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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

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PHYSICAL REVIEW X 12, 041023 (2022)

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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˜

xðηÞ¼ηAajxif ˜

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;1is 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

Δβ¼ηmp−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

QUICK QUANTUM STEERING: OVERCOMING LOSS AND NOISE …PHYS. REV. X 12, 041023 (2022)

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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.

QUICK QUANTUM STEERING: OVERCOMING LOSS AND NOISE …PHYS. REV. X 12, 041023 (2022)

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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.

VATSHAL SRIVASTAV et al. PHYS. REV. X 12, 041023 (2022)

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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˜

xÞ˜

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;pÞ¼m1−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)

041023-8

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˜

x¼8

<

:

˜

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

abi¼Nηpδab

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

ab2

VarðCx

abÞþ∂β

∂Sx

ab2

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

ΔdΔVarðβQÞ

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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