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Quantifying high dimensional entanglement with
two mutually unbiased bases
Paul Erker
1,2,3
, Mario Krenn
4,5
, and Marcus Huber
6,1,7,5
1
Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
2
Faculty of Informatics, Universit
`
a della Svizzera italiana, Via G. Buffi 13, 6900 Lugano, Switzerland
3
Facolt
`
a indipendente di Gandria, Lunga scala, 6978 Gandria, Switzerland
4
Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090 Vienna, Austria.
5
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090
Vienna, Austria.
6
Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
7
ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels, Barcelona, Spain
July 27, 2017
We derive a framework for quantifying
entanglement in multipartite and high di-
mensional systems using only correlations
in two unbiased bases. We furthermore de-
velop such bounds in cases where the sec-
ond basis is not characterized beyond be-
ing 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 param-
eters.
Entanglement has long been recognized as the
key concept that takes quantum communication
beyond the classically possible. It finds applica-
tions in secure key distribution [1], super-dense
coding [2] and improves communication capaci-
ties 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 entangle-
ment in photon pairs naturally emerging from
down-conversion processes. Prominent examples
are path entangled photons in waveguides [47],
access to the entangled angular momentum of
photon pairs [815] and energy-time binned
photons [1618]. Such high dimensional entan-
glement endows each photon pair with more
shared information, greatly improving the effi-
ciency of known protocols and enabling secure
quantum communication at noise levels that
would be prohibitive for qubit systems [1923].
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 character-
izing the potential for accommodating many
dimensions [2529]. 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 oper-
ational meaning to the amount of information
encoded in the entanglement. For example
|ψi =
1
2+k
2
|0, 0i + |1, 1i +
P
k
i=1
|i, ii
with
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
2
1). On the
Accepted in Quantum 2017-07-25, click title to verify 1
arXiv:1512.05315v2 [quant-ph] 26 Jul 2017
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 entan-
glement 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 [30, 3338]. 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 ex-
ploited for high dimensional experiments [4042].
In this work we combine the advantage of both
approaches and quantify high dimensional entan-
glement purely from correlations in two mutually
unbiased bases (MUBs), and show that the re-
sulting quantification can readily be implemented
experimentally with modern cameras. In fact,
for sufficiently pure states the entire high dimen-
sional entanglement can be certified with this
minimal access. Since with only two measure-
ment settings the word mutually seems super-
fluous we will sometimes only refer to unbiased
bases. A natural candidate for two such bases
are discretised position and momentum correla-
tion. They are known to be readily accessible
at a high quality and have thus been used to
ascertain entanglement before [4345]. There-
fore, 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 entangle-
ment using two local unbiased measurement set-
tings.
Before we do, let us introduce the relevant con-
cepts. Sets of basis vectors {|v
k
i
i} are called (mu-
tually) unbiased, if they are both orthonormal
hv
k
i
|v
k
j
i = δ
ij
and their overlaps are unbiased
|hv
k
i
|v
k
0
j
i|
2
=
1
d
. MUBs can be used for an ef-
ficient 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 be-
tween maximally entangled states and the quan-
tum state under investigation. For mixed
states its regularised asymptotic version quan-
tifies 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 de-
terministically create using only local opera-
tions and classical communication (LOCC)? This
asymptotic conversion rate
N
k
is then found
in the limit N . For pure states it
corresponds to the entropy of entanglement
E
oF
(|ψ
AB
i) := S(Tr
A/B
(|ψ
AB
ihψ
AB
|)), where
S(ρ) = Tr(ρ log(ρ)). For general quantum
states Entanglement of Formation can be eval-
uated via a convex roof construction as the min-
imal average entanglement across all possible de-
compositions E
oF
(ρ) = inf
D(ρ)
P
i
p
i
E
oF
(|ψ
i
i).
Even if the whole state ρ 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 ap-
ply our methods to already existing experimental
data (e.g. from [40, 41]). For bipartite systems
the existing method makes use of the sum over
all diagonal correlations in m different MUBs, i.e.
C
m
(ρ) =
P
m
k=1
P
d1
i=0
hv
k
i
(v
k
i
)
|ρ|v
k
i
(v
k
i
)
i. The
simplest example are polarisation encoded states,
for which m = 3 MUBs can be used to com-
pute C
3
=
hH,H i+hV,V i+hD,D i+hA,A i+hL,L i+hR,R i
hH,H i+hH,V i+hV,H i+hV,V i
,
where hX, Y i denotes coincidence counts in X
and Y for the first and second photon, respec-
tively and H, V, D, A, L, R denote the six differ-
Accepted in Quantum 2017-07-25, click title to verify 2
ent polarisation states. This quantity is known
to be bounded for separable states by C
m
(ρ
sep
)
1+
m1
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. Im-
portantly, for large classes of states (and all pure
entangled states), C
2
is already sufficient to de-
tect 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 ide-
alised two qubit case: The eigenstates of the
Pauli matrices σ
i
form MUBs. If we now mea-
sured correlations in such as e.g. hσ
x
σ
x
i and
hσ
y
σ
y
i and found both of these values to be
(close to) 1, positivity of the density matrix im-
plies that hσ
z
σ
z
i 1. So despite only having
measured two out of the three defining correla-
tions 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 exam-
ple 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 mo-
tivates two questions that naturally follow from
this rather idealised setting:
Can we make these considerations analyti-
cal, noise robust and quantitative?
Are two measurements still sufficient for any
dimension?
We affirmatively answer both questions by intro-
ducing the following quantity
B(ρ) = N
h
d
d1
X
i=0
hv
2
i
(v
2
i
)
|ρ|v
2
i
(v
2
i
)
i
!
1
X
m6=n,m6=l
l6=o,n6=o
q
hv
1
m
v
1
n
|ρ|v
1
m
v
1
n
ihv
1
l
v
1
o
|ρ|v
1
l
v
1
o
i
X
i6=j
q
hv
1
i
v
1
j
|ρ|v
1
i
v
1
j
ihv
1
j
v
1
i
|ρ|v
1
j
v
1
i
i
i
(1)
where we have used N =
q
2
d(d1)
, and the ad-
ditional 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(ρ) is indeed a di-
rect lower bound to the generalised concurrence
and thus a lower bound on the Entanglement
of Formation can be easily computed whenever
B(ρ) 0, via
E
oF
(ρ) log(1
B(ρ)
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 im-
mediate consequence is that with just two global
measurement settings we can certify that the
maximally entangled state |φ
+
i =
1
d
(
P
d1
i=0
|iii)
has an entanglement of formation of E
oF
=
log(d), i.e. all the entanglement can be quan-
tified exactly through B(|φ
+
ihφ
+
|) =
q
2(1
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 experi-
mentally 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 cre-
ated 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 classifi-
cation, capturing necessary resources, but leaving
open the sufficiency entirely. Here we can make
use of a multipartite generalization of entangle-
ment of formation proposed in [36, 54, 55]. The
operational interpretation is simply the minimal
Accepted in Quantum 2017-07-25, click title to verify 3
necessary average entanglement across every cut
for creating this state via LOCC. It does not re-
veal a deeper structure, but is nonzero for every
genuinely multipartite entangled state and zero if
there exists a decomposition into at least bisep-
arable states. Formally it can be defined as
E
GME
:= inf
D(ρ)
X
i
p
i
min
A
i
S(Tr
A
i
(|ψ
i
ihψ
i
|)) . (3)
While, again, two mutually unbiased bases can
suffice for some states to quantify all multipar-
tite entanglement exactly, the multitude of po-
tential correlation structures giving rise to Gen-
uine Multipartite Entanglement (GME) makes
the construction of one universal criterion impos-
sible. While the criteria themselves constitute
genuine lower bounds and need no assumption
about the underlying state, their capability of
revealing multipartite entanglement will be tai-
lored 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 writ-
ten as |ψi =
P
r
S
i=0
λ
i
|v
i
v
i
i. This Schmidt decom-
position 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 accom-
plished 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 crite-
rion 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 infor-
mation 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 the-
oretically impact the bound. The first is dephas-
ing noise, i.e. a noise model where the intended
(maximally entangled) target state is mixed with
its dephased version:
ρ
dp
(p) = p|φ
+
ihφ
+
| +
(1 p)
d
(
d1
X
i=0
|iiihii|) . (4)
In this case B(ρ
dp
) = N(d(p +
(1p)
d
2
1), which
implies entanglement detection up to a noise
threshold of p
crit
=
1
d+1
, which actually increases
noise resistance with the dimension d. The other
extreme case is of course white noise, which pro-
vides the toughest challenge for entanglement de-
tection.
ρ
wn
(p) = p|φ
+
ihφ
+
| +
(1 p)
d
2
. (5)
Here B(ρ
wn
) = N(d(p+
(1p)
d
2
1
d2
d
(d1)
2
(1
p)), which leads to a decreasing noise resistance
of p
crit
=
d
2
3d+2
d
2
2d+2
which approaches 1 for d .
Both of these noise models are somewhat unreal-
istic, 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 es-
timate 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 unbi-
ased bases.
The photon pairs are created in a non-
linear spontaneous parametric down-conversion
(SPDC) crystal [58], which is the working horse
in quantum optical experiments. The two pho-
tons, which are deterministically separated (for
instance when they have opposite polarization in
a type-II SPDC, or different wavelengths), illumi-
nate 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 pho-
tons and photon pairs. (This is in contrast to
electron multiplying charge-coupled device (EM-
CCD) cameras, which have high efficiencies but
high dark count rates. There, quantum corre-
lations 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 mo-
mentum correlations - thus in strong position
anti-correlations. The strength of spatial corre-
lations can be quantified with the Federov-Ratio
F , which is the ratio between the marginal and
conditional probability [58, 61]. Experimentally,
one can reach very large values - realistically F
Accepted in Quantum 2017-07-25, click title to verify 4
LASER
ICCD
ppKTP
DM
C-Polarizer
FL
IL
IL
A
1000 px
1000 px
σ
= 100 px
7 8 9
0
1 2 3 4
5
6
7 8 9
0
1 2 3 4
5
6
7 px
white noise
cross-correlations
100
σ=
25
σ=
100
σ=
25
σ=
p
e
e
I
IVIII
II
0.00 0.02 0.04 0.08 0.1
1.5
2.0
2.5
3.0
3.5
0.00
0.005
0.020.015
0.01
1.0
1.5
2.0
2.5
3.0
3.5
p
B C
0.06
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 σ = 100 pixels, and orange σ = 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.
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 re-
duce 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 pa-
rameters, fig.(1B), one can nonetheless find con-
figurations 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 × 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 consid-
ered). With this information, we find a good
configuration which reduces as much as possible
Accepted in Quantum 2017-07-25, click title to verify 5
the negative influence of the other two effects.
Specifically, we introduce large empty areas be-
tween 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 config-
uration leads to 2.4 e-bits, mainly reduced due to
white noise. If we decrease the number of pixels
further to 3×3 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 chal-
lenge 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 de-
tected photon pairs. In order to collect a suitable
amount of correlated photons, one could expect
the measurement time to be 100200 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 en-
tanglement.
While in principle any amount of bipartite en-
tanglement can be certified for pure states, realis-
tic noise assessment shows that with this limited
number of local observables and current technol-
ogy 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 de-
sign. 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 ques-
tions. Especially quantum key distribution pro-
tocols 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 eas-
ily attacked if the source or measurement devices
are hacked. It may be possible to use entangle-
ment quantifiers, such as the one presented here,
to certify security of the source for high dimen-
sional 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 experi-
mental 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 partic-
ular 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[6567].
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 Kl¨ockl for productive discussions at LIQ-
UID 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 fund-
ing from the Swiss National Science Founda-
tion (SNF) and the National Centres of Compe-
tence in Research ”Quantum Science and Tech-
nology” (QSIT). MH furthermore acknowledges
Accepted in Quantum 2017-07-25, click title to verify 6
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 ex-
perimental 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
|ψi = N
1
x
max
X
x=0
w
x
|x, xi
(6)
where w
x
can be found numerically and |x, xi 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 neigh-
boring 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:
|x, xi = N
2
c
x
|x, xi +
X
yNB(x)
c
x,y
|x, yi + |y, xi
(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 × N -
areas, as shown in Fig. 1C in the main text.
Accepted in Quantum 2017-07-25, click title to verify 9
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:
ρ = p|ψihψ| +
1 p
d
2
d
2
, (8)
where |ψi 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. Two dark counts; one in Region 1 (where photon A usually appears), one in Region 2
2. One dark count, one real count
3. Two real counts
Now a simple calculation shows:
all counts =
¯
P (D
1
D
2
) + (9)
+ P
1
D
1
D
2
¯
1
¯
2
+ D
1
2
¯
D
2
¯
1
+
1
D
2
¯
D
1
¯
2
+
1
2
¯
D
1
¯
D
2
+
+ P
2
D
1
D
2
¯
1
2
¯
2
2
+ 2D
1
2
¯
D
2
¯
1
2
¯
2
+ 2
1
D
2
¯
D
1
¯
2
2
¯
1
+ 4
1
2
¯
D
1
¯
D
2
¯
1
¯
2
+
+ ...
=
X
n=0
P
n
D
1
D
2
¯
1
n
¯
2
n
+ nD
1
2
¯
D
2
¯
1
n
¯
2
n1
+ n
1
D
2
¯
D
1
¯
2
n
¯
1
n1
+ n
2
1
2
¯
D
1
¯
D
2
¯
1
n1
¯
2
n1
where the last sum has a simple closed form and
¯
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 = P
1
1
2
¯
D
1
¯
D
2
(10)
p
good
=
good counts
all counts
(11)
We use D
1
= D
2
(dark counts in the two areas are the same) and
1
=
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
Accepted in Quantum 2017-07-25, click title to verify 10
reader of the lower bounds for the concurrence (i.e. the square root of the linear entropy) developed
in [54, 55] as
I :=
s
2
d(d 1)
X
m6=n
|hmm|ρ|nni|
| {z }
I
1
q
hmn|ρ|mnihnm|ρ|nmi
| {z }
I
2
inf
D(ρ)
X
i
p
i
q
(2(1 Tr(ρ
i
A
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
(ρ) = log(Tr(ρ
2
)) and the fact that the family of Renyi entropies is
monotonically decreasing in α, i.e. that for S
α
:=
1
1α
Tr(ρ
α
) it holds that S
α
S
β
α β. This
directly implies that entanglement of formation, defined as
E
oF
:= inf
D(ρ)
X
i
p
i
S(ρ
i
A
) (13)
is directly lower bounded by
E
oF
log(1
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
i
1
i} = {|ii}
hmn|ρ|mni =
N
m,n
P
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 |hv
2
i
|ji|
2
=
1
d
,
that means we can write the second basis as
|v
k
i :=
1
d
d1
X
m=0
e
k
m
|mi. (16)
Now we can evaluate a specific sum of correlations in the mutually unbiased basis:
Σ := C
2
C
1
=
X
k
hv
k
v
k
|ρ|v
k
v
k
i =
P
k
N
k,k
P
i,j
N
i,j
(17)
we first notice that |v
k
v
k
i =
1
d
P
m,n
e
i(φ
k
m
φ
k
n
)
|mni, such that
Σ =
1
d
2
X
k
X
m,n,l,o
e
i(φ
k
m
φ
k
n
+φ
k
o
φ
k
l
)
hmn|ρ|loi (18)
Now we can use that
P
d1
m=0
e
i(φ
k
m
φ
k
0
m
)
= 0 k 6= k
0
due to the fact that hv
2
k
|v
2
k
0
i = 0 k 6= k
0
. The sum
Σ can be split in three terms Σ = Σ
1
+ Σ
2
+ Σ
3
:
m = l and n = o: This amounts to Σ
1
=
1
d
X
m,n
hmn|ρ|mni
| {z }
=Tr(ρ)=1
=
1
d
Accepted in Quantum 2017-07-25, click title to verify 11
m = n and l = o: This is exactly the desired sum, i.e. Σ
2
=
1
d
P
m6=l
hmm|ρ|lli
For the term m = l and n 6= o:Here we find a pre-factor of
P
d1
k=0
e
i(φ
k
o
φ
k
n
)
= 0. The same for
m = n and l 6= o, n = o and m 6= l and l = o and m 6= n.
The remaining terms that do not vanish fulfill with m 6= n, m 6= l, n 6= o and l 6= o. terms for
which
P
k
e
i(φ
k
m
φ
k
n
+φ
k
o
φ
k
l
)
= c
m,n,l,o
yield Σ
3
=
1
d
2
P
m6=n,m6=l
l6=o,n6=o
c
m,n,l,o
<e[hmn|ρ|loi]
Now we can make use of additional experimental data taken in the first basis and use the Cauchy-
Schwarz-inequality to show
dΣ
3
X
m6=n,m6=l
l6=o,n6=o
q
hmn|ρ|mnihlo|ρ|loi 0 (19)
Finally since <e[z] |z| we can now state the main result here being:
I
1
d
Σ
1
d
1
d
X
m6