High-Dimensional Pixel Entanglement: Efficient Generation
and Certification
Natalia Herrera Valencia
1
, Vatshal Srivastav
1
, Matej Pivoluska
2,3
, Marcus Huber
4,5
, Nicolai Friis
4
,
Will McCutcheon
1
, and Mehul Malik
1,4
1
Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, UK
2
Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia
3
Institute of Computer Science, Masaryk University, Brno, Czech Republic
4
Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Vienna, Austria
5
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
Photons offer the potential to carry large
amounts of information in their spectral, spa-
tial, and polarisation degrees of freedom.
While state-of-the-art classical communica-
tion systems routinely aim to maximize this
information-carrying capacity via wavelength
and spatial-mode division multiplexing, quan-
tum systems based on multi-mode entangle-
ment usually suffer from low state quality,
long measurement times, and limited encod-
ing capacity. At the same time, entangle-
ment certification methods often rely on as-
sumptions 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 en-
tanglement 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 di-
mensions, and an entanglement-of-formation
of up to 4 ebits. Furthermore, our measure-
ment 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 de-
velopment of quantum technologies, resulting in rev-
olutionary concepts in quantum communication such
as superdense coding [1] and device-independent se-
Natalia Herrera Valencia: Email address: nah2@hw.ac.uk
Mehul Malik: Email address: m.malik@hw.ac.uk
curity [2, 3], as well as enabling fundamental tests of
the very nature of reality [4–6]. 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 of-
fered by photonic quantum systems has provided the
means for quantum communication with record ca-
pacities [7, 8], noise-resistant entanglement distribu-
tion [9, 10], robust loophole-free tests of local real-
ism [11, 12], and scalable methods for quantum com-
putation [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 O(d
4
) single-outcome pro-
jective measurements [14, 15], making this task ex-
tremely 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 cru-
cial to certify entanglement without compromising the
security and validity of the applications by introduc-
ing assumptions on the state, e.g., purity of the gener-
ated state [19] or conservation of quantities [20]. Re-
cent work has shown how measurements in mutually
unbiased bases (MUBs) allow the efficient certifica-
tion of high-dimensional entanglement [21, 22], with-
out any such assumptions on the underlying quantum
state. Here we employ and extend this method to im-
prove upon the quality and speed of high-dimensional
entanglement certification.
High-dimensional entanglement has been demon-
strated in multiple photonic platforms, with encod-
ings in the orbital angular momentum (OAM) [22,
23], 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 dif-
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 1
arXiv:2004.04994v4 [quant-ph] 23 Dec 2020
ficulty of measuring coherent superpositions of multi-
ple 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 poten-
tial [16]. Path-encoding in integrated photonic cir-
cuits offers yet another promising avenue for real-
ising high-dimensional entanglement [24]. However,
the precise fabrication and control of
d(d−1)
2
Mach-
Zehnder interferometers required for universal opera-
tions in d dimensions poses significant practical chal-
lenges as the dimension is increased [25].
Meanwhile, techniques for the creation, manipula-
tion 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 ampli-
tude modes. However, such measurements necessarily
suffer from loss [28] and have limited quality [29, 30],
resulting in long measurement times and reduced en-
tanglement quality in large dimensions [22]. An al-
ternative choice of basis is the discretised transverse
position-momentum, or “pixel” basis [31], where the
lack of efficient single-photon detector arrays neces-
sitates scanning through localised position or mo-
mentum modes, or subtracting a large noise back-
ground [32]. Such measurements are also subject to
extreme loss, and as a result require long measure-
ment 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 com-
bines three distinct improvements in the genera-
tion and measurement of spatially entangled modes.
Working in the discretized transverse position-
momentum DoF, we first tailor our spatial-mode ba-
sis by adapting it to the characteristics of the two-
photon state generated and measured in our experi-
mental setup. Second, we implement a recently devel-
oped 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 op-
timisation, spatial-mode measurement, and certifica-
tion tools allow us to certify high-dimensional entan-
glement with a record quality, speed, and dimension,
reaching state fidelities up to 98%, certified entangle-
ment 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 sponta-
neous parametric down-conversion (SPDC) is charac-
terised by its joint-transverse-momentum-amplitude
or JTMA (Fig. 1 a), which is well approximated by
the function [33, 34]
F (
~
k
s
,
~
k
i
) = exp
n
−
1
2
|
~
k
s
+
~
k
i
|
2
σ
2
P
o
sinc
n
1
σ
2
S
|
~
k
s
−
~
k
i
|
2
o
,
(1)
where
~
k
s
and
~
k
i
are the transverse momentum com-
ponents of the signal and idler photons. The param-
eters σ
P
and σ
S
are the widths of the minor and ma-
jor axis of the JTMA, which are dependent on the
pump transverse momentum bandwidth and the crys-
tal characteristics, respectively. In effect, σ
P
indicates
the degree of momentum correlation, while σ
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(
~
k
s
,
~
k
i
) = exp
n
−
1
2
|
~
k
s
|
2
σ
2
C
o
exp
n
−
1
2
|
~
k
i
|
2
σ
2
C
o
F (
~
k
s
,
~
k
i
),
(2)
where σ
C
denotes the collection bandwidth, that is,
the size of the back-projected detection mode. The ef-
fect of the collection on the correlations is illustrated
in Fig. 1, where the Gaussian functions attributed to
the SMF modes decrease the probability of detect-
ing higher-order modes associated with the edges of
the JTMA. Finally, we include the hologram func-
tions Φ
s
(
~
k
s
) and Φ
i
(
~
k
i
) used for performing projec-
tive measurements on the signal and idler photons, re-
spectively, which leads to the two-photon coincidence
probability
Pr(Φ
s
, Φ
i
) =
Z
d
2
~
k
s
Z
d
2
~
k
i
Φ
s
(
~
k
s
)Φ
i
(
~
k
i
)G(
~
k
s
,
~
k
i
)
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 shap-
ing [35, 36] and entanglement witnesses that adapt to
the non-maximally entangled nature of the state [22].
Below, we introduce a new approach to engineer-
ing high-quality spatial-mode entanglement based on
three distinct improvements: tailoring the spatial-
mode measurement basis, precise two-photon mea-
surements via intensity-flattening, and a modified wit-
ness for high-dimensional entanglement.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 2
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 ((0, k
sy
), (0, k
iy
)) 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{−
1
2
k
sy
σ
C
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
R
d
2
~
k
i
F (
~
k
s
,
~
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 dis-
cretized transverse-momentum, or macro-pixel basis.
This basis provides several advantages over other spa-
tial 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 [29, 30] and their superposi-
tions, measurements with phase-only modulation are
lossless in theory and produce a count rate that is in-
dependent 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 deter-
mined by circle packing formulas, this basis is com-
patible with state-of-the-art quantum communication
technologies based on multi-core fibres [37–40] and
was recently employed for high-dimensional entangle-
ment transport through a commercial multi-mode fi-
bre [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
|Φi =
1
√
d
P
m
|mm i, where |m i stands for one pho-
ton 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 cer-
tification. In contrast with Procrustean filtering tech-
niques that achieve a similar goal by adding mode-
dependent loss [23, 43], 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 corre-
lation albeit with lower amplitudes, while the inner
pixel modes show weaker correlation with higher am-
plitudes. 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 aris-
ing from non-vanishing pump transverse bandwidth
σ
P
, while simultaneously equalizing pixel probability
amplitudes and maximizing photon count rates. This
proceeds as follows: the diameter of the circular re-
gion containing the macro-pixel basis is determined
by the width σ
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 dimen-
sion 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 ap-
proximating a state with equal Schmidt coefficients
extremely well.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 3
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
◦
(the 45
◦
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 com-
posed 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 σ
C
that depends on the
specific characteristics of the fiber and coupling op-
tics used. As a result, higher order modes are rel-
atively suppressed, which has an especially adverse
effect when measuring coherent high-dimensional su-
perpositions 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 collec-
tion 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 or-
der modes are relatively suppressed. This effectively
allows us to make the approximation G(
~
k
s
,
~
k
i
) ≈
F (
~
k
s
,
~
k
i
) in Eq. (2), equating the collected JTMA
with the generated one. We implement this tech-
nique 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 demon-
strated 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 mea-
surements in the chosen standard basis, comple-
mented by measurements in a basis that is unbiased
with respect to it, in order to efficiently certify high-
dimensional entanglement [22]. Here, we briefly intro-
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 4
duce this witness, identify its limitations when applied
to experiment, and discuss our modifications that sig-
nificantly 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 trans-
verse spatial (or momentum) modes. When perform-
ing single-outcome projective measurements (as de-
scribed above) in such a basis, the detector count rates
are restricted by the limited projection onto the col-
lection 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 unbiased
1
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. Mea-
surements in a spatially delocalised unbiased basis can
thus collect photons incident on any of the pixels.
As a result, ‘unbiased’ measurements take O(d) less
time individually, or O(d
2
) less time in the bipar-
tite 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 re-
spect to each other (and with respect to the original
standard basis), without resorting to measurements
in the standard basis at all. This construction is ex-
tremely beneficial and permits us to lower the mea-
surement times for certifying high-dimensional entan-
glement significantly.
The method described in Ref. [22] allows one to
certify high-dimensional entanglement by estimating
a lower bound on the fidelity F (ρ, Φ) of a mea-
sured state ρ with a chosen target state |Φi =
1
√
d
P
m
|mm i. By performing two-photon measure-
ments in the standard basis {|mni} and a second
basis {| ˜m
k
˜n
∗
k
i} that is unbiased with respect to the
standard basis, one can obtain a lower bound to the
fidelity F (ρ, Φ). As explained in Ref. [22], the fidelity
can in turn be used to bound the Schmidt number
of ρ from below. That is, the certified entangle-
ment dimensionality is the maximal d
ent
such that
(d
ent
− 1)/d ≥ F (ρ, Φ). The unbiased bases can be
constructed in a standard manner by following the
prescription by Wootters and Fields [44], i.e.,
|
˜
j
k
i =
1
√
d
d−1
X
m=0
ω
jm+km
2
|m i, (4)
where ω = exp(
2πi
d
) is the principal complex d−th
root of unity, k ∈ {0, . . . , d−1} labels the chosen basis,
and j ∈ {0, . . . , d − 1} labels the basis elements. We
1
For 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.
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 {|
˜
j
k
i}
j
together with the standard basis
{|m i}
m
forms a complete set of d + 1 mutually unbi-
ased bases (MUBs) with the property that the overlap
between any two basis states chosen from any two dif-
ferent 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 |Φ i =
1
√
d
P
m
|mm i. Such states
are invariant under transformations (U ⊗U
∗
) for any
unitary operator U, and thus have the same form in
any WF basis, allowing us to express our target state
as |Φ i =
1
√
d
P
m
| ˜m
k
˜m
∗
k
i. The fidelity of our ex-
perimental state to this target state F (ρ, Φ) can then
be expressed in terms of k-th WF basis and be split
into two contributions, F (ρ, Φ) = F
1
(ρ, Φ) + F
2
(ρ, Φ),
where
F
1
(ρ, Φ) :=
1
d
X
m
h ˜m
k
˜m
∗
k
|ρ | ˜m
k
˜m
∗
k
i, (5a)
F
2
(ρ, Φ) :=
1
d
X
m6=n
h ˜m
k
˜m
∗
k
|ρ |˜n
k
˜n
∗
k
i. (5b)
While the first term F
1
(ρ, Φ) can be calculated
directly from measurements in k-th WF basis
{| ˜m
k
˜n
∗
k
i}, we show in Appendix A.III how one can
determine a lower bound for F
2
(ρ, Φ) with measure-
ments in a second WF basis (labeled by k
0
6= k in the
construction above) {| ˜m
k
0
˜n
∗
k
0
i}, allowing us to calcu-
late a lower bound for the fidelity to the maximally
entangled state given by
F (ρ, Φ) ≥
˜
F (ρ, Φ) =
1
d
X
m
h ˜m
k
˜m
∗
k
|ρ | ˜m
k
˜m
∗
k
i
+
X
m
h ˜m
k
0
˜m
∗
k
0
|ρ | ˜m
k
0
˜m
∗
k
0
i −
1
d
(6)
−
X
m6=m
0
m6=n
n6=n
0
n
0
6=m
0
˜γ
mnm
0
n
0
q
h ˜m
0
k
˜n
0∗
k
|ρ | ˜m
0
k
˜n
0∗
k
ih ˜m
k
˜n
∗
k
|ρ | ˜m
k
˜n
∗
k
i,
where the term ˜γ
mnm
0
n
0
vanishes whenever (m−m
0
−
n + n
0
) mod (d) 6= 0, and is equal to
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 entangle-
ment dimensionality. In particular, this allows us to
bypass the use of measurements in the standard ba-
sis 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 ρ whose entanglement is being certified.
Nevertheless, the witness is valid for any choice of
target state, and thus requires no assumptions about
the state ρ. We do make certain assumptions about
our local measurement devices, which means that our
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 5
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 re-
quired for practical security. Assumptions on states
and devices can enter in different form, including
assumptions on certain properties of the state (pu-
rity, conservation of energy or momentum), to assum-
ing certain device properties (perfect measurements,
background/dark count subtraction). Our only as-
sumption 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 re-
lation. This assumption can and has been tested
by preparing eigenstates of one basis and measur-
ing in another basis, to estimate if all possible re-
sults 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 back-
ground 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 im-
plementation. Moreover, while entanglement wit-
nesses 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 opti-
misation, spatial-mode measurements, and entangle-
ment certification in a two-photon entanglement ex-
periment. 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 de-
picted 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 opti-
mally located in the setup (see Appendix A.I). A
grating-stabilised CW laser at 405 nm is loosely fo-
cused onto a 5 mm ppKTP crystal to produce photon
pairs at 810 nm entangled in their transverse position-
Table 1: Fidelities F (ρ, Φ
+
) to the maximally
entangled state, obtained via mea-
surements in all mutually unbiased
bases in dimension d
d d
ent
F (ρ, Φ
+
) E
oF
(ebits)
3 3 98.2 ± 0.9 % 1.4 ± 0.1
5 5 97.5 ± 0.5 % 2.2 ± 0.1
7 7 96.4 ± 0.7 % 2.6 ± 0.1
11 11 93.9 ±0.7 % 3.1 ± 0.1
13 13 94.1 ±0.6 % 3.3 ± 0.1
17 17 94.3 ±0.3 % 3.6 ± 0.1
19 18 94.4 ±0.4 % 3.8 ± 0.1
momentum via type-II spontaneous parametric down-
conversion (SPDC). Single-outcome projective mea-
surements in the optimized macro-pixel basis (or any
of its MUBs) are performed on the photon pairs us-
ing diffractive holograms implemented on spatial light
modulators (SLMs), where each macro-pixel mode is
defined by displaying a grating over the correspond-
ing 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 Fig-
ure 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 effec-
tively “switches on” the selected modes so they effi-
ciently couple to single-mode fibers. After the SLMs,
two intensity-flattening telescopes (IFT) are used for
ensuring precise, generalized spatial-mode measure-
ments across the entire modal bandwidth of interest
(see Appendix A.I).
Our modified entanglement witness allows charac-
terizing the generated state by lower bounding its fi-
delity to the maximally entangled state through mea-
surements in any two mutually unbiased bases. How-
ever, 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 mea-
sure 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 ≈ 94% or greater in all subspaces, certifying
entanglement dimensions of d
ent
= d for d = 3, ..., 17
(see Table 1). In d = 19, we obtain an estimate of
exact fidelity of 94.4 ± 0.4%, certifying an entangle-
ment dimensionality of d
ent
= 18 (and just below the
fidelity bound of 94.7% for d
ent
= 19). We also cal-
culate the entanglement-of-formation (E
oF
), which is
an entropic measure of the amount of entanglement
needed to create our state [47, 48] (please see Ap-
pendix A.IV for more details).
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 6
Figure 3: Experimental data in 19 dimensions. Normalised two-photon coincidence counts showing correlations in the
standard 19-dimensional pixel basis {| mni}
mn
and its 19 mutually unbiased bases {|
˜
i
k
˜
j
∗
k
i}
i,j
. With these measurements,
we obtain a fidelity of F (ρ, Φ
+
) = 94.4 ± 0.4% to a 19-dimensional maximally entangled state, certifying 18-dimensional
entanglement.
Our E
oF
reaches a value of 3.8 ± 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
[16, 18] 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 assum-
ing 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 cer-
tifying 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 ± 2)% using a total data acquisi-
tion time of 3.6 minutes (corresponding to 722 single-
outcome measurements), certifying an entanglement
dimensionality of d
ent
= 18 ± 1. In comparison, prior
experiments on OAM entanglement required a mea-
surement time two orders of magnitude larger for
certifying just d
ent
= 9 in 11 dimensions [22]. A
summary of results for dimensions 19 and higher is
shown in Table 2. As can be seen, the entanglement
quality starts dropping above d ≈ 30, which is a re-
sult of the limited resolution of our devices and the
SPDC generation bandwidth itself. Nevertheless, we
are able to certify an entanglement dimensionality of
d
ent
= 55 ± 1 (See Fig. 4) in a 97-dimensional space
and an entanglement-of-formation of E
oF
= 4.0 ± 0.1
ebits in a 31-dimensional space, which both consti-
tute 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
oF
val-
ues 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 stan-
dard basis measurements would require ∼14 years of
measurement time for data of similar quality.
4 Conclusion
We have demonstrated the certification of pho-
tonic high-dimensional entanglement in the transverse
position-momentum degree-of-freedom with a record
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 7
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 {|
˜
i
k
˜
j
∗
k
i}
i,j
. Using these measurements, we obtain a
fidelity bound of
˜
F (ρ, Φ
+
) = 56 ± 1% that is above the bound B
54
= 54/97 = 0.5567, thus certifying an entanglement
dimensionality of d
ent
= 55 ± 1.
quality, measurement time, and entanglement dimen-
sionality. 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 ≈ 94%
or above in local dimensions of 19 or below, certify-
ing 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 reduc-
tion 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-
Table 2: Fidelity bounds
˜
F (ρ, Φ
+
) obtained
via measurements in two mutually
unbiased bases in dimension d
d d
††
ent
F (ρ, Φ
+
) E
oF
(ebits)
19 18(+1) 93 ± 2 % 3.6 ± 0.2
23 22(-1) 92 ± 2 % 3.6 ± 0.2
29 27(-1) 90 ± 2 % 3.8 ± 0.2
31 29(-1) 92 ± 2 % 4.0 ± 0.1
37 32(-1) 84 ± 1 % 3.4 ± 0.1
51
†
38(-1) 73 ± 1 % 2.8 ± 0.1
97 55(±1) 56 ± 1 % 1.9 ± 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 cer-
tified by the mean values of the corresponding fidelity
bounds, while other Schmidt-number thresholds within
the confidence interval are shown in parenthesis.
of-formation of 4 ebits, which to our knowledge is
the highest amount certified without any assump-
tions on the state. While we have used projective
single-outcome measurements in our experiment, re-
cent progress on generalized multi-outcome measure-
ment devices [50–52] 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 sup-
port from the QuantERA ERA-NET Co-fund
(FWF Project I3773-N36) and the UK Engineer-
ing and Physical Sciences Research Council (EP-
SRC) (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 × 2 mm × 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.
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.
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”
0 200 400 600 800 1000
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Propagation distance (mm)
Beam size (mm)
ω
�
= μ
Figure A.1: Gaussian beam propagation for the pump. The collimated UV pump (beam waist ω
0
= 950µm) 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
p
= 188µm. We place the crystal such
that this final beam waist is at its longitudinal center.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 12
0 200 400 600 800
-1.0
-0.5
0.0
0.5
1.0
Propagation distance (mm)
Beam Size (mm)
ω
= μ
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 ω
SPDC
= 188 µm, 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 ω
SLM
= 343 µm on the SLM after propagation from the crystal.
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.
0 200 400 600 800 1000 1200 1400
-4
-2
0
2
4
Propagation Distance (mm)
Beam Size (mm)
ω
�
= μ
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 ω
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 ω
c
= 3723 µm, which determines the maximum mode size that we can use for the pixel
holograms without introducing large amounts of loss or measurement imprecision.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 13
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 H
AB
= H
A
⊗ H
B
, with equal local dimensions
dim(H
A
) = dim(H
B
) = d, and an a priori unknown state ρ 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 (ρ, Φ) to the target state |Φ i =
P
n
λ
n
|nn i
AB
, given by:
F (ρ, Φ) = Tr(|ΦihΦ|ρ) =
d−1
X
m,n=0
hmm|ρ|nni. (A.1)
The entanglement dimensionality can be deduced from the fidelity taking into account that for any state ρ of
Schmidt number r ≤ d, the fidelity of Eq. (A.1) is bounded by:
F (ρ, Φ) ≤ B
r
(Φ) :=
r−1
X
m=0
λ
2
i
m
, (A.2)
where the sum runs over the i
m
with m ∈ {0, ..., d − 1} such that λ
i
m
≥ λ
i
0
m
∀m ≥ m
0
. Hence, any state with
F (ρ, Φ) > B
r
(Φ) 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 λ
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 |Φ
+
i, i.e., where
λ
n
= 1/
√
d ∀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 {|mni}
m,n=0,...,d−1
, measures locally
w.r.t. this basis, obtaining estimates for the matrix elements {hmn |ρ |mn i}
m,n
from the coincidence counts
{N
ij
}
i,j
in the chosen basis via
hmn |ρ |mn i =
N
mn
P
i,j
N
ij
. (A.3)
Then, one measures in M of d possible mutually unbiased bases {|
˜
i
k
˜
j
∗
k
i}
i,j
, where the label k ∈ {0, 1, . . . , 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,
|
˜
j
k
i =
1
√
d
d−1
X
m=0
ω
jm+km
2
|m i, (A.4)
where ω = exp(
2πi
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 {|
˜
j
k
i}
j
together
with the standard basis {|m i}
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 {h
˜
i
k
˜
j
∗
k
|ρ |
˜
i
k
˜
j
∗
k
i}
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 (ρ, Φ) by splitting it in two contributions, F (ρ, Φ) = F
1
(ρ, Φ) + F
2
(ρ, Φ), where
F
1
(ρ, Φ) :=
1
d
X
m
hmm |ρ |mm i, (A.5a)
F
2
(ρ, Φ) :=
1
d
X
m6=n
hmm |ρ |nn i. (A.5b)
The first term F
1
(ρ, Φ) can be calculated directly from the measurements in the standard basis, while the term
F
2
(ρ, Φ) can be bounded from below by the quantity
˜
F
2
(ρ, Φ) ≤ F
2
(ρ, Φ), given by
˜
F
2
:=
d−1
X
j=0
h
˜
j
k
˜
j
∗
k
|ρ |
˜
j
k
˜
j
∗
k
i −
1
d
−
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
˜γ
mm
0
nn
0
p
hm
0
n
0
|ρ |m
0
n
0
ihmn |ρ |mn i, (A.6)
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 14
where we have noted that
P
d−1
m,n=0
hmn |ρ |mn i = 1 by construction due to Eq. (A.3), and the prefactor ˜γ
mm
0
nn
0
is given by
˜γ
mm
0
nn
0
=
(
0 if (m − m
0
− n + n
0
) mod (d) 6= 0
1
d
otherwise.
(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
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.,
d−1
X
j=0
h
˜
j
k
˜
j
∗
k
|ρ |
˜
j
k
˜
j
∗
k
i =: Σ = Σ
1
+ Σ
2
+ Σ
3
, (A.8)
which are given by
Σ
1
:=
1
d
X
m,n
hmn |ρ |mn i =
1
d
, (A.9a)
Σ
2
:=
1
d
X
m6=n
hmm |ρ |nn i, (A.9b)
Σ
3
:=
1
d
2
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
Re
c
mnm
0
n
0
hm
0
n
0
|ρ |mn i
, (A.9c)
where we have introduced the quantity
c
mnm
0
n
0
:=
X
j
ω
j(m−m
0
−n+n
0
)+k(m
2
−m
02
−n
2
+n
02
)
. (A.10)
We can then bound the real part appearing in the summands of Σ
3
by their modulus, i.e.,
Re
c
mnm
0
n
0
hm
0
n
0
|ρ |mn i
≤ |c
mnm
0
n
0
hm
0
n
0
|ρ |mn i| = |c
mnm
0
n
0
| · |hm
0
n
0
|ρ |mn i| (A.11)
and use the Cauchy-Schwarz inequality and the spectral decomposition ρ =
P
i
p
i
|ψ
i
ihψ
i
| such that
|hm
0
n
0
|ρ |mn i| = |
X
i
√
p
i
hm
0
n
0
|ψ
i
i
√
p
i
hψ
i
|mn i| (A.12)
≤
s
X
i
p
i
hm
0
n
0
|ψ
i
ihψ
i
|m
0
n
0
i
s
X
i
p
i
hmn|ψ
i
ihψ
i
|mn i =
p
hm
0
n
0
|ρ |m
0
n
0
ihmn |ρ |mn i.
Finally, one notes that for any single basis choice (labelled by k), one has
|c
mnm
0
n
0
| = |
X
j
ω
j(m−m
0
−n+n
0
)
|, (A.13)
which vanishes whenever (m − m
0
− n + n
0
) mod (d) 6= 0, and is equal to d otherwise, resulting in ˜γ
mm
0
nn
0
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.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 15
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 {|˜n
k
i}
n
labelled by k as the new ‘standard basis’. Then we can express
the vectors of the original standard basis {|m i}
m
as
|m i =
X
n
h ˜n
k
|m i |˜n
k
i =
1
√
d
X
n
ω
−nm−k m
2
|˜n
k
i =
1
√
d
X
n
c
(k, st.)
mn
|˜n
k
i, (A.14)
where c
(k, st.)
mn
= ω
−nm−k m
2
. Similarly, we can choose a decomposition into the conjugated basis vectors, i.e.,
|m i =
X
n
h ˜n
∗
k
|m i |˜n
∗
k
i =
1
√
d
X
n
c
(k, st.)∗
mn
|˜n
∗
k
i. (A.15)
We can then write the analogous expression to Σ in Eq. (A.8) as
Σ
(k, st.)
=
X
m
hmm |ρ |mm i =
1
d
2
X
n,n
0
q,q
0
X
m
c
(k, st.)∗
mn
c
(k, st.)
mn
0
c
(k, st.)
mq
c
(k, st.)∗
mq
0
h˜n
k
˜n
0∗
k
|ρ |˜q
k
˜q
0∗
k
i, (A.16)
which we can also split up into three terms, i.e.,
Σ
(k, st.)
= Σ
(k, st.)
1
+ Σ
(k, st.)
2
+ Σ
(k, st.)
3
, (A.17)
where
Σ
(k, st.)
1
:=
1
d
X
m,n
h ˜m
k
˜n
∗
k
|ρ | ˜m
k
˜n
∗
k
i =
1
d
, (A.18a)
Σ
(k, st.)
2
:=
1
d
X
m6=n
h ˜m
k
˜m
∗
k
|ρ |˜n
k
˜n
∗
k
i, (A.18b)
Σ
(k, st.)
3
:=
1
d
2
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
c
(k, st.)
mnm
0
n
0
h ˜m
0
k
˜n
0∗
k
|ρ | ˜m
k
˜n
∗
k
i, (A.18c)
while all other contributions to the sum in Eq. (A.16) vanish. The coefficient c
(k, st.)
mnm
0
n
0
appearing in Σ
(k, st.)
3
is
given by
c
(k, st.)
mnm
0
n
0
=
X
j
c
(k, st.)∗
jm
0
c
(k, st.)
jn
0
c
(k, st.)
jm
c
(k, st.)∗
jn
=
X
j
ω
−j(m−m
0
−n+n
0
)
, (A.19)
and since |c
(k, st.)
mnm
0
n
0
| = |c
mnm
0
n
0
|, the coefficient ˜γ
mm
0
nn
0
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
F (ρ, Φ) ≥
1
d
X
m
h ˜m
k
˜m
∗
k
|ρ | ˜m
k
˜m
∗
k
i +
X
m
hmm |ρ |mm i −
1
d
−
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
˜γ
mm
0
nn
0
q
h ˜m
0
k
˜n
0∗
k
|ρ | ˜m
0
k
˜n
0∗
k
ih ˜m
k
˜n
∗
k
|ρ | ˜m
k
˜n
∗
k
i. (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
0
as a mutually unbiased basis to construct our fidelity bound.
As before, we express these bases w.r.t. to each other as
| ˜m
k
0
i =
X
n
h ˜n
k
| ˜m
k
0
i |˜n
k
i =
1
d
X
n,p
ω
p(m−n)+p
2
(k
0
−k)
|˜n
k
i =
1
√
d
X
n
c
(k, k
0
)
mn
|˜n
k
i, (A.21)
where we have defined
c
(k, k
0
)
mn
:=
1
√
d
X
p
ω
p(m−n)+p
2
(k
0
−k)
. (A.22)
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 16
In order to continue, let us recall the formula for quadratic Gauss sums for odd roots of unity.
g(a : d) =
d−1
X
i=0
ω
ai
2
=
a
d
ε
d
√
d, (A.23)
where d is odd,
a
d
∈ {−1, 0, 1} is the Jacobi symbol and
ε
d
=
(
1 if d ≡ 1 mod 4
i if d ≡ 3 mod 4.
(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:
G(a, b, c) =
c−1
X
n=0
e
2πi
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:
G(a, b, c) = ε
c
√
c
a
c
e
−2πi
ψ(a)b
2
c
, (A.26)
where
ε
c
=
(
1 if c ≡ 1 mod 4
i if c ≡ 3 mod 4.
(A.27)
a
c
∈ {1, 0, −1} is the Jacobi symbol (in our case 1 or −1, because a and c are coprime) and ψ(a) is a number
such that 4ψ(a)a ≡ 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 = −
b
2a
and k = −
b
2
4a
to get
G(a, b, c) =
c−1
X
n=0
e
2πi
a(n−h)
2
+k
c
= e
2πi
k
c
c−1
X
n=0
e
2πi
a(n−h)
2
c
= e
−2πi
ψ(a)b
2
c
c−1
X
n=0
e
2πi
an
2
c
= e
−2πi
ψ(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 Z
c
:= {0, . . . , c−
1}.
In our case we can set a = k
0
−k, b = (m −n), and c = d, where d is an odd prime and gcd(k
0
−k, d) = 1 (as
both k 6= k
0
and k, k
0
∈ {0, . . . , d − 1}). Consequently, all conditions are met and we have
c
(k, k
0
)
mn
:=
1
√
d
X
p
ω
p(m−n)+p
2
(k
0
−k)
= ε
d
k
0
− k
d
ω
−ψ(k
0
−k)(m−n)
2
. (A.29)
We also need the relation between the two conjugated bases, i.e.,
| ˜m
∗
k
0
i =
X
n
h ˜n
∗
k
| ˜m
∗
k
0
i |˜n
∗
k
i =
1
√
d
X
n
c
(k, k
0
)∗
mn
|˜n
∗
k
i. (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,
Σ
(k, k
0
)
=
X
m
h ˜m
k
0
˜m
k
0
|ρ | ˜m
k
0
˜m
k
0
i =
1
d
2
X
n,n
0
q,q
0
X
m
c
(k, k
0
)∗
mn
c
(k, k
0
)
mn
0
c
(k, k
0
)
mq
c
(k, k
0
)∗
mq
0
h˜n
k
˜n
0∗
k
|ρ |˜q
k
˜q
0∗
k
i, (A.31)
which can again be split up into three terms, i.e.,
Σ
(k, k
0
)
= Σ
(k, k
0
)
1
+ Σ
(k, k
0
)
2
+ Σ
(k, k
0
)
3
, (A.32)
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 17
where
Σ
(k, k
0
)
1
:=
1
d
X
m,n
h ˜m
k
˜n
∗
k
|ρ | ˜m
k
˜n
∗
k
i =
1
d
, (A.33a)
Σ
(k, k
0
)
2
:=
1
d
X
m6=n
h ˜m
k
˜m
∗
k
|ρ |˜n
k
˜n
∗
k
i, (A.33b)
Σ
(k, k
0
)
3
:=
1
d
2
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
c
(k, k
0
)
mnm
0
n
0
h ˜m
0
k
˜n
0∗
k
|ρ | ˜m
k
˜n
∗
k
i, (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
(k, k
0
)
mnm
0
n
0
, which, with help of Eq. (A.29) can be written as
c
(k, k
0
)
mnm
0
n
0
=
X
j
c
(k, k
0
)∗
jm
0
c
(k, k
0
)
jn
0
c
(k, k
0
)
jm
c
(k, k
0
)∗
jn
= (ε
∗
d
)
2
ε
2
d
k
0
− k
d
4
X
j
ω
ψ( k
0
−k)[(j−m
0
)
2
−(j−n
0
)
2
−(j−m)
2
+(j−n)
2
]
= ω
ψ(k
0
−k)(m
0
2
−m
2
−n
0
2
+n
2
)
X
j
ω
ψ( k
0
−k)2j(m−m
0
−n+n
0
)
. (A.34)
It remains to be noted that |c
(k, k
0
)
mnm
0
n
0
| = |c
mnm
0
n
0
|. We can hence write the fidelity bound in the usual form
F (ρ, Φ) ≥
1
d
X
m
h ˜m
k
˜m
∗
k
|ρ | ˜m
k
˜m
∗
k
i +
X
m
h ˜m
k
0
˜m
∗
k
0
|ρ | ˜m
k
0
˜m
∗
k
0
i −
1
d
−
X
m6=m
0
,m6=n
n6=n
0
,n
0
6=m
0
˜γ
mnm
0
n
0
q
h ˜m
0
k
˜n
0∗
k
|ρ | ˜m
0
k
˜n
0∗
k
ih ˜m
k
˜n
∗
k
|ρ | ˜m
k
˜n
∗
k
i. (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
oF
) of our bipartite state from the
measurement data for any two MUBs. Here, the E
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
oF
is given by
E
oF
≥ log
2
(d) − H(A
1
|B
1
) − H(A
2
|B
2
), (A.36)
where H(A
i
|B
i
) is the conditional Shannon entropy for the i-th mutually unbiased basis (MUB), i.e.,
H(A
i
|B
i
) = H({ρ
(i)
jk
}) − H({ρ
(i)
j
}), for i = 1, 2, (A.37)
with
ρ
(i)
jk
= hjk|ρ|jki
i
, ρ
(i)
j
=
X
k
hjk|ρ|jki
i
. (A.38)
Since we know the terms {ρ
(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
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
oF
increases until approximately d = 31, where it reaches a maximum value of 4.0 ±0.1 ebits. As the
dimension is increased further, experimental imperfections and the inherent dimensionality limits of the source
bring this number back down.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 18
Figure A.4: Entanglement of formation (E
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
oF
= 4.0 ± 0.1 ebits for d = 31) is indicated by dotted lines.
Accepted in Quantum 2020-12-18, click title to verify. Published under CC-BY 4.0. 19