Optimal experimental demonstration of error-tolerant quantum witnesses

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by  Kunkun Wang

Optimal experimental demonstration of error-tolerant quantum witnesses
Kunkun Wang,1 George C. Knee,2,∗ Xiang Zhan,1 Zhihao Bian,1 Jian Li,1 and Peng Xue1,3,†
1Department of Physics, Southeast University, Nanjing 211189, China
2Department of Physics, University of Warwick,
Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
3State Key Laboratory of Precision Spectroscopy,
East China Normal University, Shanghai 200062, China
Testing quantum theory on macroscopic scales is a longstanding challenge that might help to
revolutionise physics. For example, laboratory tests (such as those anticipated in nanomechanical
or biological systems) may look to rule out macroscopic realism: the idea that the properties of
macroscopic objects exist objectively and can be non-invasively measured. Such investigations are
likely to suffer from i) stringent experimental requirements, ii) marginal statistical significance and
7 iii)logicalloopholes. Weaddressalloftheseproblemsbyrefiningtwotestsofmacroscopicrealism,
1 or‘quantumwitnesses’,andimplementingtheminamicroscopictestonaphotonicqubitandqutrit.
0 The first witness heralds the invasiveness of a blind measurement; its maximum violation has been
2 shown to grow with the dimensionality of the system under study. The second witness heralds the
invasivenessofagenericquantumchannel,andcanachieveitsmaximumviolationinanydimension
b
–itthereforeallowsforthehighestquantumsignal-to-noiseratioandmostsignificantrefutationof
e
F the classical point of view.

0
PACSnumbers: 42.50.Xa,42.50.Lc,42.50.Ex,42.50.Dv,03.65.Ta
2

] I. INTRODUCTION with control experiments [13, 16, 25, 31, 32]. Further-
h
p more, improvements to the protocol have been sought:
- Since the birth of quantum mechanics it has been dif- for example, an alternative test of MR described, vari-
t
n ficult to reconcile the principle of quantum superposi- ously, as a quantum witness, no-signalling in time [33–
a tionwiththeintuitiveexperienceofmacroscopicobjects, 35], or non-disturbance condition [9, 13, 26]. Compared
u which appear to always inhabit explicit states indepen- to the original LG test which needs to involve at least
q
dently of observation. Leggett and Garg [1–3] formu- three possible measurement times and the measurement
[
lated a possible solution by defining macroscopic realism oftwo-timecorrelations,thequantum-witnesstestenjoys

2 (MR),aworldviewcombiningtwoassumptions: (MRps) many advantages: Because only instantaneous expecta-
v Macroscopic realism per se (that a macroscopic object tion values are required, it can usually be violated for a
0 willinhabitexactlyoneofitspossiblestatesatalltimes) much wider parameter regime [35–37], and is more ro-
3
and (NIM) Non-invasive measurability (that the object bust to imperfections [16]. Furthermore, it was recently
6
4 isnotinfluencedbyappropriatelycarefulmeasurements). shown that Fine’s theorem (derived initially for local re-
0 Fromtheseassumptions,theyderivedLeggett-Garg(LG) alism) does not apply for MR [35]. Notwithstanding the
. inequalities, which are used to test for the quantum be- argument of Ref [34], which offers a different perspec-
1
0 havior of a system undergoing coherent evolution [1, 4– tive involving quasi-probabilities, LG inequalities do not
7 11]. The LG inequalities have been tested for a wide formanoptimaltightboundaryforMR.Incontrast, the
1 rangeofquantummechanicalsystems,suchasdefectcen- quantum witness condition [38] is both necessary and
: tersindiamond[12,13],superconductingcircuits[14–16], sufficient for MR [35]. There are however, additional
v
i photons [17–21], atoms in optical lattices [22], nuclear improvements that may be made: as we show, a larger
X magneticresonance[23–25]andphosphorusimpuritiesin ‘quantum signal-to-noise ratio’ may be had, and logical
r silicon [26], and violations have been observed as quan- loopholes may be narrowed by altering the experimen-
a
tum theory predicts. Alongside the major experimen- tal protocol. Here we continue in the pursuit of rigor-
talchallengeofapplyingthetesttomacroscopicsystems ous and amenable protocols for testing MR, reporting
(whichcouldtheninprinciplehaveabearingondynami- a microscopic experimental demonstation with a view to
cal collapse theories [27]), proof-of-principle studies have spurring-ontestswhichwillnon-triviallyconstrainfuture
sofarconcentratedontheimplementationofnoninvasive theories of physics.
measurements. Approaches include using weak [14, 28]
Maroney and Timpson classified MRps into three
null-result [16, 21, 29] or quantum non-demolition mea-
types [9]. Two out of these three, termed Operational-
surements [30]; but latterly efforts have focussed on de-
Eigenstate-Support and Supra-Eigenstate-Support, in-
termining and accounting for measurement clumsiness
volve hidden variables in a non-trivial way and cannot
be ruled out by any experimental test on a two-level
system [39]. However, a third type, termed Eigenstate-
∗ [email protected] Mixture macrorealism, which has it that all superposi-
† [email protected] tions are in fact statistical mixtures in a preferred ba-
2
sis, can be ruled out by violating the LG inequality or a when such an operation is chosen: see Fig. 1(a). To ob-
quantum witness condition. It is therefore this notion of tain a classical condition on V, consider the measurably
MRps that we adopt for the remainder of this paper. zero (or near zero) effect of a general channel on certain
‘fixed point’ preparation states. Quantum theory pre-
dicts that a modulation of the global quantum phase of
II. FIRST WITNESS a state, for example, should have such a null effect. At-
tempting to interpret a superposition of such fixed-point
Considertwoobservables,AandB,measuredattimes states (where, according to quantum mechanics the vari-
t = 0 and t = T > 0, respectively. The measurement of ousstatesnowpickupdefiniterelativephases)asamere
observable A is a blind measurement (the measurement classical or incoherent mixture (i.e. our assumption of
is performed but the result is not recorded) shown in MRps) leads to the condition
Fig. 1(a). The outcomes of the first measurement (of
V =0. (5)
A) are written {a } for i = 1,...,M, with corresponding
i
probabilities P(a ); b is a particular outcome of the later
i According to quantum theory, once the channel is re-
measurement (of B). Based on the joint measurement of
laxed from a blind measurement to some general map E,
these two observables the probability of obtaining result
we have V = Tr(Πb[Φ(ρ)−Φ(E(ρ))]) for Πb the pro-
binthelatermeasurementisP(cid:48)(b)=(cid:80)M P(b|a )P(a ) ρ
i=1 i i jector corresponding to outcome b of the later measure-
with P(b|a ) the conditional probability of the outcome
i ment, and Φ describing the time evolution of density op-
b given the earlier result a . The probability of outcome
i erator ρ from t = 0 to t = T. It is easy to see that
b without the prior measurement of A is written P(b).
for pure state ρ = |ψ(cid:105)(cid:104)ψ|, E(ρ) = U ρU†, and choosing
The first of our two quantum witnesses (derived in Refs. 0 0
Πb = Φ(ρ) = U ρU† (U ,U are unitary operators), we
[38, 40] and elsewhere), is defined as [41] 1 1 0 1
can achieve
W :=P(b)−P(cid:48)(b). (1)
V =1−|(cid:104)ψ|U |ψ(cid:105)|2. (6)
ρ 0
BasedonthetenetsofMR,thepresenceoftheblindmea-
Therefore, if U orthogonalises |ψ(cid:105), the V witness can
surementofAshouldnotaffectthesubsequentevolution 0
reach its algebraic maximum (V = 1) in any dimen-
of the system. One then has max
sion. To achieve this maximum in general, (as can be
W =0, (2) seenbyinspectingEq.(6))thechannelmustapplyapure
phase to each fixed-point state, such that when a super-
our first quantum witness condition. It can be derived position of these states is injected the phases combine
underthesameassumptionsastheLGinequalites[2,16]. to form an orthogonal superposition. A blind measure-
Equation (2) can be violated by a quantum-mechanical ment is equivalent to a random phase channel: this is at
system; the theoretical upper bound on the violation is the root of the more modest maximum violations shown
given by [41] in (3).
The reader may object that the assumption of non-
1
W =1− , (3) invasive measurability (NIM) has become non-invasive
max M operability (NIO): namely, that a suitably careful oper-
ation (that need not be interpreted as a ‘measurement’)
where the number of blind-measurement outcomes M ≤
canbemadewithoutaffectingthefutureevolutionofthe
N, the dimension of the system under study. The max-
system. Byempiricallytestingforinvasiveness,however,
imum violation can be found in the von Neumann mea-
we shall see that in the end such auxiliary assumptions
surement limit when M =N and W =1−1/N.
max
will play no role in the conditions that we test; so it is
not necessary to consider the subtleties associated with
III. SECOND WITNESS replacing ‘measurement’ with ‘operation’.
We propose a further improvement to LG’s approach.
IV. ERROR TOLERANCE
Our second quantum witness V consists in replacing the
blindmeasurement(above)withagenericquantumchan-
The conditions W = 0 and V = 0 are not suitable for
nel:
experimental test: since almost any real experiment will
V :=P(b)−P(cid:48)(cid:48)(b). (4) find a violation (because of finite statistical or technical
errors),onemayquestionthemeaningofinferencesmade
P(cid:48)(cid:48)(b) denotes the probability of getting outcome b in from data recorded in such an experiment. It is better
the later measurement of B at t = T, when a generic to construct the compound conditions
quantum channel has been performed at t = 0. Since
the generic channel could be, but need not be, a blind mini(Wi)≤Wσ ≤maxi(Wi) (7)
measurement,weseethatP(cid:48)(b)isaspecialcaseofP(cid:48)(cid:48)(b) min (V )≤V ≤max (V ), (8)
i i σ i i
3
FIG. 1. (a) Generic procedure for testing our two quantum-witnesses. We need four processes, including state preparation,
stateevolution(e.g. Φ(ρ)=U ρU†)andprojectivemeasurement(PM)ofB. Wealsorequiretheoptionalapplicationofeither
1 1
i) blind measurement (BM) of A (for testing witness W) or ii) some generic channel E(ρ), here chosen as a phase modulation
U ρU† (for testing witness V). (b) Experimental setup. The heralded single photons are created via type-I SPDC in a BBO
0 0
crystalandareinjectedintotheopticalnetwork. Qubitstatesarepreparedviaapolarizingbeam-splitter(PBS)andhalf-wave
plates (HWPs). The phase channel U2D can be implemented by a set of wave plates (not shown), while a quartz crystal
0
(QC) is used to realize a blind measurement of A. For the qutrit, the polarizing beam-splitter, half-wave plate and beam
displacerareusedforstatepreparation. U3D isimplementedwithhalf-waveplatesandquarter-waveplates(QWPs),theblind
0
measurementagainwiththequartzcrystal(notshown),andU3D canbeimplementedwithacascadedinterferometernetwork.
1
The projective measurement of B is realized via a beam displacer which maps the basis states of the qubit/qutrit to distinct
spatial modes.
where the W (V ) correspond to the witness measured Assuming ancillary tests have been performed to de-
i i
in the fixed-point states |ψ (cid:105) (which constitute a pre- termineW (V ),andfoundeachtobe(forexample)close
i i i
ferred basis), and W (V ) is a witness measured in the to zero, all that remains is the testing of W (V ). By
σ σ σ σ
state |ψ (cid:105) = (cid:80) α |ψ (cid:105), where (cid:80) |α |2 = 1. That the judiciouslychoosingU and|ψ (cid:105),themaximumviolation
σ i i i i i 0 σ
W (V ) can be nonzero constitutes an important qual- of(8)isgenerallygreaterthanthemaximumviolationof
i i
ityoferror-tolerancenotpresentinconditions(2)or(5): (7),andfurthermoreindependentofthedimensionofthe
one requires the signature of quantumness to be signifi- system. This is clearly more experimentally favourable;
cant w.r.t the control quantities W or V . Recalling our there is a greater robustness to imperfection, and viola-
i i
interpretation of MRps as Eigenstate-Mixture macrore- tions will emerge sooner from statistical noise, i.e. with
alism, the use of min and max functions to bound W fewer experimental trials. Note that with the introduc-
σ
andV allowsforthemostgeneralMRpsexplanationin- tion of the control quantities V , it might seem that a
σ i
terpreting the superposition σ as an arbitrarily weighted violation of (7) or (8) with magnitude greater than one
incoherent mixture of the fixed-point states [16]. We is possible – e.g. by arranging all V =−1 and V =+1.
i σ
therefore used only MRps (without using NIM or NIO) Such violations are, however, not permitted by any de-
toderive(7)and(8)anditisthisassumptionalonethat terministic operation allowed in quantum mechanics (as
is being tested. we show in Appendix A). Next we will give details and
results of an experimental demonstration of violation of
these witness conditions.
Interestingly,oncetheseconditionsareadoptedasnec-
essary and sufficient conditions for MRps, the focus has
shifted from the intrinsic properties of Wσ or Vσ to their V. OPTIMAL VIOLATIONS
relation to the W or V (respectively). Testing MRps
i i
is then merely asking the question, ‘does a certain mea-
It is simple to find preparations, channels and mea-
sured quantity for preparation σ lie in the convex hull of
surementswhichsaturatethemaximumviolationsofour
thesamequantitiesmeasuredfortheclassicalstatesψ ?’.
i two witnesses. In our experimental demonstration, we
Furthermore, it is not even required that the W (V ) are
i i choose the fixed-point states to be an orthonormal ba-
small in magnitude, although this may be expedient for
sis {|ψ (cid:105),|ψ (cid:105)} for the qubit and {|φ (cid:105),|φ (cid:105),|φ (cid:105)} for the
0 1 0 1 2
large violations. In that case, it may be tempting to as-
qutrit, which all satisfy W = V = 0 because they form
i i
sumethemtobezerowhenquantumtheorypredicts: but
the eigenvectors of i) the blind measurement observable
measuring the control quantities experimentally instead
A, and ii) of the unitary channel U . These basis states
0
has the advantage of forming a more logically watertight
willbeencodedinacombinationofpolarizationandpath
argumentcontra macrorealism,sincethedegreeof‘clum-
degreesoffreedomofsinglephotons(seeFig.1(b)). Writ-
siness’[32]isdeterminedandneednotbeassumed. Oth-
ing matrices in these bases, we choose
erwise, the idea that the object under study is not quan-
(cid:18) (cid:19) (cid:18) (cid:19)
tum at all, but merely a classical system subjected to 1 0 1 1
U2D = and U2D = (9)
clumsy operations, remains a substantial loophole. 0 0 -1 1 1 −1
4
for the qubit, and wave plates is used to apply a rotation on two modes of
the qutrit state and the other is used to compensate the
1 0 0 
optical delay. The beam displacers are used to translate
U03D =0 ei23π 0  and (10) between polarization encoding and spatial-mode encod-
0 0 ei43π ing. In this way, wave plates acting on the two polariza-
(cid:113) √ √  tion modes propagating in the same spatial mode can be
2 6 − 6
3 6 6 used to effect two-(spatial)-mode transformations. See
U3D = 0 √2 √2  (11) Appendix B for further details. In Appendix C, we
1 (cid:113) (cid:113)2 (cid:113)2  present a scheme that generalises this approach to any
1 − 1 1
3 3 3 desired dimension N.
for the qutrit. The measurement operator is chosen Toimplementtheblindmeasurement,notethatafterit
as Πb= |ψ (cid:105)(cid:104)ψ | or |φ (cid:105)(cid:104)φ | respectively. The su- is applied, the system is a mixture with diagonal density
1 1 2 2 √ (cid:80)
operatorρ =Γ(ρ)= (cid:104)ψ |ρ|ψ (cid:105)|ψ (cid:105)(cid:104)ψ |. Note
perposition states are |ψσ(cid:105)√= (|ψ0(cid:105) − |ψ1(cid:105))/ 2 and blind m m m m m
that Γ(ρ) is a completely dephasing channel. Thus the
|φ (cid:105) = (|φ (cid:105)−|φ (cid:105)+|φ (cid:105))/ 3, which achieve W2D =
σ 0 1 2 σ blind measurement can be realized by a quartz crystal,
1/2,W3D =2/3,V2D =V3D =1.
σ σ σ inserted into the lower spatial mode so as to destroy the
spatial coherence of the photons. The coherence length
of the photons is L ≈ λ2/∆λ, where λ is the central
VI. EXPERIMENTAL RESULTS c
wavelength of the source and ∆λ is the spectral width
of the source [46]. Hence the thickness of the quartz
The state of the qubit can be represented by two or-
crystal should be at least 23.97mm: In our experiment,
thogonal polarization states of single photons, which are
it is about 28.77mm. To test the violation of our second
generated via a type-I spontaneous parametric down-
quantum witness condition (8), we replace the quartz
conversion(SPDC)[42]–seeFig. 1(b). Thepolarization-
crystal by wave plates with certain setting angles which
degenerate photon pairs at a wavelength of 801.6nm are
are used to realize the unitary evolution U2D or U3D.
produced using a 0.5mm-thick β-barium-borate (BBO) 0 0
In Fig. 2(a), the experimentally determined values
crystal pumped by a 90mW diode laser, which is filtered
of our witnesses are shown. Due to the high preci-
out by a 3nm bandpass filter. The signal photon is her-
sion nature of our laboratory setup, we found all quan-
alded for evolution and measurement by detection of a
tities to be close to their predicted values. In par-
trigger photon: coincidences are registered by avalanche
ticular the fixed point preparations gave witness val-
photodiodeswitha7nstimewindow. Apolarizingbeam
ues close to zero, and we found W2D = 0.4980 ±
splitter (PBS) and a half-wave plate (H’ ), set at certain σ
0 0.0060(35sd),V2D = 0.9998 ± 0.0004(80sd),W3D =
angles, are used to prepare the input states. U2D is re- σ σ
1 0.6700±0.0080(44sd),V3D =0.9820±0.0020(72sd). The
alizedbyahalf-waveplate(H’ )at22.5◦. Abirefringent σ
1 numberofstandarddeviations(sd)ofviolation,givenby
calcite beam displacer is used to map the basis states of (cid:112)
(Ξ −max Ξ )/ Var(Ξ )+Var(max Ξ)(Ξ=W,V),is
the qubit to two spatial modes and to accomplish the σ i i σ i i
shown parenthetically. Note how using the second wit-
projective measurement of B. The probability of the
ness lead to violations of MRps with higher statistical
photons being measured in |ψ (cid:105) is estimated by normal-
1
significance. In Fig. 2(b), these data are shown along-
izingphotoncountsineachspatialmodetototalphoton
side theoretical predictions for the maximum violations
counts. The count rates are corrected for differences in
of both witnesses.
detector efficiencies and losses before the detectors. We
assume that the ensemble of prepared photons is fairly
representedbythesampleofdetectedphotons(fairsam-
pling assumption). Total coincidence counts are about VII. CONCLUSION
13,000 over a collection time of 6s.
For a qutrit, the basis states |φ (cid:105), |φ (cid:105), and |φ (cid:105) are ThoughviolationoftheLGinequalityhasbecomethe
0 1 2
encoded by the horizontal polarization of the photon in standard laboratory proof of ‘quantumness’, ruling out
theuppermode,andthehorizontal/verticalpolarization MR in the same way that violation of Bell’s inequality
of the photon in the lower mode. After passing through rules out local realism, the LG inequality is only nec-
a polarizing beam-splitter and half-wave plate (H ), the essary (but not sufficient) for MR. In this paper, we
0
heraldedsinglephotonspassabeamdisplacer(BD )fol- employed the necessary and sufficient quantum-witness
1
lowed by two half-wave plates (H and H ) [43], one in condition,whichallowsforgreaterstatisticalsignificance
1 2
each spatial mode [44]. By tuning their angles, the pho- given fewer experimental resources. We significantly in-
tonsarepreparedinoneoftheinputstates. Theunitary creasethestatisticalsignificanceyetfurther,bychanging
channel U3D (belonging to SU(3)) can be decomposed from a blind measurement to a unitary channel. In both
1
into three unitary channels, each of which applies a ro- cases, we measured the classical disturbance introduced
tation to just two of the basis states, leaving the other by our operations, which tightens the logical loopholes
unchanged. Each of them can be realized by two half- inthedemonstration,aswellasremovingtheneedtoas-
wave plates and a beam displacer [45]. One of the half- sumeanykindofnon-invasiveness. Wehaverecordedex-
5
(a) (b)
σ
σ
σ σ σ σ
qubit qutrit DimensionN
FIG.2. (a)ExperimentallydeterminedvaluesforourtwoquantumwitnessesW andV,forthefixed-pointpreparations0and
1 (also 2) and for the superposition preparation σ of a photonic qubit (and qutrit). Theoretical predictions ate represented
by dashed and dotted lines. Error bars indicate the statistical uncertainty which is obtained based on assuming Poissonian
statistics. (b) Experimental results showing maximum violations of both quantum witness conditions for two-level and three-
level photonic systems. The blue (red) dashed (dashed-dotted) line represents the theoretical predictions of the maximum
violations of the first (second) quantum witness condition.
perimentalviolationsinbothaphotonicqubitandqutrit. Aboveweusedthetrace-preservingpropertyofE. There-
Our results agree well with the theoretical maximum vi- fore we have that max V ≥0 and min V ≤0, and thus
i i i i
olations,andshowcaseperhapsthefinal,finetuned,pro-
V −maxV ≤+1 (A4)
tocolfortestingmacroscopicrealism per se, whichgives σ i
i
the greatest possible chance for finding convincing vi-
V −minV ≥−1, (A5)
σ i
olations in truly macroscopic systems, while remaining i
error-tolerant and evading the clumsiness loophole. and the maximum violation of our condition has a mag-
nitudeofunity. SinceE isanyquantumchannel,itcould
be the completely dephasing channel. So this proof also
ACKNOWLEDGMENTS establishes
W −maxW ≤+1 (A6)
σ i
WewouldliketothankNeillLambertandCliveEmary i
for helpful discussions. We acknowledge support by Wσ−minWi ≥−1, (A7)
i
NSFC (Nos. 11474049 and 11674056) and NSFJS (No.
BK20160024). G. C. K. is supported by the Royal Com- although tighter bounds were proved by Schild and
mission for the Exhibition of 1851. Emary [41]. We have made the assumption that E is
a quantum channel; that is, it belongs to the subset of
quantumoperationsthatpreservethetraceofthedensity
operator. Trace-decreasing operations are indeterminis-
Appendix A: Maximum violation of the witnesses
ticandareonlypossibleviapost-selection. Weleavethe
discussion of maximum violations for that complemen-
Our second witness is defined
tary subset, and their interpretation, for future work.
V :=Tr(Πb(cid:48)(ρ −E(ρ ))), (A1)
ρ σ σ
Appendix B: Experimental realization of the unitary
where Πb(cid:48) = Φ†(Πb) is related to a measurement in the
evolutions and projective measurements for two-
preferred basis Πb(cid:48) by the channel Φ. Clearly and three-level systems
−1≤V ≤1 (A2)
ρ To test the violation of first quantum witness condi-
tion, we insert a quartz crystal to destroy spatial coher-
because it is the difference of two probabilities. Now, by
using the completeness relation (cid:80) |ψ (cid:105)(cid:104)ψ | = I for an ence. Whereas, to test the violation of second quantum
i i i witness condition, we replace the quartz crystal with a
orthonormal set of basis vectors, one can see that
set of wave plates which are used to realize the unitary
(cid:88)V =Tr(cid:0)Πb(cid:48)[(cid:80) |ψ(cid:105)(cid:104)ψ|−E((cid:80) |ψ(cid:105)(cid:104)ψ|)](cid:1) channels U02D or U03D. For implementation of U02D, we
i i i i i i i use a half-wave plate to realize the Pauli operator σ on
z
i the qubit state. For a qutrit case, the evolution U3D is a
=Tr(cid:0)I(cid:2)Πb(cid:48)−E†(Πb(cid:48))(cid:3)(cid:1) channelthataddsaphaseei4π/3 onthestate|φ (cid:105),0ei2π/3
2
=0. (A3) on|φ (cid:105)andkeeps|φ (cid:105)unchanged,andcanberealizedby
1 0
6
wave plates with certain setting angles inserting to the phase of the output.
proper spatial modes (upper and lower modes). Figure 3 shows the experiment setup to achieve N-
Intheprojectivemeasurementstage,abeamdisplacer dimensional unitary evolution. If N is an even number,
is used to map the basis states of qutrit to three spa- the basis states are encoded by the horizontal / vertical
tialmodesandtoaccomplishtheprojectivemeasurement polarization in n=N/2 spatial modes, i.e.
Πb = |φ (cid:105)(cid:104)φ |. For photon detection, we record pair-
2 2
wisecoincidencesbetweentheheraldingdetectorandany |1(cid:105)&|2(cid:105),|3(cid:105)&|4(cid:105),...,|N−1(cid:105)&|N(cid:105)∼H1&V1,H2&V2,...,Hn&Vn.
one of the three detectors in the measurement appara-
Else if N is odd, basis state |1(cid:105) is encoded by the hor-
tus (heralded single clicks). Simultaneous registrations
izontal polarization in the first spatial mode, and the
in the heralding detector and two of the three detectors
remaining basis states are encoded by the horizontal /
for measurement give the three-body coincidences. Such
vertical polarization in (N −1)/2 spatial modes: i.e.
coincidencesmightoccurwhenthesourcegeneratesmore
than one photon per pulse, but we find the frequency of
|1(cid:105),|2(cid:105)&|3(cid:105),...,|N −1(cid:105)&|N(cid:105)∼H ,H &V ,...,H &V
these to be negligible in our experiment. 1 2 2 n n
We estimate the relative detection efficiency of each with n = N+1. According to Eq. (C1), if we can com-
detector Di assuming that the sum of the photon counts bine the diff2erent spatial modes into one spatial mode
at the detectors does not depend on the configuration and apply arbitrary U(2) in this mode in sequence, we
of optical components before the detector. We then canimplementanarbitraryunitaryoperatoracrossmany
use these efficiencies to correct each count rate N˜i = spatial modes. An arbitrary transformation of the po-
Ni/Di. For the probability of detection we then have larization of a photon can be realized by a set of wave
Pi = N˜i/(cid:80)3i=0,1,2N˜i, where N˜i is the corrected num- plates which contain two half-wave plates (blue rectan-
ber of heralded clicks at detector i. The corrections are gle in Fig. 3) and one quarter-wave plate (orange rect-
made to eliminate the effects of different efficiencies of angle). As shown in Fig. 3, the first E can be
N,N−1
the detectors. achieved by inserting a set of wave plates in the last
optical mode. Then by applying a half-wave plate at
45◦ in all of the other modes, after passing through the
first beam displacer (BD ), we can combine the levels
Appendix C: Generalization to arbitrary N 1
of N and N − 2 together. By inserting another wave
plate set, we can implement E . We can use the
N,N−2
Just as shown in Fig. 1 (a) of the main text, there are same procedure to realize the following E . After this
i,j
four processes in this experiment, including state prepa- setup, the output labelling is reversed. So that we have
ration, operation of U0 (or blind measurement), state e.g. (H(cid:48) &V(cid:48) ),...,(H(cid:48)&V(cid:48)),(H(cid:48)&V(cid:48)) corresponding to
n n 2 2 1 1
evolution U1 and projective measurement of B. In order the basis states of (|N −1(cid:105)&|N(cid:105))...(|3(cid:105)&|4(cid:105)),(|1(cid:105)&|2(cid:105))
to extend the dimension of the system to N-dimension, . The maximum number of beam displacers needed to
the primary problem is to find a setup to implement ar- buildthisN-dimensionalunitaryoperatoris2N−4when
bitrary unitary channels, since these can be combined to N is an even number and 2N −3 for odd. This number
enable full state preparation and measurement (as well growsonlypolynomiallywiththenumberofdimensions.
as U0 and U1). For the blind measurement, it can be In order to extract the maximum violation of our
realized by inserting quartz crystals with different thick- witnesses, it suffices to prepare the maximally coher-
√
nesses in different optical modes. ent state [48] |ψ (cid:105) = (cid:80) |ψ (cid:105)/ N and make the final
σ i i
It has been proven that any N ×N unitary array can measurement include Π such that U†Π U = |ψ (cid:105)(cid:104)ψ |
bedecomposedasasequenceofU(2)transformationson b 1 b 1 σ σ
is a projector onto this state [41]. For the W wit-
two-dimensional subspaces of the N-dimensional Hilbert
ness, the blind measurement of course projects onto
space,withthecomplementarysubspaceunchanged[47]:
the |ψ (cid:105) states; for the V witness, one may use U =
i 0
(cid:80)N ei(k)2π/N|ψ (cid:105)(cid:104)ψ |. In this way, a phase from each
U(N)=(E ·E ...E ·S)−1. (C1) k=0 k k
N,N−1 N,N−2 2,1 basis state contributes to the violation. It may be pos-
sible to design other schemes where only phases from
Here Ei,j is an N-dimensional identity array with the e.g. macroscopically distinct basis states contribute: see
elements of Ei,i,Ei,j,Ej,i, and Ej,j replaced by the cor- Ref [49] for a similar analysis of measurement schemes
i,j i,j i,j i,j
responding U(2) array elements. S is used to adjust the for the LG inequality.
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