Conclusive quantum steering with superconducting transition edge sensors

1 MB·English

by  Devin H. Smith

Conclusive quantum steering with superconducting transition edge sensors
Devin H. Smith,1,2,∗ Geoff Gillett,1,2 Marcelo P. de Almeida,1,2 Cyril Branciard,2
Alessandro Fedrizzi,1,2 Till J. Weinhold,1,2 Adriana Lita,3 Brice Calkins,3 Thomas
Gerrits,3 Howard M. Wiseman,4 Sae Woo Nam,5 and Andrew G. White1,2
1Centre for Engineered Quantum Systems and Centre for Quantum Computation and Communication Technology,
University of Queensland, Brisbane, QLD 4072, Australia
2School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
3National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA
4Centre for Quantum Computation and Communication Technology,
Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia
5National Institute of Standards and Technology, 325 Broadway, Boulder CO 80305, USA
Quantumsteeringallowstwopartiestoverifysharedentanglementevenifonemeasurementdevice
is untrusted. A conclusive demonstration of steering through the violation of a steering inequality
2
is of considerable fundamental interest and opens up applications in quantum communication. To
1
dateallexperimentaltestswithsinglephotonstateshavereliedonpost-selection,allowinguntrusted
0
devicestocheatbyhidingunfavourableeventsinlosses. Hereweclosethis“detectionloophole”by
2
combining a highly efficient source of entangled photon pairs with superconducting transition edge

n sensors. We achieve an unprecedented ∼62% conditional detection efficiency of entangled photons
a andviolateasteeringinequalitywiththeminimalnumberofmeasurementsettingsby48standard
J
deviations. Our results provide a clear path to practical applications of steering and to a photonic

1 loophole-free Bell test.
1

] Quantumentanglementenablesunconditionallysecure Bell inequalities [12], one can derive experimental cri-
h
p communication and powerful devices such as quantum teria [13] to demonstrate steering: steering inequalities
- computers. In their strongest form, the correlations as- impose limits on the observable correlations that can be
t
n sociated with entanglement rule out locally causal world explained without the need to invoke quantum steering.
a views in Bell tests [1]. In a weaker regime, quantum An experimental violation of a steering inequality was
u correlations can still be harnessed to steer quantum recently reported in [14].
q
states, demonstrating that two parties, Alice and Bob, We define the steering task as depicted in Fig. 1: Al-
[
share entanglement even if one party is untrusted [2]. ice and Bob receive quantum states from a source and

3 Quantum steering was originally introduced by Erwin measurethemusingrandomlychosenmeasurementsfrom
v Schr¨odinger [3], in reaction to the Einstein, Podolsky a prearranged set; if the observed correlations violate a
9 and Rosen (EPR) “paradox” [4]; it describes the abil- steeringinequality,thenAliceandBobwillbeconvinced
2
ity to remotely prepare different ensembles of quantum that their shared states were entangled. This holds true
8
statesbyperformingmeasurementsononeparticleofan evenifAliceandBobtrustneitherthesourcenorAlice’s
0
. entangled pair, demonstrating the paradox. Depending measurement device.
1
on the measurement and its random outcome, the re- Similarly to the case of Bell inequalities, a conclusive
1
1 mote system is prepared in a different state; however, violationof asteeringinequality requiresthat theexper-
1 the unconditioned remote state remains unaffected, thus imentdoesnotsufferfromanyrelevantloopholes. When
: preventinganypossiblesuperluminalsignalling. Interest- one party has untrusted equipment, the so-called detec-
v
ingly, steering and Bell nonlocality were recently found tionloophole[15]inparticulariscritical: ifAliceandBob
i
X to be inextricably linked with Heisenberg’s uncertainty have to post-select their data on Alice’s detected events,
r principle [5]. then low efficiencies enable her measurement devices to
a The original form of the EPR paradox was demon- cheat by dropping unfavourable results—in the context
strated experimentally with atomic ensembles [6], of quantum key distribution, for instance, this would al-
continuous-variablestates[7,8],andposition-momentum low the untrusted supplier of the devices to access the
entangled single photons[9]. More recently, quantum key. The fair sampling assumption invoked in [14] for
steering was redefined in a quantum information con- instance is not satisfactory for such untrusted devices.
text in [2], promising new applications such as quan-
Here we close the detection loophole by using an en-
tum communication using untrusted devices [10]. This
tangled photon source with high pair collection effi-
newformalisationalsoallowsastrictcomparisonbetween
ciency [16, 17] and highly-efficient transition edge sen-
the concepts of Bell nonlocality, steering and entangle-
sors [18]. We test a steering inequality which naturally
ment [2]. In analogy to entanglement witnesses [11] and
accountsforAlice’snon-detectedevents,andviolateitby
at least 48 standard deviations with the minimum num-
ber of two measurement settings and by more than 200
standard deviations for measurements in three different
∗ [email protected] bases.
2
Averaging this over Alice’s results, we define
E(cid:2)(cid:104)Bˆ (cid:105)2 (cid:3) ≡ (cid:88) P(A =a) (cid:104)Bˆ (cid:105)2 . (1)
i Ai i i Ai=a
a=±1,0
As shown in the Methods section, if the correlation P
could be explained by the source sending Bob an un-
entangled two-level system (qubit)—that is, in the ter-
minology of [2], if the correlation admits a “local hid-
den state” model—and if Bob implements qubit mea-
FIG. 1. Steering Gedankenexperiment. Alice and Bob re- surements in two or three mutually unbiased bases, for
ceive particles from a black box (the source, S) and want to instance the Pauli Xˆ, Yˆ and Zˆ operators, then the fol-
establish whether these are entangled. From a prearranged lowing inequality holds:
set,theyeachchoosemeasurementstobeperformedontheir
respective particles. Bob’s measurement implementation is N
trusted, but this need not be the case for Alice’s; her mea- S ≡ (cid:88) E(cid:2)(cid:104)Bˆ (cid:105)2 (cid:3) ≤ 1. (2)
surementdeviceisalsotreatedasablackboxfromwhichshe N i Ai
i=1
gets either a “conclusive”, A = ±1, or a “non-conclusive”
i
outcome, A = 0. To demonstrate entanglement, Alice and Note that the upper bound above depends crucially on
i
Bob need to show that she can steer his state by her choice Bob’s measurement settings, which in experimental im-
of measurement. They can do so through the violation of a plementation will not be perfectly orthogonal, nor per-
steering inequality: whenever Bob’s apparatus detects a par- fectly projective; we detail in the Methods section how
ticle, Alice needs to provide her measurement result. If the theboundmustbecorrectedtoaccountforexperimental
recordedcorrelationsoftheirmeasurementresultssurpassthe
imperfections.
boundimposedbythesteeringinequality,AliceandBobhave
Quantum mechanics allows a violation of inequal-
conclusively proven the entanglement of their particles.
ity(2),whichthusimpliessteering. Togetafirstinsight,
suppose that Alice and Bob share Werner states of vis-
ibility V, ρ = V |ψ−(cid:105)(cid:104)ψ−|+(1−V)11/4, where |ψ−(cid:105) is
RESULTS theBellsingletstate,andthatAliceimplementsthesame
measurements as Bob. Then, due to the anti-correlation
of the singlet state, (cid:104)Bˆ (cid:105) = ∓V: this illustrates
A quadratic steering inequality for qubits i Ai=±1
that Alice can steer Bob’s state to be aligned with her
measurement axis, limited by the visibility of the shared
Todemonstratesteering,AliceandBobneedtobeable state. If Alice has a probability η of getting a conclusive
to freely choose and perform different measurements; we outcomewheneverBobgetsone, thenE(cid:2)(cid:104)Bˆ (cid:105)2 (cid:3)=ηV2.
considerthecasewhereeachofthemcanperformN =2 This implies that the steering inequality (2)iwAiill be vio-
or 3 measurements, labeled i,j = 1,2 or 3, with a pri- lated if
ori binary outcomes A ,B = ±1, see Fig. 1. Since Bob
i j
trusts his measuring device his measurement can be de- ηV2 >1/N. (3)
scribed by a well characterised quantum observable Bˆ .
j
Satisfying the requirements given by Eq. (3) in a pho-
Heconsidersonlythecaseswherehismeasurementgives
tonic architecture is challenging. For the minimal set of
him a conclusive result, that is, when at least one of his
N=2 measurement settings, an experimental test of the
detectors clicks — if both click, Bob outputs a random
steering inequality (2) requires, even for a pure entan-
result. In contrast, Alice’s devices are not trusted and
gled singlet state with visibility V=1, that Alice detects
hermeasurementapparatusisconsideredablackbox. It
a signal more than η>50% of the times Bob requests a
returns outcomes A = ±1, indicating conclusive mea-
i
response. To reach these requirements, the experimental
surement results, or A = 0 when no or both detectors
i
apparatus has to be carefully optimised.
fire. Because Alice must output a result whenever Bob
registers an event, these inconclusive results cannot be
discarded from further analysis.
Experimental setup
The correlation observed by Alice and Bob can be de-
scribed by the probability distribution P(A = a,B =
i j Weperformedourexperimentusingentangledphotons
b), witha=±1or0, andb=±1. IfBobreceivesastate
created in a polarisation Sagnac source based on spon-
that is not entangled to Alice’s the set of possible corre-
taneous parametric downconversion [16, 17], see Fig. 2.
lations will be restricted, as the inequality below shows.
Thissourcedesignmeetstwocrucialrequirements;ahigh
First, we define Bob’s expectation value for a measure-
entangled-pair collection efficiency and near-ideal polar-
ment conditioned on Alice’s result:
isation entanglement.
We followed [17] in the basic design of our source.
(cid:104)Bˆ (cid:105) ≡ P(B =+1|A =a) − P(B =−1|A =a). To maximise the conditional coupling between the Alice
i Ai=a i i i i
3
ton number resolution and high detection efficiency of
TES up to 95% at 1550nm, while being virtually free of dark
LP
counts [18]. Our detectors were optimised for 810nm
L2
ppKTP Alice L2 PBS SPAD with an optical cavity similar to that presented in an
earlier work [18], with an estimated detection efficiency
BD for 820nm photon in the 1550nm single-mode SMF-28
BP HWP fibre connected to this cavity to be larger than 97%. In
LP
L2 QWP practice, the measured detection efficiencies of our two
LD
Bob TES were 1.50 and 1.56 times higher than the efficiency
L1 of our reference SPAD at 820nm.The dominant source
PC Source ofopticalloss,whichleadstotheseless-than-optimalfig-
ures, was a splice between the single mode 820nm fibres
connected to the source and the fibres connected on the
FIG. 2. Experimental scheme. Polarisation-entangled two- TES, which were single mode at 1550nm.
photon states are generated in a periodically poled 10mm The TESs were operated between 40mK and 75mK
KTiOPO (ppKTP)crystalinsideapolarisationSagnacloop
4 and yielded analog output pulses with a rise time of
[16, 17]. The continuous wave, grating-stabilised 410nm
∼320ns and jitter of ∼78ns. In order to detect coin-
pumplaser(LD)isfocussedintothiscrystalwithanaspheric
cidences between the TES signals and the TTL pulses
lens(L1,f=4.0mm)anditspolarisationissetwithafibrepo-
generated by the SPADs each amplified TES signal was
larisationcontroller(PC)andahalf-waveplate(HWP),con-
trollingtheentangledoutputstate[16]. Bobfiltershisoutput digitised with a constant fraction discriminator; because
photon with a long-pass glass filter (LP) and a 3nm band- the TESs rethermalise after each detection event, with a
passfilter(BP),beforecollectingitwithanasphericlens(L2, relaxationperiodof∼2µs,thenon-numberresolvingdis-
f=18.4mm) into a single-mode fibre. He performs his mea- criminators were set to impose a dead time of the same
surementinanexternalfibrebridge,withacombinationofa length to avoid false detections during the TES relax-
quarter-waveplate(QWP),HWP,apolarisingbeamdisplacer ation period. To match the delay caused by the TES
(BD) and multi-mode-fibre-coupled single-photon avalanche
detection system, Bob’s SPAD signals were delayed by
diodes (SPADs). To minimise loss, Alice performs her mea-
∼450ns. Coincident events were then detected with a
surement directly at the source using a QWP, HWP and a
field-programmable gate array with a timing window of
polarisingbeamsplitter(PBS),followedbyaLPfilterandfi-
98ns.
bre collection with focussing optics identical to Bob’s, finally
The long dead time period imposed by our electron-
detecting her photons with highly efficient, superconducting
transition edge sensors (TESs) [18]. ics leads to a rate-dependent loss and we therefore oper-
atedthesourceatcomparativelylowratesofphoton-pair
creation. We achieved optimal conditional detection ef-
ficiency at a laser pump power of 250µW, generating
and Bob’s collection apparatus, we optimised the pump
∼12kHz of single photons in each TES channel. At this
and collection spots based on [19], with the optimum
rate, the loss due to dead time was ∼2.5%.
found at using pump spot and collection mode diam-
eters of 200µm and 84µm in the crystal, respectively.
With these parameters, we achieved typical pair detec-
Experimental violation of our steering inequality
tion efficiency of 40% measured with standard single-
photon avalanche diodes (SPADs), whose detection ef-
We produced the polarisation-entangled singlet state
ficiency was estimated to be 50% at 820nm, implying √
|ψ−(cid:105) = (|HV(cid:105) − |VH(cid:105))/ 2, where |H(cid:105) and |V(cid:105) rep-
a collection efficiency of 80%. Due to the asymmetry
resent single horizontal and vertical polarised photons
of the steering task, the source and detection system do
respectively, and performed separate measurements in
not have to be symmetric. For example, in our setup
2 and 3 different bases (N=2,3, with measurements of
Alicedoesnotemploynarrow-bandfilters;thischoicein-
creasesheroverallbackground,butreducesherloss,thus Xˆ, Yˆ and Zˆ). The discrete probability distribution for
increasing the detection efficiency conditioned on Bob’s Alice and Bob’s correlations, P(Ai=a,Bj=b), is shown
measurement. in Fig. 3 (a). From these, we first estimated an aver-
aged heralding efficiency η and entangled state visibility
Another key requirement is high photon detection ef-
V and compared them to the theoretical minimum re-
ficiency. The upper bound of the conditional detection
quirements, Eq. (3), in Fig. 3 (b). The plot indicates
probabilityηislimitedbytheperformanceofAlice’spho-
that we should expect a conclusive, detection-loophole-
tondetectors,andthereforewouldnoteveninaloss-less,
free demonstration of steering.
noise-free case allow us to meet the requirements of eq.
Indeed, for the steering parameter S defined in (2),
(3)withourSPADsandtwomeasurementsettings;inour 3
we obtained
experiment, Alice thus employs superconducting tung-
sten transition edge sensors [18] (TESs). TESs utilise
S =1.7408±0.0017,
3
a layer of superconducting tungsten kept in the transi-
tion temperature range and offer a combination of pho- where the uncertainty (one standard deviation) was
4
a)
DISCUSSION
0.4 0.4 0.4
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
Our highly efficient system allows us to firmly close
-1 -1 -1
the detection loophole in our demonstration of quantum
0 0 0
1 -1 1 1 -1 1 1 -1 1 steering, achieving the highest ever reported heralding
efficiency for entangled photons, η ∼ 62%. Our exper-
imental violation of inequality (2) has a quite intuitive
1
b) interpretation: it shows that Alice can, at her will, steer
0.9 Bob’s qubit state to be preferably polarised along any of
the three axes of the Bloch sphere, see Fig. 3 (a).
0.8
0.7 While we have closed the detection loophole, we
y 0.97
t have not addressed the locality and freedom of choice
bili 0.6 loopholes[20] in this work; closing these would require
si
Vi 0.5 0.96 Alice and Bob’s choice and implementation of measure-
ments to be space-like separated, as demonstrated in
0.4 a very recent experiment reported in [21]. For practi-
cal purposes in quantum communication, however, these
0.3 0.950.52 0.54 0.56 0.58 0.6 0.62 0.64 loopholes are typically not problematic[22]: it is a nec-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 essary assumption that Alice and Bob can choose their
measurements independently of the state preparation,
and that no unwanted information leaks from Alice and
Bob’s laboratories.
FIG. 3. Experimental results. (a) Probability distributions
P(A =a,B =b)fortheS measurementsXˆ,Yˆ andZˆ,calcu- Besidesthecriteriaemployedherethereareothersthat
i i 3
latedbynormalisingtheregisteredcoincidenteventsforeach can be used to demonstrate steering[13, 23]. If Alice
measurement setting to the total count numbers. The green cannot achieve the high heralding efficiencies obtained
and blue bars represent correlations that indicate the qual- in our experiment, some of these may be advantageous:
ity of the shared entangled state. The orange bars represent as recently shown in [24], generalising the linear crite-
events that Alice failed to detect. Error bars are too small
ria of [14] allows for steering with arbitrarily high losses.
to be seen on this scale. For S , we used the data obtained
2 These however require a larger number of different mea-
fromthemeasurementsofXˆ andYˆ. (b)Theoreticalvisibility
surement settings; the experiment reported in [24] used
required to violate steering inequalities for N=3 (red dashed
up to N=16 measurements. Our choice to test inequal-
line)andN=2(blackline)foragivenefficiencyη. Ourmea-
ity (2) was motivated by its simplicity in how it natu-
surement clearly violates this bound, with an averaged vis-
ibility of V=0.9678±0.0005 at a mean heralding efficiency rally accounts for Alice’s detection inefficiencies, and by
of η=0.6175±0.0008 for the N=3 measurements (red) and itsminimalityinthenumberofsettings. NotethatN=2
V=0.9601±0.0006, η=0.6169±0.0008 for N=2 (black). All isthenumberofsettingsinitiallydiscussedbyEPR;itis
errorbars(onestandarddeviation)werecalculatedassuming also the canonical number of settings in applications to
Poissonian photon-counting statistics. The correction of the quantum cryptography [10].
analytic bounds of (3) due to measurement imprecision (see
theMethodssection)isshownbythedash-dottedredlinefor
S and the dashed black line for S . Increasing Alice’s detection efficiency above 66% will
3 2
infact enable steering to be used for quantum key distri-
butionwhereonepartydistruststheirapparatus[10];our
calculated by standard propagation of the Poissonian experiment thus constitutes an important step towards
photon-counting statistics. The corrected bound, due to practicalapplicationsofquantumsteering. Furthermore,
imprecision in Bob’s measurements and as calculated in our results imply that a fully loophole-free photonic Bell
theMethodssection,was1.062±0.003. Thiscorresponds test seems to be within arm’s reach. While the symmet-
toaviolationoftheinequality(2)bymorethan200stan- ricphotonpairdetectionefficiencyforoursetupissome-
dard deviations. what lower than the conditional detection probability η,
For N = 2, the corresponding corrected bound of in- it is not far below the 66.¯6 % limit required to violate a
equality (2) was 1.0291±0.0019. We obtained the value Clauser-Horneinequality[25]withnon-maximallyentan-
gled states [26]. Although still a technological challenge,
S =1.1410±0.0014 it is now becoming conceivable to surpass this efficiency
2
in the near future, while simultaneously addressing the
fortheexperimentalsteeringparameter,yieldingaviola- locality and freedom-of-choice loopholes such as demon-
tionofthesteeringinequalityby48standarddeviations. strated in [20].
5
METHODS measures mutually unbiased observables on qubits. In
a practical experiment however, Bob will not measure
Proof of inequality (2) along perfectly mutually unbiased bases, and his oper-
ators may not act on a 2-dimensional system only. We
show now that the parameter S can still be used to
Inequality (2) is equivalent to previously derived vari- N
demonstrate steering, but the upper bound in (2) must
ance criteria [13, 27]; for completeness, we give here a
be adapted according to Bob’s actual measurement.
simple proof.
Let us start by giving a more accurate description of
If the observed correlation can be explained by the
themeasurementBobperformsinourexperiment. First,
source sending non-entangled states to Alice and Bob,
he uses quarter- and half-wave plates, that define a di-
then the probability distribution P can be decomposed
rection (i.e., a unit vector)(cid:126)b on the Bloch sphere, rep-
in the form
resenting his choice of basis. The beam displacer (BD)
(cid:88)
P(A =a,B =b)= q P (A =a)PQ(B =b), then separates the H and V polarisations: a fraction
i j λ λ i ρλ j
t(cid:39)1 of the H polarisation goes to its first output chan-
λ
nel, and later on to the “+1” detector, while a fraction
whereλdescribesthesourcepreparation,usedwithprob- 1−t (cid:28) 1 (in our experiment, 1−t ≤ 10−5) goes to the
(cid:80)
ability q (such that q ≥ 0, q = 1 — note that
λ λ λ λ secondoutputchannel,andtothe“-1”detector. Wecan
the sum could in principle be continuous and infinite):
assume that, symmetrically, a fraction t of the V polari-
it specifies Alice’s response function P (A =a) imple-
λ i sationgoestothesecondoutputchannel,whileafraction
mented by her (untrusted) measurement device, and the
1−t goes to the first output channel, as we utilise a cal-
stateρ senttoBob. Bob’sresponsefunctionPQ(B =b)
λ ρλ j cite beam displacer as our polarising element; its intrin-
is then as quantum mechanics predicts when the observ-
sic birefringence maps polarisation into different spatial
able Bˆ is measured on ρ .
j λ modes, which is a fundamentally symmetric effect [28]
From the above decomposition, and defining
other polarising elements rely on other effects, not nec-
essarily symmetric, requiring a slightly more thorough
P (A =a)
q ≡q λ i , analysis. We finally denote by η and η the overall de-
λ|Ai=a λ P(A =a) + −
i tection efficiencies of the +1 and -1 detectors (SPADs),
(cid:80) respectively, including all losses in Bob’s lab, including
such that, as before, q ≥ 0 and q = 1,
we get P(B =b|A =a)=λ|(cid:80)Ai=aq PQ(Bλ=λb)|Aai=nad coupling and detection losses.
i i λ λ|Ai=a ρλ i For a single photon state entering Bob’s lab, repre-
(cid:104)Bˆ (cid:105) = (cid:88)q (cid:104)Bˆ (cid:105) . sented by a vector(cid:126)u in the Bloch sphere, the probability
i Ai=a λ|Ai=a i ρλ that it gives a click on the +1 or -1 detector is then
λ
By the convexity of the square, PB(±|(cid:126)b)= η2±(cid:2)1±(2t−1)(cid:126)b·(cid:126)u(cid:3).
(cid:104)Bˆ (cid:105)2 ≤ (cid:88)q (cid:104)Bˆ (cid:105)2 , It follows that
i Ai=a λ|Ai=a i ρλ
λ P (+|(cid:126)b)+P (−|(cid:126)b) = η++η− + η+−η− (2t−1)(cid:126)b·(cid:126)u
B B 2 2
which leads to
η +η (cid:12)η −η (cid:12)
E(cid:2)(cid:104)Bˆ (cid:105)2 (cid:3) ≤ (cid:88)q (cid:104)Bˆ (cid:105)2 . ≥ + 2 − −(cid:12)(cid:12) + 2 −(cid:12)(cid:12) = min(η+,η−)
i Ai λ i ρλ
λ
and
Now, for any 1-qubit state ρ , and for 3 mutually un-
λ
biased observables Bˆi, one has |PB(+|(cid:126)b)−PB(−|(cid:126)b)|=(cid:12)(cid:12)η+−2η−+η++2η− (2t−1)(cid:126)b·(cid:126)u(cid:12)(cid:12)
(cid:88)3 (cid:104)Bˆ (cid:105)2 ≤ 1. ≤ (cid:12)(cid:12)η+−η−(cid:12)(cid:12)+ η++η− (cid:12)(cid:12)(cid:126)b·(cid:126)u(cid:12)(cid:12)
i ρλ 2 2
i=1 = max(η ,η )(cid:104)δ+(1−δ)|(cid:126)b·(cid:126)u|(cid:105)
+ −
Together with the previous inequality and the normali-
(cid:80)
sation λqλ = 1, we obtain inequality (2) for N = 3; with δ ≡ |η+−η−| . Hence, defining w ≡ max(η+,η−),
the case for N =2 follows trivially. we get 2max(η+,η−) min(η+,η−)
Accounting for experimental imperfections in Bob’s |(cid:104)B(cid:105)|= |PB(+|(cid:126)b)−PB(−|(cid:126)b)| ≤w(cid:2)δ+(1−δ)|(cid:126)b·(cid:126)u|(cid:3)
P (+|(cid:126)b)+P (−|(cid:126)b)
measurements B B
and by convexity,
As already highlighted, inequality (2) is highly depen-
dant on Bob’s measurements: it is only valid when Bob (cid:104)B(cid:105)2 ≤w2(cid:2)δ+(1−δ)((cid:126)b·(cid:126)u)2(cid:3).
6
Consider now N = 2 or 3 measurement directions(cid:126)b , paredtoarateof∼12kHztotaldetectedeventsforBob.
i
such that |(cid:126)b ·(cid:126)b |≤(cid:15) for all i(cid:54)=j, for some (cid:15)>0 quanti- These will however only introduce some white noise into
i j
fying the nonorthogonality of the N directions. One can Bob’sdata,andcannotincreasetheboundinthesteering
show that for all (cid:126)u in the Bloch sphere, inequality.
Note finally that all the above calculations assumed
N that Bob received qubit states, encoded in the polarisa-
(cid:88) ((cid:126)bi·(cid:126)u)2 ≤ 1+(N −1)(cid:15). tion of single photons. This is however not guaranteed,
i=1 and the source could indeed send multiphoton states.
Thesepossiblyleadtodoubleclicks;intheexperiment,a
Indeed the worst case is obtained when the N vectors(cid:126)bi negligible fraction of about 1 in 104 of Bob’s events were
are such that(cid:126)bi ·(cid:126)bj = (cid:15) for all i (cid:54)= j, and when (cid:126)u is a double-click events. However, we believe that a careful
unit vector equidistant to the(cid:126)b , which gives the upper analysis along similar lines as in [29] should show that if
i
bound above. Bob gives a random result when he gets a double click
Following the proof of inequality (2), we now obtain, (rather than discarding the event), multiphoton states
for Bob’s actual measurements, the steering inequality arethenalsoboundtosatisfythesteeringinequality(4).
IndeeddoubleclickswillthenreduceAliceandBob’scor-
S ≤w2(cid:2)1+(N −1)(δ+(cid:15)−δ(cid:15))(cid:3). (4) relations,andwillnothelpAlice’suntrustworthydevices
N
to increase the steering parameter S .
N
In our experiments, the ratio of detection efficiencies
in Bob’s two detectors was found to be w = η /η =
+ −
1.0115±0.0007. WeestimatedtheorthogonalityofBob’s ACKNOWLEDGEMENTS
measurements (i.e., (cid:15)) by inserting a large ensemble of
different linear polarisation states into Bob’s measure- We thank M.A. Broome for assistance with the exper-
mentdevice,fullycharacterisingthetwowave-platesand imental preparations. We acknowledge financial support
the relative coupling. For the N=3 measurement set- fromtheARCDiscoveryandFederationFellowprograms
tings, we can take epsilon to be the maximum of all and an IARPA-funded US Army Research Office con-
three scalar products |(cid:126)b ·(cid:126)b |; we found (cid:15)=0.0134±0.0007 tract. This research was conducted by the Australian
i j
in that case. For the test with N=2 settings, we used Research Council Centres of Excellence for Engineered
the two most orthogonal settings, Xˆ and Yˆ, which gave Quantum Systems (Project number CE110001013) and
(cid:15)=(1.3±1.5)×10−4. From (4), this yields bounds of Quantum Computation and Communication Technology
1.0291±0.0019 and 1.062±0.003 for S and S , respec- (Project number CE110001027). The NIST contribution
2 3
tively, as quoted in the main text. was supported by the NIST Quantum Information Ini-
Other experimental imperfections include dark counts tiative, and is a work of the US Government and as such
and background of the detectors at tens of hertz, com- this article is not subject to US Copyright.
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