle$ $\displaystyle+\Re
e\langle 011|\rho|200\rangle+\Re e\langle 012|\rho|120\rangle+\Re e\langle
012|\rho|201\rangle+\Re e\langle 020|\rho|101\rangle$ $\displaystyle+\Re
e\langle 020|\rho|212\rangle+\Re e\langle 021|\rho|102\rangle+\Re e\langle
021|\rho|210\rangle+\Re e\langle 022|\rho|100\rangle$ $\displaystyle+\Re
e\langle 022|\rho|211\rangle+\Re e\langle 100|\rho|211\rangle+\Re e\langle
101|\rho|212\rangle+\Re e\langle 102|\rho|210\rangle$ $\displaystyle+\Re
e\langle 110|\rho|221\rangle+\Re e\langle 111|\rho|222\rangle+\Re e\langle
112|\rho|220\rangle+\Re e\langle 120|\rho|201\rangle$ $\displaystyle+\Re
e\langle 121|\rho|202\rangle+\Re e\langle
122|\rho|200\rangle-\underbrace{\sum_{i}\langle i|\rho|i\rangle}_{=1}\\}$

which results in the following bound

$\displaystyle C_{GME}$ $\displaystyle\geq
B_{GME}(\rho):=\frac{2}{\sqrt{3}}[(\frac{3}{2}C_{3,3}+1$
$\displaystyle-\sqrt{\langle 001|\rho|001\rangle\langle
220|\rho|220\rangle}-\sqrt{\langle 002|\rho|002\rangle\langle
110|\rho|110\rangle}-\sqrt{\langle 002|\rho|002\rangle\langle
221|\rho|221\rangle}$ $\displaystyle-\sqrt{\langle 010|\rho|010\rangle\langle
121|\rho|121\rangle}-\sqrt{\langle 010|\rho|010\rangle\langle
202|\rho|202\rangle}-\sqrt{\langle 011|\rho|011\rangle\langle
122|\rho|122\rangle}$ $\displaystyle-\sqrt{\langle 011|\rho|011\rangle\langle
200|\rho|200\rangle}-\sqrt{\langle 012|\rho|012\rangle\langle
120|\rho|120\rangle}-\sqrt{\langle 012|\rho|012\rangle\langle
201|\rho|201\rangle}$ $\displaystyle-\sqrt{\langle 020|\rho|020\rangle\langle
101|\rho|101\rangle}-\sqrt{\langle 020|\rho|020\rangle\langle
212|\rho|212\rangle}-\sqrt{\langle 021|\rho|021\rangle\langle
102|\rho|102\rangle}$ $\displaystyle-\sqrt{\langle 021|\rho|021\rangle\langle
210|\rho|210\rangle}-\sqrt{\langle 022|\rho|022\rangle\langle
100|\rho|100\rangle}-\sqrt{\langle 022|\rho|022\rangle\langle
211|\rho|211\rangle}$ $\displaystyle-\sqrt{\langle 100|\rho|100\rangle\langle
211|\rho|211\rangle}-\sqrt{\langle 101|\rho|101\rangle\langle
212|\rho|212\rangle}-\sqrt{\langle 102|\rho|102\rangle\langle
210|\rho|210\rangle}$ $\displaystyle-\sqrt{\langle 110|\rho|110\rangle\langle
221|\rho|221\rangle}-\sqrt{\langle 112|\rho|112\rangle\langle
220|\rho|220\rangle}-\sqrt{\langle 120|\rho|120\rangle\langle
201|\rho|201\rangle}$ $\displaystyle-\sqrt{\langle 121|\rho|121\rangle\langle
202|\rho|202\rangle}-\sqrt{\langle 122|\rho|122\rangle\langle
200|\rho|200\rangle})$ $\displaystyle-\sqrt{\langle 001|\rho|001\rangle\langle
110|\rho|110\rangle}-\sqrt{\langle 010|\rho|010\rangle\langle
101|\rho|101\rangle}-\sqrt{\langle 100|\rho|100\rangle\langle
011|\rho|011\rangle}$ $\displaystyle-\sqrt{\langle 112|\rho|112\rangle\langle
221|\rho|221\rangle}-\sqrt{\langle 121|\rho|121\rangle\langle
212|\rho|212\rangle}-\sqrt{\langle 122|\rho|122\rangle\langle
211|\rho|211\rangle}$ $\displaystyle-\sqrt{\langle 002|\rho|002\rangle\langle
220|\rho|220\rangle}-\sqrt{\langle 020|\rho|020\rangle\langle
202|\rho|202\rangle}-\sqrt{\langle 022|\rho|022\rangle\langle
200|\rho|200\rangle}]$ (42)

Using this criterion with the qutrit generalization of the $GHZ$ state
$|GHZ\rangle=\frac{1}{\sqrt{3}}(|000\rangle+|111\rangle+|222\rangle)$, the
noise resistance is still $p_{crit}=\frac{32}{59}$ and thus also practically
feasible for experiments (such as e.g. [57]).

##  4 Step-by-step calculation in three dimensions

As an instructive manual for applying the criterion we have simulated
coincidences in a $3\times 3$ experiment, including a conservative estimation
of the involved noise. If we label the potential outcomes in one basis by
standard

$\displaystyle\textnormal{Basis
1}:\left(\begin{array}[]{ccc}|1\rangle_{1}&|2\rangle_{1}&|3\rangle_{1}\end{array}\right)_{1}$
(44) $\displaystyle\textnormal{Basis
2}:\left(\begin{array}[]{ccc}|1\rangle_{2}&|2\rangle_{2}&|3\rangle_{2}\end{array}\right)_{2}$
(46)

This leads to $3x3$ correlation matrices for both bases:

$\displaystyle Corr_{i}=\left(\begin{array}[]{ccccccccc}\langle
1,1\,|_{i}\rho|1,1\rangle_{i}&\langle 1,2\,|_{i}\rho|1,2\rangle_{i}&\langle
1,3\,|_{i}\rho|1,3\rangle_{i}\\\ \langle 2,1\,|_{i}\rho|2,1\rangle_{i}&\langle
2,2\,|_{i}\rho|2,2\rangle_{i}&\langle 2,3\,|_{i}\rho|2,3\rangle_{i}\\\ \langle
3,1\,|_{i}\rho|3,1\rangle_{i}&\langle 3,2\,|_{i}\rho|3,2\rangle_{i}&\langle
3,3\,|_{i}\rho|3,3\rangle_{i}\end{array}\right).$ (50)

Here we use only a single index $i$ as both photons are assumed to be measured
in the same basis for both matrices. I.e. $\langle
1,1\,|_{i}\rho|1,1\rangle_{1}$ can be computed through the number of
coincidence events $N_{11}$ when both parties are measuring in the first basis
(simply by computing $\frac{N_{11}}{\sum_{i,j=1}^{3}N_{ij}}$). While this
procedure will work for any quantum setup with two mutually biased bases
available, let us assume for now that we perform the measurements in the
position (basis 1) and momentum basis (basis 2), and using simulated data from
the noise model of section (1) we get the following correlation matrices:

$\displaystyle\textnormal{Corr}_{1}=\left(\begin{array}[]{ccc}1015&23&9\\\
17&947&8\\\ 9&28&1008\\\ \end{array}\right)$ (54)
$\displaystyle\textnormal{Corr}_{2}=\left(\begin{array}[]{ccc}1053&21&7\\\
29&1017&25\\\ 5&15&1023\\\ \end{array}\right).$ (58)

To calculate $B(\rho)$ in equation (1) from the main text, which leads to the
EoF, we need correlations in two unbiased bases.
$C_{1}=\frac{\textnormal{Diag}(\textnormal{Corr}_{1})}{\textnormal{Total}(\textnormal{Corr}_{1})}$
is the sum of diagonal elements divided by all counts of the correlation
matrix (54), which gives $C_{1}=\frac{2970}{3064}=0.9693$.
$C_{2}=\frac{\textnormal{Diag}(\textnormal{Corr}_{2})}{\textnormal{Total}(\textnormal{Corr}_{2})}=0.9681$
gives the correlations in both bases. In the calculation of B($\rho$), we need
$C_{2}$, which is simply the correlation in basis $2$.

The next term in B($\rho$) is subtracted from our correlations. The sum over
the four variables $(m,n,l,o)$ goes each from $1$ to $3$, with the
restrictions $m\neq n$,$m\neq l$,$l\neq o$,$n\neq o$, which leads to $18$
terms. For these combinations of cross-talk elements, only the first basis is
used. The first term for $m=1,n=2,l=2,o=1$ gives $\sqrt{\langle
1,2\,|\rho|1,2\rangle\langle 2,1\,|\rho|2,1\rangle}=\frac{\sqrt{23\cdot
17}}{N}=0.00645$, where $N=3064$ is the total number of counts of $Corr_{1}$
to normalize the density matrix. Similarly, the second term $m=1,n=2,l=2,o=3$
is $\sqrt{\langle 1,2\,|\rho|1,2\rangle\langle
2,3\,|\rho|2,3\rangle}=\frac{\sqrt{23\cdot 8}}{N}=0.0044$; the third term
$m=1,n=2,l=3,o=1$: $\sqrt{\langle 1,2\,|\rho|1,2\rangle\langle
3,1\,|\rho|3,1\rangle}=\frac{\sqrt{23\cdot 9}}{N}=0.00470$ and so on. With
that, we find that (with d=3)

$\displaystyle M_{1}=\sum_{\begin{subarray}{c}m\neq n,m\neq l\\\ l\neq o,n\neq
o\end{subarray}}\sqrt{\langle
v_{m}^{1}v_{n}^{1}|\rho|v_{m}^{1}v_{n}^{1}\rangle\langle
v_{l}^{1}v_{o}^{1}|\rho|v_{l}^{1}v_{o}^{1}\rangle}=0.0852$ (59)

The final sum has indices $i$ and $j$ each going from $1$ to $3$, with the
restriction $i\neq j$. The first term with $i=1$ and $j=2$ is $\sqrt{\langle
1,2\,|\rho|1,2\rangle\langle 2,1\,|\rho|2,1\rangle}=\frac{\sqrt{55\cdot
56}}{N}=0.00645$, the second term with $i=1$, $j=3$ is $\sqrt{\langle
1,3\,|\rho|1,3\rangle\langle 3,1\,|\rho|3,1\rangle}=\frac{\sqrt{9\cdot
9}}{N}=0.00294$. This leads to

$\displaystyle M_{2}=\sum_{i\neq j}\sqrt{\langle
v_{i}^{1}v_{j}^{1}|\rho|v_{i}^{1}v_{j}^{1}\rangle\langle
v_{j}^{1}v_{i}^{1}|\rho|v_{j}^{1}v_{i}^{1}\rangle\,}=0.02856$ (60)

Finally this leads to

$\displaystyle B(\rho)=\sqrt{\frac{2}{d(d-1)}}\left(d\cdot
C_{2}-1-M_{1}-M_{2}\right)$ $\displaystyle=\sqrt{\frac{1}{3}}\left(3\cdot
0.9681-1-0.0852-0.02856\right)=1.0338.$ (61)

The entanglement of formation is defined in the main text in eq. (2), and if
we apply our values, we find

$\displaystyle E_{oF}(\rho)\geq-\log(1-\frac{B(\rho)^{2}}{2})=1.1$ (62)

That means, the state has at least $1.1$ bits of nonlocal information. This
obviously implies an entanglement dimensionality of more than two (as the
limit for qubits is 1). For demonstration purposes we can also find a lower
bound for the entanglement dimensionality (Schmidt number) through the
straightforward approach

$\displaystyle D\geq\lceil 2^{E_{oF}(\rho)}\rceil=\lceil 2.4\rceil=3,$ (63)

therefore we also show that the Schmidt number of the state is at least $3$.