Quick Quantum Steering: Overcoming Loss and Noise with Qudits

Abstract

A primary requirement for a robust and unconditionally secure quantum network is the establishment of quantum nonlocal correlations over a realistic channel. While loophole-free tests of Bell nonlocality allow for entanglement certification in such a device-independent setting, they are extremely sensitive to loss and noise, which naturally arise in any practical communication scenario. Quantum steering relaxes the strict technological constraints of Bell nonlocality by reframing it in an asymmetric manner, with a trusted party only on one side. However, tests of quantum steering still require either extremely high-quality entanglement or very low loss. Here we introduce a test of quantum steering that harnesses the advantages of high-dimensional entanglement to be simultaneously noise robust and loss tolerant. Despite being constructed for qudits, our steering test is designed for single-detector measurements and is able to close the fair-sampling loophole in a time-efficient manner. We showcase the improvements over qubit-based systems by experimentally demonstrating quantum steering in up to 53 dimensions, free of the fair-sampling loophole, through simultaneous loss and noise conditions corresponding to 14.2-dB loss equivalent to 79 km of telecommunication fiber, and 36% of white noise. We go on to show how the use of high dimensions counterintuitively leads to a dramatic reduction in total measurement time, enabling a quantum steering violation almost 2 orders of magnitude faster obtained by simply doubling the Hilbert space dimension. Our work conclusively demonstrates the significant resource advantages that high-dimensional entanglement provides for quantum steering in terms of loss, noise, and measurement time, and opens the door toward practical quantum networks with the ultimate form of security.

Full text

Quick Quantum Steering: Overcoming Loss and Noise with Qudits
Vatshal Srivastav ,1,* Natalia Herrera Valencia,1Will McCutcheon ,1Saroch Leedumrongwatthanakun ,1
S´ebastien Designolle ,2Roope Uola,2Nicolas Brunner,2and Mehul Malik 1,†
1Institute of Photonics and Quantum Sciences, Heriot-Watt University,
Edinburgh EH14 4AS, United Kingdom
2Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
(Received 26 April 2022; revised 29 September 2022; accepted 21 October 2022; published 30 November 2022)
A primary requirement for a robust and unconditionally secure quantum network is the establishment of
quantum nonlocal correlations over a realistic channel. While loophole-free tests of Bell nonlocality allow
for entanglement certification in such a device-independent setting, they are extremely sensitive to loss and
noise, which naturally arise in any practical communication scenario. Quantum steering relaxes the strict
technological constraints of Bell nonlocality by reframing it in an asymmetric manner, with a trusted party
only on one side. However, tests of quantum steering still require either extremely high-quality
entanglement or very low loss. Here we introduce a test of quantum steering that harnesses the advantages
of high-dimensional entanglement to be simultaneously noise robust and loss tolerant. Despite being
constructed for qudits, our steering test is designed for single-detector measurements and is able to close the
fair-sampling loophole in a time-efficient manner. We showcase the improvements over qubit-based
systems by experimentally demonstrating quantum steering in up to 53 dimensions, free of the fair-
sampling loophole, through simultaneous loss and noise conditions corresponding to 14.2-dB loss
equivalent to 79 km of telecommunication fiber, and 36% of white noise. We go on to show how the use of
high dimensions counterintuitively leads to a dramatic reduction in total measurement time, enabling a
quantum steering violation almost 2 orders of magnitude faster obtained by simply doubling the Hilbert
space dimension. Our work conclusively demonstrates the significant resource advantages that high-
dimensional entanglement provides for quantum steering in terms of loss, noise, and measurement time,
and opens the door toward practical quantum networks with the ultimate form of security.
DOI: 10.1103/PhysRevX.12.041023 Subject Areas: Photonics, Quantum Physics
Quantum Information
I. INTRODUCTION
In today’s digital landscape riddled with threats such as
cyberattacks and information leaks, the advent of secure
quantum communication has strongly impacted modern
technological progress. Intense research advances have
been carried out in the past two decades to achieve a
secure and robust implementation of quantum communi-
cation between two distant parties [1–5]. The ultimate form
of security is provided in the scenario when the two parties,
Alice and Bob, can verify entanglement between them in a
device-independent (DI) manner [6], i.e., without requiring
any trust in their devices or the channels themselves. A
requirement for this form of entanglement distribution is a
test of quantum nonlocality, such as the loophole-free
violation of a Bell inequality, which has been demonstrated
over short-distance scales of up to a kilometer [7–9].
However, inevitable loss due to propagation and environ-
mental noise restrict the maximum distance over which
entanglement can be certified in a DI manner, making such
protocols vulnerable to attacks associated with the fair-
sampling loophole (where an eavesdropper could exploit an
unjustified fair-sampling assumption) [10,11]. In other
words, we must associate our inability to measure every
photon that was created with the actions of a malicious
eavesdropper “listening in”on the quantum conversation.
Closing the fair-sampling loophole is technologically
demanding, normally requiring extremely high overall
system detection efficiencies, which naturally imposes
practical limitations over realistic long-distance channels.
Quantum steering is an alternative scenario that relaxes
the rigid technical requirements of device-independent
entanglement certification. Here, one can assume that a
trusted measurement device exists only on one side
[12–14]. In this asymmetric one-sided device-independent
(1SDI) setting, entanglement is certified when the untrusted
*vs54@hw.ac.uk
†m.malik@hw.ac.uk
Published by the American Physical Society under the terms of
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PHYSICAL REVIEW X 12, 041023 (2022)
Featured in Physics
2160-3308=22=12(4)=041023(13) 041023-1 Published by the American Physical Society

party (say, Alice) is able to condition or “steer”the state of
the trusted party (Bob) through her local measurements.
Note that the fair-sampling loophole is a threat only on
Alice’s side due to her untrusted measurement apparatus, as
well as any loss or noise introduced in the untrusted
channel. Since Bob’s measurement devices are trusted,
he is exempt from the loophole. Figure 1(a) illustrates an
example of such a scenario, where Alice and the entangle-
ment source are located in different (untrusted) geographic
locations and are connected by an inaccessible (untrusted)
undersea fiber-optic channel.
The experimental detection of steering is conveniently
achieved via the violation of so-called steering inequalities
[15,16]. Experimental violations of such inequalities using
qubit entanglement with the fair-sampling loophole closed
were reported recently [17–19]. By increasing the number
of measurement settings used by both parties, the threshold
heralding efficiency required to demonstrate Einstein-
Podolsky-Rosen (EPR) steering with qubit entanglement
can be lowered arbitrarily in the limit of infinitely many
measurement settings. However, this is only possible if
Alice and Bob share a high-quality qubit-entangled state,
with higher loss requiring increasingly higher-state quali-
ties [18]. In addition to loss, any realistic quantum channel
will be prone to sources of noise such as dark and
background counts, copropagating classical signals, and
imperfect measurement devices. Furthermore, performing a
large number of measurements can be impractical, leading
to extremely long measurement times in an already lossy
scenario [18,20].
It has been established that high-dimensional entangle-
ment can overcome several limitations of qubit-entangled
systems [21,22]. Such entangled “qudits”can exhibit
stronger correlations than qubit entanglement [23,24]
and can tolerate lower heralding efficiency thresholds for
tests of nonlocality [25]. Qudits also offer advantages in
quantum key distribution [26–28], leading to higher key
rates and robustness to noise [29–33]. Notably, the large
dimensionality of photonic platforms has enabled entan-
glement distribution with greater noise resistance [34,35]
and higher information encoding capacities [36–40] than
qubits. In the realm of quantum steering, the use of qudits
has enabled demonstrations of genuine high-dimensional
steering as well as increased noise robustness [41,42].
However, these prior demonstrations do not address the
critical issue of loss and reconstruct multioutcome mea-
surements via multiple single-outcome measurements,
opening them up to the fair-sampling loophole. While
the noise robustness offered by high-dimensional entan-
glement makes it a strong contender for the realization
of device-independent quantum-communication protocols
[6,43], many practical considerations have hindered its
adoption—general multioutcome measurements of high-
dimensional quantum states of light are notoriously
difficult to realize, e.g., requiring arrays of cascaded,
unbalanced interferometers [44] or complex spatial-
mode-transformation devices [45,46]. In addition, they
are prone to impractically long measurement times and
suffer from loss and noise due to the lack of ideal multi-
outcome detectors.
Here, we overcome many of the challenges associated
with high-dimensional photonic systems through simulta-
neous advances in theory and experiment, allowing us to
demonstrate quantum steering with the fair-sampling loop-
hole closed under extreme conditions of loss and noise.
First, we formalize a set of linear steering inequalities
requiring only a single detector at each party, unlike
standard multioutcome measurements that require ddetec-
tors for measuring qudits. These inequalities are especially
formalized for high dimensions by binarizing projective
measurements onto mutually unbiased bases (MUBs) [47]
and exhibit the same loss tolerance and noise robustness as
FIG. 1. 1SDI entanglement distribution scenario. The illustra-
tion in (a) depicts an example of a fiber-based 1SDI quantum
network between two distant stations, Alice and Bob. A source
distributes entangled photon pairs to Alice and Bob via optical
fiber links. In the 1SDI scenario, the source, Alice’s base station,
and the fiber link between them are untrusted, for example, due to
their geographic location or compromised devices. On the other
hand, Bob’s station is trusted, and hence, imperfections on his
side are ignored. A suitable strategy to certify entanglement in
this scenario is given by quantum steering (b), where a bipartite
state ρAB is shared between Alice and Bob. Alice’s measurements
and the channel between her and the source are affected by loss
(η) and noise (p) and considered to be untrusted. By performing
local measurements xwith outcomes a, Alice conditions the
shared state ρAB to an assemblage σajxat Bob. Bob can perform
suitable measurements on his shared state and check if the state is
steerable (thus, entangled) by violating a steering inequality.
VATSHAL SRIVASTAV et al. PHYS. REV. X 12, 041023 (2022)
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their counterparts designed for multioutcome detectors
[48]. In contrast, however, these inequalities provide the
strong advantages of needing significantly fewer techno-
logical resources and being free from strong assumptions
on detector noise, such as background or accidental count
subtraction. We violate these steering inequalities exper-
imentally with photon pairs entangled in their discrete
transverse position momentum, in the presence of noise and
at a record-low heralding efficiency of 0.038 0.001
(approximately 14.2 dB), which is equivalent to the optical
loss of a 79-km-long telecommunications fiber [49].
Despite the number of single-outcome measurements
scaling with dimension Oðd2Þ, we show that the total
measurement time in large dcan be reduced considerably
thanks to the high statistical significance of a violation
in high dimensions for a fixed number of counts. We
experimentally verify this by comparing the violations at
two different dimensions (d¼23, 41) taken at two differ-
ent acquisition times under a fixed channel loss (approx-
imately 12.1 dB). As a result, we are able to dramatically
reduce the total measurement time from 2.53 h for d¼23,
to 2.5 min for d¼41. In both dimensions, we violate the
steering inequality by 10 standard deviations. Below, we
elaborate on the theoretical formulation of our binarized
linear steering inequality, followed by a detailed discussion
of the experimental implementation and results.
II. THEORY
In quantum steering, the untrusted party Alice conditions
the shared bipartite state ρAB through her measurement
operators fAajxg, where xdenotes her choice of measurement
and aher outcome. By doing so, she creates the conditional
(non-normalized) states σajx¼trA½ðAajx⊗1BÞρAB, also
known as an assemblage. Since the trusted party Bob has
access to the assemblage, he can check if σajxcan be
produced without the use of entanglement via a so-called
local hidden state (LHS) model [12]. A steering inequality
[15,16] allows Bob to detect assemblages that do not follow
any LHS model and thus demonstrate steering. Formally, it
consists of a set of (unnormalized) measurements fFajxgon
Bob’s side and a value βLHS such that, for all unsteerable
assemblages we have that
β≡X
a;x
trðFajxσajxÞ≤βLHS;ð1Þ
where βLHS is the maximum value of the functional βfor any
unsteerable, i.e., LHS assemblage. When using entangle-
ment and appropriate measurements, higher functional
values βQcan be obtained. Hence, steering is demonstrated
whenever the above inequality in Eq. (1) is violated, that is,
when βQ>βLHS.
Here, we formalize a set of linear steering inequalities
designed especially for single-outcome projective measure-
ments, which are particularly suitable for single-photon
detection systems that are widely implemented on photonic
platforms [50,51]. First, we begin with a steering inequality
in which Alice and Bob measure d-outcome projective
measurements in a given d-dimensional MUB, construc-
tions of which are known to exist for prime and prime
power dimensions [47]. Alice (Bob) measures Aajx(AT
ajx)
with an outcome a¼0;…;d−1in a basis given by x¼
0;…;m with m≤d; see Eq. (B1). We then binarize these
measurements such that Alice’s measurements become
˜
A˜
aj˜
xðηÞ¼ηAajxif ˜
a¼1;
1−ηAajxif ˜
a¼0;ð2Þ
where ηis the one-sided heralding efficiency at Alice. The
one-sided heralding efficiency is defined as the probability
that detecting a photon at the trusted party (Bob) heralds the
presence of a photon at the untrusted party (Alice). Note
that the binarized measurements
˜
A˜
aj˜
xare labeled by new
indices ˜
a¼0, 1 corresponding, respectively, to a no-click
or click event at the detector and ˜
x¼ða; xÞ, reflecting the
new set of measurement settings that includes every
projector from each basis. Bob’s measurements, which
are used to evaluate the steering inequality, are defined in a
similar fashion
˜
F˜
aj˜
x¼(AT
ajxif ˜
a¼1;
cð1−AT
ajxÞif ˜
a¼0;ð3Þ
where c¼1=ðffiffiffi
d
p−1Þis a constant chosen to facilitate
the derivation of a closed-form expression for an upper
bound
˜
βon the corresponding LHS bound given by
˜
βLHS ≤1þmðffiffiffi
d
pþ1Þ≡
˜
β; see Eq. (A8) in Appendix A.
Therefore, obtaining a value of the steering inequality
larger than
˜
βdemonstrates steering.
Second, in order to study the effects of noise suffered
by the shared state ρAB, one can consider the isotropic
state that includes a fraction of added white noise, specifi-
cally, ρABðpÞ≡pjϕdihϕdjþð1−pÞ1d2=d2, where jϕdi≡
Pd−1
i¼0jiii=ffiffiffi
d
pis the maximally entangled state in dimen-
sion d, and p∈½0;1is the noise mixing parameter.
Finally, we use Eq. (1) to evaluate the functional βQin
terms of one-sided heralding efficiency ηand mixing
parameter p; see Eq. (A9). To check the violation of the
steering inequality, we calculate the difference between
βQðη;pÞand
˜
β(upper bound on
˜
βLHS), which is given as
Δβ¼ηmp−1−p
ffiffiffi
d
p−1:ð4Þ
The condition Δβ>0must be satisfied in order to
demonstrate steering, which also leads to a critical value ηcr
for the one-sided heralding efficiency η. The critical
heralding efficiency ηcr generally depends on the noise
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parameter p, the number of MUB measurement settings m,
and the dimension of the Hilbert space d[see Eq. (A10)]. In
the noise-free case (p¼1), the critical one-sided heralding
efficiency can be reduced to ηcr ¼1=m, the lowest possible
value [52], which can also be obtained with qubits [18].
However, any experiment features noise, i.e., with a mixing
parameter p<1, originating from various technical imper-
fections ranging from detector dark counts or multipair
emission to misalignments in the system [35]. As the
amount of noise increases, it can be shown that the value of
ηcr required to demonstrate steering increases; see
Eq. (A10). For qubits (d¼2), the required critical effi-
ciency ηcr to demonstrate steering reaches unity when the
noise parameter p∼0.71 for m¼2measurements, while
steering is impossible for p≤1=2[12,53].
In contrast, Fig. 2shows that by increasing the number of
MUB measurement settings m, which is only possible in
the qudit regime (d>2), one can still demonstrate steering
in the presence of substantial loss and noise in the channel
as compared to qubit-based systems. For example, using
53 MUB settings, one can tolerate a heralding efficiency as
low as η¼0.044 and a noise parameter of p¼0.5. This
enables one to find the perfect trade-off between loss in the
channel and noise in the system. Additionally, with our
binarized steering inequalities, projective measurements
with only two outcomes (photon detected or not detected)
not only show the same loss tolerance and noise robustness
as their multioutcome counterparts for steering [48], but are
also more feasible to implement experimentally with two
single-click photon detectors that are typically available in
every photonics laboratory.
A key practical requirement for demonstrating these
methods is the acquisition of sufficient data to report results
within a desired confidence interval. By considering the
variance of our estimator for βQ, as well as the correspond-
ing expected violation Δβ, we can determine the required
measurement time. To explore this dependence, we provide
an explicit derivation of the required measurement time as a
function of the dimension in Appendix C.
III. EXPERIMENT
To experimentally demonstrate noise-robust EPR steer-
ing with the fair-sampling loophole closed, we use pairs of
photons entangled in their discrete transverse position-
momentum degree of freedom, also known as pixel
entanglement [54,55]. As shown in Fig. 3(a), two spatially
entangled photons at 1550 nm are generated in a 5-mm
periodically poled pottasium titanyl phosphate (ppKTP)
crystal through the process of type-II spontaneous para-
metric down-conversion using a Ti:sapphire femtosecond
pulsed laser with 500-mW average power. After being
separated by a polarizing beam splitter, each photon from
the entangled pair is directed to the two parties Alice and
Bob, who can perform local projective measurements by
using a spatial light modulator (SLM) to display tailored
phase-only holograms. These holograms allow the meas-
urement of a general superposition of spatial modes, i.e., a
state from any MUB. Only photons carrying the correct
states are efficiently coupled into single-mode fibers, which
lead to two superconducting nanowire detectors connected
to a coincidence logic. This measurement system allows
Alice and Bob to perform the local measurements in
Eqs. (2) and (3) and evaluate the elements of the functional
in Eq. (1) in terms of the normalized coincidence counts
between them and of the normalized exclusive single
counts measured by Bob; see Eq. (B6) in Appendix B.
To close the fair-sampling loophole on Alice’s side, her
measurement basis is designed using hexagonal pixels of
equal size with zero spacing between them. This enables
Alice to maximize the detection efficiency of her SLM
and channel. Bob’s measurement basis design is informed
by prior knowledge of the two-photon joint-transverse-
momentum amplitude (JTMA) [54], which allows him to
tailor his pixel mask in order to optimize the resultant one-
sided heralding efficiency in the experiment ηexp (see
Appendix B). The choice of the phase-only pixel basis
gives the added advantage that projective measurements in
mutually unbiased bases provide the highest possible SLM
heralding efficiency, since they do not require any ampli-
tude modulation in their holograms [56]. Furthermore, the
two-photon state encoded in the pixel basis can be designed
FIG. 2. Noise robustness and loss tolerance. Dependence of
the critical heralding efficiency ηcr for Alice with respect to the
noise parameter pin the bipartite shared state ρAB ðpÞwhen using
m¼dMUB measurement settings to demonstrate EPR steering
in dimension d. By increasing the number of measurement
settings m(or increasing the dimensions d), the critical efficiency
ηcr decreases substantially, even at significant noise levels
(p<1). For example, with m¼53 MUB settings, one can
tolerate a heralding efficiency as low as η¼0.044 and a noise
parameter p¼0.5(equal mixture of maximally entangled state
and white noise). In contrast, demonstrating steering with qubit
entanglement (d¼2) requires significantly higher heralding
efficiencies ηcr and a noise parameter p>0.5in the impractical
limit of infinite measurement settings (q→∞, dashed line) [18].
Thus, high-dimensional entanglement enables us to access
regimes of noise and loss that are inaccessible by qubit entan-
glement, even in the best-case scenario.
VATSHAL SRIVASTAV et al. PHYS. REV. X 12, 041023 (2022)
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to be close to a maximally entangled state in dimension d
owing to the knowledge of the JTMA of the generated two-
photon state, which maximizes its entanglement of for-
mation [54,55]. In our experiment, we utilize the crosstalk
between individual pixels as a reliable measure of the
system noise parameter [pexp; see Eq. (B10)]. This measure
is valid because the amount of crosstalk between discrete
spatial modes does not change substantially across the
MUBs and thus behaves isotropically (see Appendix Bfor
more details).
IV. RESULTS
In our experiment, we demonstrate detection-loophole-
free steering in up to dimension d¼53 by performing
binarized projective measurements in a number of MUBs
ranging from mmin ¼12 to mmax ¼53. We exclude the
computational (hex-pixel) basis in every dimension due to its
much higher loss. The system noise parameter (pexp) ranges
from pexp ¼0.823 to 0.625 and is shown in Fig. 4(a) for
every dimension. In dimensions d≥17, we introduce addi-
tional loss on Alice’s channel by decreasing the diffraction
efficiency of SLM-A and thus reducing the one-sided
heralding efficiency ηexp →ηcrðm¼dÞuntil the state
becomes just unsteerable [Fig. 4(b)]. For these reduced
ηexp, we are able to demonstrate steering with m¼d
MUB measurement settings. Using m¼53 MUB settings,
for example, we are able to tolerate a record-low one-sided
heralding efficiency of ηexp ¼0.038 0.001, and violate
our steering inequality by more than 8 standard deviations.
Note that this violation is obtained under noise conditions
corresponding to pexp ¼0.641. With lower noise, the one-
sided heralding efficiency can be lowered further.
We use our result in d¼41 to highlight the tolerance to
loss enabled by the use of high dimensions. Figure 4(c)
shows the critical one-sided heralding efficiency ηcrðmÞ
required to demonstrate steering in d¼41 using m
measurement settings, as a function of noise parameter
p.Asmis increased, the critical efficiency at a given noise
level can be significantly reduced. For example, at the fixed
noise level pexp ¼0.625, we are able to demonstrate
steering with a one-sided heralding efficiency of ηexp ¼
0.175 using m¼11 measurement settings. Note that this is
possible as long as ηexp satisfies ηexp >ηcrðmÞ.Aswe
increase the number of measurement settings up to m¼41,
Alice’s channel efficiency can be further reduced by a
FIG. 3. Experimental setup. (a) A Ti:sapphire pulsed laser at 775 nm is used to pump a nonlinear ppKTP crystal to generate a pair of
photons entangled in their transverse position momentum via type-II spontaneous parametric down-conversion. The pump photons are
filtered by a dichroic mirror (DM) and the down-converted photons are separated with a polarizing beam splitter (PBS). One photon
from the entangled photon pair is sent to the untrusted party, Alice, who generates holograms on a spatial light modulator (SLM-A) to
perform the projective measurements
˜
A˜
aj˜
x; see Eq. (2). Bob receives the other photon along with the outcome ˜
afrom Alice, forming the
assemblage σ˜aj˜x. He then performs the projective measurements
˜
F˜aj˜x[Eq. (3)] according to the steering inequality defined in Eq. (1).
Each of the photons in the selected mode are collected by a combination of telescope lenses (L4 and L5) into single-mode fibers (SMFs)
and are detected by superconducting nanowire detectors (SNSPDs). Coincident detection events corresponding to joint two-photon
measurements are recorded by a coincidence counting logic (CC) with a coincidence window of 0.2 ns. (b),(c) Experimental data
showing coincidence counts between Alice and Bob and exclusive single counts measured on Bob’s side in dimension d¼19 using
MUB measurement settings x¼0,1.
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factor of 4 to ηexp ¼0.044 and still demonstrate quantum
steering. This clearly demonstrates the increased robustness
to loss enabled by high dimensions in detection-loophole-
free steering violations. In general, the amount of loss and
noise tolerated can be optimized by working along the
critical one-sided heralding efficiency curves shown in
Fig. 4(c) for a given number of measurement settings.
A common problem encountered in experiments with
qudits is that the total measurement time (T) increases
drastically with the dimension of the Hilbert space, as the
total number of single-detector measurements scales as
md2. For our binarized steering inequality, we show that
this is indeed not the case when one considers a steering
violation within a fixed confidence interval. Consider the
example shown in Fig. 5, where for a given amount of noise
and loss, a steering violation is obtained only in dimensions
d≥23.Asdis increased, the total measurement time T
required to violate the steering inequality by 10 standard
deviations decreases dramatically till it attains a minimum,
and then gradually increases again. This allows us to
minimize Tover larger dimensions at which the resilience
against loss and noise is substantially high and still
demonstrate steering in a practical measurement time
(see Appendix Cfor a detailed derivation). We verify this
result in our experiment by comparing the experimental
data taken at two different dimensions (d1¼23,d2¼41)
using m¼dsettings, at a fixed one-sided heralding
efficiency ηexp ¼0.062 0.006 and noise parameter
pexp ¼0.775, but with two very different acquisition times
FIG. 4. Experimental results. (a) The experimental noise mix-
ing parameter pexp as a function of the state dimension d
corresponding to a (1−p) fraction of white noise present in
the state. (b) The critical one-sided heralding efficiency required
to demonstrate steering decreases as a function of d, as shown for
the noise-free case (ηwith p¼1, solid black curve) and nonideal
state quality (ηcr with pexp <1, dashed red curve). In dimensions
3≤d<17 (black crosses), our experimental one-sided heralding
efficiency is under the critical value [ηexp <ηcrðm¼dÞ]; thus,
the state does not demonstrate steering. In dimensions d≥17,we
are able to demonstrate steering with similarly low heralding
efficiencies ηexp (green squares) by using m¼dMUB settings.
The experimental heralding efficiency can be lowered further
(blue dots) by adding loss in Alice’s channel, allowing us to
demonstrate EPR steering in d¼53 with a record low ηexp ¼
0.038 and a noise parameter pexp ¼0.641. (c) To showcase the
loss tolerance of our binarized steering inequality, we focus on
the case at d¼41.Atηexp ¼0.175, one can demonstrate steering
with fewer measurement settings (m¼11). However, when
Alice’s channel efficiency is reduced by a factor of 4 to
ηexp ¼0.044, EPR steering can still be demonstrated by using
more measurement settings (m¼41).
FIG. 5. Dependence of total measurement time (T) on dimen-
sion (d). The total measurement time T(on log scale) required to
violate the steering inequality by 10 standard deviations can be
calculated from Eq. (C7) as a function of the prime dimension d
for m¼dmeasurement settings, for a fixed one-sided heralding
efficiency η¼0.062 and noise parameter p¼0.775. The total
time taken to obtain a steering violation using m¼23 measure-
ment settings in dimension d¼23 is 2 orders of magnitude larger
than that with m¼d¼41.
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(tac
1¼750 ms, tac
2¼2.2ms) (see Table I). For d1¼23, the
total measured singles counts per acquisition window are
around 104, while for d2¼41, counts are lowered over
100 times to about 70. The coincidence counts in d2¼41
are just a few counts per second, on the order of the
crosstalk (see Fig. 6). While these two cases both demon-
strate a violation of the steering inequality by 10 standard
deviations, d2¼41 achieves a substantial reduction in the
total measurement time (T1¼2.53 handT2¼2.53 min).
Note that the response time of the spatial light modulators is
excluded from the total measurement time T,asthisisa
technical limitation that can be addressed by using a fast phase
modulation device such as a digital mirror device [57].
V. CONCLUSION AND OUTLOOK
We introduce a set of noise-robust EPR steering inequal-
ities in high dimensions, specially designed for single-
outcome detectors, that showcase the same loss tolerance as
their multioutcome counterparts. We report the violation of
our EPR steering inequality with the fair-sampling loophole
closed, with a record-low one-sided heralding efficiency of
η¼0.038 and a noise mixing parameter of pexp ¼0.641,
which is equivalent to the loss in a 79-km-long telecom-
munication fiber and 35.9% white noise. We are able to
achieve this level of noise and loss tolerance by harnessing
spatial entanglement in Hilbert space dimensions up
to d¼53. The amount of loss and noise that these
inequalities can tolerate can be increased substantially by
increasing the entanglement dimension d. For instance,
in d¼499, the maximum tolerable loss to demonstrate
steering increases to 27 dB, which is equivalent to the loss
in a 135-km-long telecom optical fiber. Similarly, the
maximum amount of white noise that can be allowed in
the system while violating the steering inequality at unit
heralding efficiency reaches 95%. Finally, we demonstrate
that the total measurement time required to demonstrate
steering can be significantly reduced through the use of
high dimensions. For a fixed amount of noise and loss, the
total integration time required to violate our steering
inequality by 10 standard deviations can be lowered
60 times through a modest increase to the Hilbert space
dimension (d¼23 to d¼41).
Our demonstration of time-efficient quantum steering
under realistic environmental conditions holds signifi-
cant promise for the future development of one-sided
device-independent quantum-communication protocols.
Any long-distance communication channel, such as one
relying on optical fiber or free-space transmission, neces-
sarily includes loss due to light leakage or scattering and
noise due to modal dispersion or atmospheric turbulence.
The steering inequalities presented here demonstrate a way
to overcome these detrimental effects through the use of
high-dimensional entanglement. It is important to note that
our methods, in particular, our single-detector steering
inequality, are not limited to the spatial degree of freedom,
but can readily be extended to other photonic properties
such as time frequency [58] or path encoding in a photonic
integrated circuit [59]. Our methods could also be helpful in
the demonstration of quantum memory networks on pho-
tonic platforms, which currently require high-efficiency
channels for transmission, thus extending the scalability
limits of long-range quantum networks [60,61].Our
resource-efficient steering protocol fulfills an important
prerequisite for establishing security between malicious
servers and clients, and future work toward greater device
independence for attacks on detector efficiency, for in-
stance [62], would further progress 1SDI quantum key
distribution [63], as well as private quantum computing and
related protocols [64]. Above all else, our demonstration of
quantum steering under prohibitive conditions of noise and
loss shows that the fundamental phenomenon of entangle-
ment can indeed transcend the limits imposed by a realistic
environment, when one makes full use of the inherently
high-dimensional photonic Hilbert space.
FIG. 6. Time-efficient steering in high dimensions. The coinci-
dence counts between Alice and Bob and exclusive single counts
measured by Bob in measurement basis x¼1in dimension
(a) d¼23 and (b) d¼41 at a fixed one-sided heralding
efficiency of ηexp ∼0.062, detected with two different acquisition
times. Interestingly, even though the counts in d¼41 are 2
orders of magnitude lower than in d¼23, we still violate the
steering inequality with 10 standard deviations in both cases.
TABLE I. Experimental measurement time Trequired for a
steering violation in two different dimensions d.
dηexp tac ThSi
23 0.063 0.001 750 ms 2.53 h ∼104
41 0.062 0.006 2.2 ms 2.53 min ∼70
QUICK QUANTUM STEERING: OVERCOMING LOSS AND NOISE …PHYS. REV. X 12, 041023 (2022)
041023-7

ACKNOWLEDGMENTS
This work is made possible by financial support from the
QuantERA ERA-NET Co-fund (FWF Project No. I3773-
N36), the UK Engineering and Physical Sciences Research
Council (Grant No. EP/P024114/1), and the European
Research Council Starting Grant PIQUaNT No. 950402.
N. B., S. D., and R. U. acknowledge financial support
from the Swiss National Science Foundation (Project
No. 192244, Ambizione Grant No. PZ00P2-202179, and
NCCR QSIT).
APPENDIX A: DETAILS FOR
˜
βAND βQðη;pÞ
First, we compute an upper bound
˜
βon the LHS bound
˜
βLHS for the Bob’s binarized measurements defined in
Eq. (3). We begin with the general definition of LHS bound
given in Ref. [13]
˜
βLHS ¼max
˜μ


X
˜a;˜x
˜
D˜μð˜
aj˜
xޘ
F˜
aj˜
x


∞
;ðA1Þ
which can be rewritten as
˜
βLHS ¼max
˜μ
˜
βLHS
˜μ;where
˜
βLHS
˜μ≡


X
˜
x
F˜μ˜xj˜x


∞
:ðA2Þ
For a given deterministic strategy ˜μ, we write ˜
x∈˜μfor all
˜
x¼ða; xÞsuch that ˜μ˜
x¼1and the complement (corre-
sponding to ˜μ˜
x¼0)is ˜
x∉˜μ. Following this notation, j˜μjis
the number of ˜
xsuch that ˜
x∈˜μ. Using the definition of
Eq. (3), we get
˜
βLHS
˜μ¼


X
˜
x∈˜μ
AajxþcX
˜
x∉˜μð1−AajxÞ


∞ðA3Þ
¼c½mðd−1Þ−j˜μj þ ð1þcÞ


X
˜
x∈˜μ
Aajx


∞
:ðA4Þ
The expression in Eq. (A4) is true using the fact that for
any positive semidefinite α1þB, where αis real (not
necessarily positive) constant and Bis positive semi-
definite with eigendecomposition Pjλjjλjihλjj, can be
written as
kα1þBk∞¼max
jλjþα;ðA5Þ
since the eigenvalues λjþαare positive by assumption.
To continue the computation after Eq. (A4), we make use
of the bound



X
˜
x∈˜μ
Aajx


∞
≤1þj˜μj−1
ffiffiffi
d
p;ðA6Þ
which is similar to results given in Refs. [65–67], though it
becomes increasingly loose particularly for j˜μj>m.We
then get
˜
βLHS
˜μ≤ð1þcÞð ffiffiffi
d
p−1Þ
ffiffiffi
d
pþcmðd−1Þ
þ1−cðffiffiffi
d
p−1Þ
ffiffiffi
d
pj˜μj;ðA7Þ
so that there is a natural choice of c¼ð1=ffiffiffi
d
p−1Þthat
allows us to get an upper bound independent of ˜μ. With this
value, we eventually obtain the desired bound
˜
βLHS ≤1þmðffiffiffi
d
pþ1Þ≡
˜
β:ðA8Þ
Next, we evaluate the quantum value βQof the functional
in Eq. (1) for binarized measurements of Alice and Bob
performed on a shared maximally entangled state influ-
enced by isotropic noise ρABðpÞin terms of one-sided
heralding efficiency η, mixing parameter p, and we get
βQðη;pÞ¼m1−p
d½ηþðd−ηÞð ffiffiffi
d
pþ1Þ
þpðηþffiffiffi
d
pþ1Þ;ðA9Þ
where dis the dimension of the Hilbert space, and mis the
number of MUB measurement settings out of dþ1MUBs
given for prime dimension d[47].
With βQðη;pÞand
˜
β, we can simply calculate Δβ[see
Eq. (4)], from where we can check that to violate the
steering inequality, ηmust be greater than a threshold ηcr,
η>1
mðp−1−p
ffiffid
pÞ
≡ηcr:ðA10Þ
For a pure maximally entangled state (p¼1), the above
threshold reduces to simply ηcr ¼ð1=mÞ.
APPENDIX B: EXPERIMENTAL DETAILS
Because of our loose pump focusing on the ppKTP
crystal, we have an initial heralding efficiency of approx-
imately 50%. Additional losses are introduced by our
measurement apparatus, where spatial light modulators
have a diffraction efficiency of approximately 70%, the
telescopes and single-mode fibers used to mode match the
collection to multiple spatial modes result in a coupling
efficiency of approximately 50%, and our superconductor
nanowire detectors have an efficiency of approximately
90%. As shown in Ref. [54], Bob’s projective measure-
ments can be optimized from the prior information of the
JTMA, such that the resultant one-sided heralding effi-
ciency for Alice increases. In our experiment, we employ
VATSHAL SRIVASTAV et al. PHYS. REV. X 12, 041023 (2022)
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the same optimization to tailor the pixel masks used for
Bob’s projective measurements, resulting in Bob’s pixel
sizes being smaller than that of Alice (see Fig. 7). This
results in the maximum one-sided heralding efficiency of
17.5% in d¼41. Additionally, the spacing between the
macropixels in Bob’s mask is chosen such that the
coincidence counts between different modes (crosstalk)
are suppressed [see Fig. 3(a)].
The holograms on SLM on Alice and Bob perform
projective MUB measurements Aajx¼jφx
aihφx
ajon the
entangled photons. They are designed according to the
prescription given in Ref. [47]
jφx
ai¼ 1
ffiffiffi
d
pX
d−1
l¼0
ωalþxl2jli;ðB1Þ
where ω¼expð2πi=dÞis a dth root of the unity. The first
basis is the computational one (individual pixel mode)
denoted fjligd−1
l¼0and the other dbases are fjφx
aigd−1
a¼0
labeled by x¼0;…;d−1.
To evaluate our steering inequality in the experiment, we
use coincidence counts (when both Alice’s and Bob’s
detectors click simultaneously) and Bob’s exclusive single
counts (when only Bob’s detectors click while Alice
measures no clicks). For xth basis outcome or projector
aand bon Alice and Bob, the coincidence counts (Cx
ab) and
exclusive single counts on Bob’s side (Sx
ab) are given as
Cx
ab ≔Na
xtr½A1jax ⊗ΠbjxρAB;ðB2Þ
Sx
ab ≔Na
xtr½A0jax ⊗ΠbjxρAB;ðB3Þ
where Na
xis the total count measured by Bob. Hence, it
must satisfy Nx
a¼PbðCx
ab þSx
abÞ.
Bob can then normalize the data by
˜
Cx
ab ≔Cx
ab=Nx
a;ðB4Þ
˜
Sx
ab ≔Sx
ab=Nx
aðB5Þ
to obtain the steering inequality elements,
tr½˜
F˜
aj˜
xσ˜
aj˜
x¼8
<
:
˜
Cx
aa;˜
a¼1;
cP
b≠a
˜
Sx
ab;˜
a¼0;ðB6Þ
and the inequality can be evaluated by
βQ¼X
a;x ˜
Cx
aa þcX
b≠a
˜
Sx
ab:ðB7Þ
Similarly, we calculate the one-sided heralding efficiency
ηexp for the xth measurement setting on Alice’s channel
from
ηexp ¼X
a;b
˜
Cx
ab:ðB8Þ
Note that in our work we perform only m¼dMUB
measurements excluding the computational basis. The one-
sided heralding efficiency ηexp for each MUB measurement
(not the computational basis) in dimensions ddoes not vary
significantly. To characterize the level of noise in the
system, we use the amount of crosstalk vbetween pixel
modes:
v¼Pa
˜
Cx
aa
Pab
˜
Cx
ab
:ðB9Þ
This estimate is valid because the vdoes not change
substantially across the MUBs, and thus it behaves isotropi-
cally. Formally, the mixing parameter pexp in the experi-
ment is given as
pexp ¼vd −1
d−1:ðB10Þ
APPENDIX C: MINIMIZING TOTAL
MEASUREMENT TIME T
The expectation (mean) number of coincidences and
exclusive single counts at Bob’s side are
FIG. 7. Optimization of one-sided heralding efficiency. The
one-sided heralding efficiency, or the probability that detecting a
photon at the trusted party (Bob) heralds the presence of a photon
at the untrusted party (Alice), can be optimized from knowledge
of the two-photon JTMA [54]. The size of Bob’s hex pixels can
be set in such a manner that the probability of coincidence counts
between Alice and Bob is increased while the probability of
single counts at Bob is decreased, effectively increasing the one-
sided heralding efficiency ηat Alice.
QUICK QUANTUM STEERING: OVERCOMING LOSS AND NOISE …PHYS. REV. X 12, 041023 (2022)
041023-9

hCx
abi¼Nηpδab
dþ1−p
d2;ðC1Þ
hSx
abi¼N
d−Cx
ab;ðC2Þ
where N¼Rtac is the total number of copies of the state
Bob receives during an acquisition window tac given the
underlying single-count rate Rdetected at Bob’s side. In
our experiment, we assume that the single-count rates R
does not vary significantly for different dimensions. Since
the statistics of the raw counts are Poissonian, their
variances are VarðCx
abÞ¼hCx
abiand VarðSx
abÞ¼hSx
abi.
We can estimate the variance of βQby
VarðβQÞ¼X
xab  ∂β
∂Cx
ab2
VarðCx
abÞþ∂β
∂Sx
ab2
VarðSx
abÞ;
ðC3Þ
which is inversely proportional to N(and therefore, Rand
tac), so we can factor out
VarðβQÞ¼fðη;p;d;mÞ
N;ðC4Þ
where fðη;p;d;mÞis a function of η,p,d, and mwhich is
independent of N(equivalent to Rand tac). We wish to find
the dimension that minimizes the total experiment time T¼
tacmd2while violating the steering inequality by 10
standard deviations. For an expected violation Δβ,we
require
Δβ≥10 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
VarðβQÞ:
qðC5Þ
From Eqs. (C4) and (C5), we can then solve for N,
N≥102fðη;p;d;mÞ
ðΔβÞ2;ðC6Þ
which is valid only when Δβ>0. We then evaluate the
total measurement time to saturate the bound,
T¼Nmd2
R¼md2102
R
fðη;p;d;mÞ
ðΔβÞ2:ðC7Þ
Interestingly, at fixed value of ηand p, for m¼dmea-
surement settings, the expression md2fðη;p;d;mÞ=ðΔβÞ2
depends nonmonotonically on dimension d. This makes the
total measurement time Tto reach a minimum at a
nontrivial dimension d. At fixed heralding efficiency
η¼0.062, noise level p¼0.775, and m¼d, the expres-
sion of Tcan be simplified and is given as
T¼d2
R½5d5
2−14d2þ100d3
2þ107d−3ffiffiffi
d
pþ0.19
ðffiffiffi
d
p−1Þð1þ0.01 ffiffiffi
d
p−0.05dÞ2:ðC8Þ
The plot in Fig. 5shows the dependence of Twith respect
to dat rate R∼105.
APPENDIX D: EXTENSION TO GENERAL
STEERING INEQUALITIES
A crucial question is whether other families of steering
inequalities defined for all dimensions will have the same
benefit of decreased measurement times from increased
dimensions. While each family of inequalities must be dealt
with individually, we can identify the prerequisite. Under
weak assumptions, we can show an optimum nontrivial
dimension that likely exists where the total measurement
time is minimum.
We start by fixing the measure of confidence, i.e., the
number of standard deviations σnof a steering violation:
βQ−βLHS
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
VarðβQÞΓ
p¼σn:ðD1Þ
Here, VarðβQÞΓis the variance of the βQmeasured
experimentally in a total of Γtrials (corresponding to
Γ¼Nmd2from Appendix C). The term βQ−βLHS is the
amount of violation for the steering inequality. It is
important to note that the variance of a maximum like-
lihood estimator, here VarðβQÞΓ, goes as 1=Γ. Therefore,
we can write
VarðβQÞΓ¼1
ΓVarðβQÞΓ¼1≡
VarðβQÞ
Γ:ðD2Þ
The number of trials Γis proportional to the total
measurement time T[see Eq. (C7)]. Using Eqs. (D1)
and (D2), we can then further simplify
T∝Γ¼σ2
n
VarðβQÞ
ðβQ−βLHSÞ2;ðD3Þ
where σnis fixed. Note that the above expression is defined
only when there is a nonzero violation of a steering
inequality, i.e., βQ−βLHS >0. The total measurement time
Twill decrease with increasing dimension dif
ΔT
Δd∝1
ΔdΔVarðβQÞ
ðβQ−βLHSÞ2<0:ðD4Þ
The condition in Eq. (D4) holds for the steering inequalities
that are defined for qudits, such as proposed in Appendix A
and in Refs. [24,41,68].
In our setup, for instance, at fixed loss and noise
(η¼0.062,p¼0.775), there are no violations for prime
dimensions d≤d0¼19. For prime dimensions d>19,
VATSHAL SRIVASTAV et al. PHYS. REV. X 12, 041023 (2022)
041023-10

violation does occur. This provides a nontrivial dependence
of total measurement time Ton dimension das in Eq. (C8).
Furthermore, a finite dimension which minimizes T
requires that it is nonmonotonic on d>d
0. This is ensured
if the above conditions hold and limd→∞T¼∞[as is the
case for Eq. (C8), where at limd→∞T¼d4]. Note that each
inequality has to be analyzed case by case. Extending these
techniques to suit all families of steering inequalities would
be an interesting direction for future work.
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