Preparation contextuality powers parity-oblivious multiplexing

0.18 MB·English

by  Robert W. Spekkens

Preparation contextuality powers parity-oblivious multiplexing
Robert W. Spekkens,1 D. H. Buzacott,2,3 A. J. Keehn,2,3 Ben Toner,4 and G. J. Pryde2,3
1DAMTP, University of Cambridge, Cambridge, United Kingdom CB3 0WA
2Centre for Quantum Computer Technology, Griffith University, Brisbane 4111, Australia
3Centre for Quantum Dynamics, Griffith University, Brisbane 4111, Australia
4Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
In a noncontextual hidden variable model of quantum theory, hidden variables determine the
outcomes of every measurement in a manner that is independent of how the measurement is im-
plemented. Using a generalization of this notion to arbitrary operational theories and to prepara-
tion procedures, we demonstrate that a particular two-party information-processing task, “parity-
obliviousmultiplexing,”ispoweredbycontextualityinthesensethatthereisalimittohowwellany
theory described by a noncontextual hidden variable model can perform. This bound constitutes
9
0 a “noncontextuality inequality” that is violated by quantum theory. We report an experimental
0 violation of this inequality in good agreement with the quantum predictions. The experimental
2 results also providethefirst demonstration of 2-to-1 and 3-to-1 quantumrandom access codes.

n
PACSnumbers: 03.65.Ta,03.67.-a,42.50.Dv,42.50.Ex,42.50.Xa
a
J

8 The Bell-Kochen-Specker theorem [1] shows that the theprobabilityofsuccessinthistaskandwedemonstrate

predictions of quantum theory are inconsistent with a a quantum protocol for parity-oblivious multiplexing for
]
h hidden variable model having the following feature: if which the probability of success exceeds the noncontex-
p A, B and C are Hermitian operators such that A and tual bound.
-
t B commute, A and C commute, but B and C do not
n Finally, we report an experimental implementation of
commute, then the value predicted to occur in a mea-
a this protocol that achieves a probability of success in
surement of A does not depend on whether B or C was
u
good agreement with the quantum result and in viola-
q measuredsimultaneously. Thisfeatureiscalled“noncon-
tion of the NC inequality.
[ textuality.” Significantly,it is only well-defined for mod-

elsofquantumtheory(andthenonlyforprojectivemea-
2 Operational theories and noncontextual mod-
v surements and deterministic models) [2]. By contrast, els. In an operational theory, the primitives of descrip-
3 Bell’s definition of a local model applies to any theory
tion are preparations and measurements, specified as in-
6 that can be described operationally [3]. Consequently,
structions for what to do in the laboratory. The theory
4
whereasonecantestwhether ornotexperimentalstatis-
1 simplyprovidesanalgorithmforcalculatingtheprobabil-
ticsareconsistentwithalocalmodel(bytestingwhether
. ity p(k|P,M) of an outcome k of measurement M given
5 or not they satisfy Bell inequalities), there is no way to
apreparationP. As anexample,inquantumtheory,ev-
0
testwhetherornotexperimentalstatisticsareconsistent
8 ery preparation P is represented by a density operator
with a noncontextual model (and no way of defining as-
0 ρ , every measurement M is represented by a positive
P
v: sociated“noncontextualityinequalities”)unless onegen- operator valued measure {EM,k}, and the probability of
eralizesthetraditionalnotionofnoncontextualityinsuch
i outcome k is given by p(k|P,M)=Tr(ρPEM,k).
X awaythatitmakesno referencetothe quantumformal-
r ism. Suggestions for such a formulation have been made In a hidden variable model of an operational theory,
a
by severalauthors [4]. A particularlynaturalgeneraliza- a preparation procedure is assumed to prepare a sys-
tion(andslightmodification)whichappliestoallmodels tem with certain properties and a measurement proce-
(deterministicornot)ofanyoperationaltheoryhasbeen dure is assumed to reveal something about those prop-
proposed in Ref. [2]. We here derive a noncontextuality erties. The set of all variables describing the system
(NC) inequality based on this notion. is denoted λ. It is presumed that for every preparation
P, there is a probability distribution p(λ|P) such that
Because information-theoretic tasks can be character-
implementing P causes the system to be prepared in
ized entirely in terms of experimental statistics, one can
physical state λ with probability p(λ|P). Similarly, it
explorewhethertheoriesthatviolateNCinequalitiesmay
is presumed that for every measurement M, there is a
provide information-theoretic advantages over theories
distribution p(k|λ,M) such that implementing M on a
that satisfy these inequalities. We prove that this is in-
systemdescribedby λ yields outcome k with probability
deed the case for a task which we call parity-oblivious
p(k|λ,M). For the hidden variable model to reproduce
multiplexing, a kind of two-party secure computation.
the predictions of the operational theory, it must satisfy
(The notionthatcontextualitymightyieldanadvantage
p(k|P,M)= dλp(k|λ,M)p(λ|P).
for multiplexing tasks was first put forward by Galv˜ao
[5].) The NC inequality we derive provides a bound on A hidden vRariable model is preparation noncontextual
2
if the following implication holds P , and for each integer y, Bob implements a binary-
x
outcomemeasurementM ,andreportsthe outcomebas
y
∀M :p(k|P,M)=p(k|P′,M)→ p(λ|P)=p(λ|P′), his output. The probability of winning is
(1)
thatis,iftwopreparationsyieldthesamestatisticsforall 1
p(b=x )= p(b=x |P ,M )
possiblemeasurementsthentheyarerepresentedequiva- y 2nn y x y
lently in the hidden variable model. Similarly, measure- y∈{X1,...,n}x∈X{0,1}n
(3)
ment noncontextuality is the condition that
where1/2nnisthepriorprobabilityforaparticularxand
∀P :p(k|P,M)=p(k|P,M )→ p(k|λ,M)=p(k|λ,M ), y. The parity-oblivious constraint requires that for ev-
′ ′
(2) erys-parity,thereis nooutcomeofanymeasurementfor
that is, if two measurements have the same statistics which posterior probabilities for s-parity 0 and s-parity
for all possible preparations then they are represented 1 are different, that is,
equivalently in the model. More details can be found
∀s∀M∀k : p(P |k,M)= p(P |k,M). (4)
in Ref. [2]. An NC inequality is any inequality on ex- x x
perimental statistics that follows from the assumption x|Xx·s=0 x|Xx·s=1
that there exists a hidden variable model that is prepa-
Noncontextualityinequality. Themaintheoretical
rationandmeasurementnoncontextual. Itis ofthe form
result of this letter is the following theorem.
f(p(k|P ,M ),p(j|P ,M ),...) ≤ C for some function f
1 1 2 2
and constant C. Theorem 2. In an operational theory that admits a
Parity-oblivious multiplexing. Suppose that Al- preparation noncontextual hidden variable model, the op-
ice and Bob wish to perform the following information- timal probability of success in n-bit parity-oblivious mul-
processingtask, which we call n-bit parity-oblivious mul- tiplexing satisfies p(b=x )≤(n+1)/2n.
y
tiplexing. Alice has as input an n-bit string x chosen
Proof. Define P to be the procedure obtained by
uniformly at random from {0,1}n. Bob has as input an s,b
choosinguniformlyatrandomanxsuchthatx·s=band
integerychosenuniformlyatrandomfrom{1,...,n}and
implementing P . Clearly, for any measurement M, the
mustoutputthe bitb=x ,thatis,the ythbitofAlice’s x
y
probabilityofoutcomek givenpreparationP issimply
input. AlicecansendasystemtoBobencodinginforma- s,b
tion about her input, however there is a cryptographic 1
p(k|P ,M)= p(k|P ,M). (5)
constraint: no information about any parity of x can be s,b 2n 1 x
−
transmitted to Bob. More specifically, letting s ∈ Par x|Xx·s=b
where Par≡{r|r ∈{0,1}n, r ≥2} is the set ofn-bit
i i Similarly, the probability of hidden variable λ given an
strings with at least two bits that are 1, no information
P implementation of Ps,b is simply
aboutx·s= x s (termedthes-parity)foranysuchs
i i i
can be transmitted to Bob (here ⊕ denotes sum modulo 1
2). This tasLk is similar to an n-to-1 quantum random p(λ|Ps,b)= 2n 1 p(λ|Px). (6)
−
access code [5, 6, 7, 8] except that it has a constraint of x|Xx·s=b
parity-obliviousness rather than a constraint on the po- Now note that one can re-express the parity-oblivious
tentialinformation-carryingcapacityofthe systemused. condition, Eq. (4), as ∀s∀M : p(k|P ,M) =
xxs=0 x
| ·
Lemma 1. Classically, the optimal probability of suc- xxs=1p(k|Px,M) (it follows Pfrom Bayes’ rule and
| ·
cess in n-bit parity-oblivious multiplexing satisfies p(b = the uniformity of the prior over x). Combining this
P
xy)≤(n+1)/2n. with Eq. (5), we infer that ∀s∀M : p(k|Ps,0,M) =
p(k|P ,M) which is simply the statement that mixed
Proof. (Fordetails,seeAppendix A.)Theonlyclassical s,1
preparations corresponding to opposite s-parities are in-
encodingsofxthatrevealnoinformationaboutanypar-
distinguishable by any measurement. But together
ity (while encoding some informationaboutx) arethose
with the assumption that the hidden variable model is
that encode only a single bit x for some i. Given that
i preparation noncontextual, Eq. (1), this implies that
y is uniformly distributed, it makes no difference which
∀s:p(λ|P )=p(λ|P ),whichstatesthatmixedprepa-
bit it is. Therefore, we may assume that Alice and Bob s,0 s,1
rations corresponding to opposite s-parities are also in-
agree that Alice will always encode the first bit, x . If
1 distinguishableatthehiddenvariablelevel. UsingEq.(6)
y = 1, which occurs with probability 1/n, then Bob can
and Bayes’ rule again, we obtain
output b = x and win. With probability (n−1)/n, we
y
have y 6= 1 and in this case Bob can at best guess the
∀s: p(P |λ)= p(P |λ). (7)
x x
value of x and wins with probability 1/2.
What isythe most general protocol that can be imple- x|Xx·s=0 x|Xx·s=1
mentedinanarbitraryoperationaltheory? Foreachin- Therefore,evenifoneknewλ,the posteriorprobabilities
put string x, Alice implements a preparation procedure for s-parity 0 and s-parity 1 would be the same, that is,
3
with Blochvectors(± 1 ,± 1 ,± 1 ) forming a cube in-
√3 √3 √3
side the Bloch sphere (see Fig. 1). Bob measures along
thexˆ,yˆorzˆaxestoobtainthefirst,secondorthirdbits.
In all cases,the guessedvalue is correctwith probability
1(1+ 1 )≃0.788675. Themixtureofthefourstatescor-
2 √3
respondingtox ⊕x =0(i.e. s-parity0fors=(1,1,0))
1 2
is identical to the mixture of the four states correspond-
ing to x ⊕x = 1 and is equal to I/2. Similarly for
1 2
the two mixtures associatedwith each of the other three
FIG. 1: Bloch representation of states and measurements in
parities, x ⊕x (s = (1,0,1)), x ⊕x (s = (0,1,1)),
quantum2-bit and 3-bit parity-oblivious multiplexing. 1 3 2 3
and x ⊕x ⊕x (s = (1,1,1)). The protocol is there-
1 2 3
fore parity-oblivious for all s-parities. Again we have a
violation of the NC inequality because for n=3 the up-
one would know nothing about any s-parity of x. The per bound on the probability of success is 2/3. It is an
argumentsofarcanbesummarizedasfollows: forprepa- openquestionwhether0.788675isthemaximumpossible
ration noncontextual models, parity-obliviousnessat the quantum violation.
operationallevelimplies parity-obliviousnessatthe level The 2-bit protocolwasoriginallypresentedas a 2-to-1
of the hidden variables. quantum random access code by Wiesner [6] and redis-
The hidden state λ provides a classical encoding of x. coveredinRef. [7],whilethe3-bitprotocolwaspresented
But, as just shown, it is one that cannot contain infor- in Ref. [8] as an instance of a 3-to-1 quantum random
mationaboutany s-parity. We recallfromlemma 1 that access code (the original idea is attributed to Chuang in
such encodings have information about at most one bit, Ref. [7]).
xi, of x. Consequently, even if Bob could determine λ Experimental results. We experimentally demon-
perfectly, he and Alice could at best achieve the optimal strate better-than-classical performance for 2-bit and 3-
probability of success achievable in a classical protocol bit parity-oblivious multiplexing by implementing the
(specifiedinlemma1),while ifBobislimited inhis abil- quantum protocols using polarization qubits. Photon
ity to determine λ (as will be the case in general in a pairs from downconversion are coupled into single mode
hidden variable model), they will do worse. optical fibers. One photon acts as a trigger, while the
Quantum case. We now consider how well one can other is used in the experiment. Alice’s state prepara-
achieveparity-obliviousmultiplexing inquantumtheory. tion consists of a fiber polarization controller, and a po-
The following is a protocol for the 2-bit case that uses larizingbeamdisplacer,rotatedto the input state angle,
a single qubit as the quantum message. Alice encodes used to ensure high-purity linearly polarized states for
her 2 bits into the four pure quantum states with Bloch the 2-bit protocol. An additional quarter wave plate is
vectors(± 1 ,± 1 )equallydistributedonanequatorial used to prepare elliptically-polarized states for the 3-bit
√2 √2
plane of the Bloch sphere, as indicated in Fig. 1 (recall protocol. Bob’s measurement consists of a polarizing
that a density operator ρ is related to its Bloch vector~r beam displacer mounted in a computer-controlled rota-
by ρ= 1(I+~r·~σ), where~σ is the vector of Paulimatri- tionmount,followedbyasinglephotoncountingmodule.
2
ces). Bobmeasuresalongthe xˆ axis ifhe wishes to learn Forourdemonstration,adetectorisplacedatonlyasin-
the first bit, and along the yˆ axis if he wishes to learn gleoutputportofthebeamdisplacerandtheprobability
thesecond. Heguessesthebitvalue0uponobtainingthe of each outcome is calculated from the relative number
positive outcome. In all cases, the guessed value is cor- of counts for a given beam displacer angle and the one
rect with probability cos2(π/8) ≃ 0.853553. Meanwhile, orthogonal to it. (Further details of the experimental
no information about the parity can be obtained by any set-up, including a figure, are provided in Appendix B.)
quantummeasurementgiventhattheparity0andparity Adjustmentofthebeamdisplacerandquarterwaveplate
1mixturesarerepresentedbythesamedensityoperator, anglesallowsmeasurementof the horizontal/verticalba-
1ρ +1ρ = 1ρ +1ρ =I/2.We havea violationof sis, the diagonal/anti-diagonal basis and the right/left-
2 00 2 11 2 01 2 10
theNCinequalityofThm.2becauseforn=2,theupper circularbasis. Validmeasurementeventsareheraldedby
bound on the probability of success is 3/4. By exploit- acoincidencecountbetweenthedirectlydetectedphoton
ing a connection with the Clauser-Horne-Shimony-Holt and the experiment photon. These experimental proce-
inequality [9], one can show that this protocolyields the dures for a given x and y define the preparation Px and
maximumpossiblequantumviolationoftheNCinequal- the measurement My respectively.
ity. We obtained probabilities p(k = x |P ,M ) by accu-
y x y
Aprotocolfor3-bitparity-obliviousmultiplexingusing mulating statistics over approximately 3.5×107 coinci-
asinglequbitproceedsasfollows. Aliceencodesherthree dence counts for each x and y in the 2-bit scheme and
bits into a set of eight pure quantum states associated 2.4× 107 in the 3-bit scheme. Using Eq. (3), we cal-
4
culated the 2-bit and 3-bit probabilities of success to By our characterization of the source, we estimate the
be p(b = x ) = 0.851929±0.000030 and p(b = x ) = probability of two photons to be 0.007±0.003 relative
y y
0.786476± 0.000017 respectively. The errors were de- to the single photon generation probability. If two pho-
termined from the Poissonian counting statistics of the tonspassthroughthepolarizersintheidealprotocol,the
parametricsourceandthesmallrepeatabilityerrorinthe maximum probability of correctly estimating the parity
wave plate settings, using standard error analysis tech- can be quite far from 1/2: it is 3/4 in the case of the
niques. These probabilities of success violate the NC in- 2-bit scheme and 2/3 for three of the four s-parities in
equalityofThm.2withahighdegreeofconfidence: 3410 the 3-bit scheme. However, the fact that this possibility
and6922standarddeviationsrespectively. Theyarealso occurs with low probability implies that the two-photon
close to the predicted quantum values of 0.853553 and contribution to the probability of correct estimation is
0.788675, achieving a violation that is 98.4% and 98.2% comparable to the one-photon contribution. (Contribu-
respectively of the gap between the NC bound and the tions from three or more photons are negligible in com-
quantum value. parison). The weighted average of these contributions is
Just as Bell inequality violations are only surprising easily calculated and the largest, among all s-parities, is
given the absence of signalling between the two wings found to be 0.504±0.002. The fact that this is within
of the experiment, the NC inequality violations are only one percent of 1/2 demonstrates that our experimental
surprising given the parity-oblivious property. However, protocols are indeed close to parity-oblivious.
whereas one can establish the absence of signalling by Given that the quantum protocols described herein
confirming that the two wings are space-like separated, are also 2-to-1 and 3-to-1 random access codes, our re-
one must directly test for transmission of information sults constitute the first experimental demonstration of
about the parity in our experiment. A consideration of a quantum advantage for these tasks as well.
how this is to be accomplished highlights two shortcom- Finally, it is worth noting that every Bell inequality is
ings in the operational definition of preparationnoncon- a specialcase of anNC inequality where allassumptions
textuality of Eq. (1): in practice one can never imple- of noncontextuality are justified by locality [2]. Conse-
ment all measurements and one never finds truly identi- quently,everyexperimentalviolationofaBellinequality
cal statistics. The first issue may be addressed by rely- demonstratestheimpossibilityofanoncontextualhidden
ing on previous experimental evidence for the existence variable model. Indeed, this is all that can be demon-
ofatomographicallycompletesetofmeasurements–one strated by those that fail to seal the locality loophole
from which the statistics of any other measurement can [10, 11]. Nonetheless, a dedicated experiment of the sort
be calculated – and testing indistinguishability relative wehavedescribedherecanachievealargeviolationwith
to this set alone, as we shall do here. The second is- high confidence at a smaller cost of experimental effort.
suemaybe addressedbypresumingakindofcontinuity:
closeness of experimental statistics implies closeness of Acknowledgements. R.W.S. thanks M. Leifer and
the representations in the model [2] (this parallels the J. Barrett for helpful discussions. This work has
problem of dealing with imperfect alignment in tradi- been supported by the Australian Research Council, an
tional proofs of contextuality [12], where continuity also IARPA-fundedUSArmyResearchOfficecontract,NWO
providesananswer[4, 13]). Inthe presentwork,we sim- VICI project 639-023-302, the Dutch BSIK/BRICKS
plydemonstratethattheexperimentalstatisticsareclose project, the EUs FP6-FET Integrated Projects SCALA
to parity-obliviouswhile yielding a large violation of the (CT-015714) and QAP (CT-015848), and the Royal So-
noncontextuality inequalities, and leave a more detailed ciety.
analysis for future work.
Wequantifytheobliviousnessofourexperimentalpro-
tocol for a particular s-parity by the maximum proba- APPENDIX A: OPTIMAL CLASSICAL
bility that Bob can correctly estimate this parity in a PROTOCOL FOR N-BIT PARITY-OBLIVIOUS
variation over all measurements. One can estimate this MULTIPLEXING
byimplementing atomographicallycompletesetofmea-
surements, then reconstructing the states ρ0 and ρ1 as- We here provide a more detailed proof of lemma 1.
sociated with s-parity 0 and s-parity 1, and finally mak- First, note that by the assumption of parity-
ing use of the fact that the maximum probability of dis- obliviousness, the classical message m sent from Alice
criminating these states is 21 + 14Tr|ρ0−ρ1|. Among all to Bob must satisfy
s-parities, we calculate the largest such probability to
be 0.5020 ± 0.0002. This calculation is not sufficient, ∀s: p(P |m)= p(P |m) (8)
x x
however, because it neglects an imperfection in the ex-
periment that also contributes to leakage of information x|Xx·s=0 x|Xx·s=1
abouttheparity,namely,thatthereisasmallprobability ByBayes’theoremandthefactthatthedistributionover
of more than one photon being sent to the experiment. inputs x is uniform, we can rewrite this as a constraint
5
on p(m|P ), namely, • For p(i) an arbitrary distribution over {1,...,n},
x
if p (m)p (m) = 0 for all i, then Alice has sim-
i,0 i,1
∀s: p(m|Px)= p(m|Px). (9) ply chosen a value i ∈ {1,...,n} according to this
x|Xx·s=0 x|Xx·s=1 distribution and encoded xi in her message.
As we will demonstrate (at the end of this section), this • For p(i) an arbitrary distribution over {1,...,n},
implies that p(m|Px) has the form if pi,b(m)pi′,b′(m) = 0 when either b 6= b′ or i 6=
i, then Alice has encoded both i and x in her
p(m|P )=p(0)p (m) ′ i
x 0 message.
n
+ p(i)[pi,0(m)δxi,0+pi,1(m)δxi,1], (10) Itremainstoproveourclaimthatthe parity-oblivious
Xi=1 constraint,Eq.(9),impliesthedecompositionofp(m|Px)
described in Eq. (10). We do this using Fourier analysis
where p(i) is a normalized probability distribution on
overZn. Letr ∈{0,1}n. Definefunctionsχ :{0,1}n→
{0,...,n}, the functions p (m),p (m) and p (m) are 2 r
0 i,0 i,1
[−1,1] where
normalized probability distributions over m, and where
δa,b is the Kronecker delta function (equal to 1 if a = b χ (x):=(−1)xr.
r ·
and 0 otherwise).
Itfollowsthatanyclassicalparity-obliviousmultiplex- These form an orthonormalset because
ing protocol can be interpreted as follows: Alice gener-
ates an integer i ∈ {0,...,n} from the distribution p(i). χr(x)χr′(x)= (−1)x·(r⊕r′) =2nδr,r′.
Upon obtaining i = 0, she sends a message m chosen Xx x∈X{0,1}n
fromthe distribution p (m) (independent of the value of
0
Moreover,noting that the dimensionality of the space of
x). Upon obtaining i ∈ {1,...,n}, she sends a message
functions on {0,1}n is 2n (a parameter for every input
m chosen from one of two distributions, depending on
string) and that there are 2n values of r, we see that the
the value of the ith bit of x : the distribution is p (m)
i,0
χ form an orthonormal basis of the function space. It
if x =0 and p (m) if x =1. r
i i,1 i
follows that we can write p(m|P ) in the Fourier series
We now determine the choice of these distributions x
that leads to a maximum probability of winning. First
p(m|P )= pˆ(m,r)χ (x).
notethatifi=0,Bobgetsnoinformationaboutx. This x r
r
isclearlynotoptimal,sowemaysetp(0)=0. Nextnote X
thattheamountthatBoblearnsaboutxi dependsonhis We infer that
ability to distinguish p (m) from p (m). To optimize
i,0 i,1
theamountthatBobcanlearn,pi,0(m)andpi,1(m)must 2npˆ(m,r)= χr(x)p(m|Px)
be chosen to be perfectly distinguishable. This is only x
X
possible if they are completely non-overlapping, that is, = p(m|P )− p(m|P ).
x x
if p (m)p (m)=0.
Ini,0 an oi,p1timal decoding, Bob simply determines x|Xx·r=0 x|Xx·r=1
whether m is in the support of p (m) or of p (m) Combining this with the parity-obliviousness condition,
y,0 y,1
and outputs b =0 or 1 accordingly. This is optimal for Eq. (9), one obtains
thefollowingreason. Themessagemonlycontainsinfor-
∀s∈Par:pˆ(m,s)=0.
mation about x if Alice happened to generate an i that
y
coincides with y and in this case Bob will output b=x
y Consequently, the only strings r for which pˆ(m,r) 6= 0
withprobability1.Whenidoesnotcoincidewithy,Bob
are those with Hamming weight 0 or 1. Denoting the
gets no information about x from m, so it is irrelevant
y Fourier coefficients of the all zero string by pˆ (m) and
0
whatheoutputs;giventhatx isequallylikelytobe0or
y that of the string with a single 1 at position i by pˆ(m),
i
1, his probability of having generatedthe correctoutput
we have
will be 1/2.
n
Finally, given that y is chosen uniformly at random,
p(m|P )=pˆ (m)+ pˆ(m)(−1)xi.
the probability of i coinciding with y is 1/n, so that x 0 i
the overall probability of a correct output is 1(1) + Xi=1
n
1− 1 (1) =(n+1)/2n. Because (−1)xi = δ −δ and 1 = δ +δ , we
n 2 xi,0 xi,1 xi,0 xi,1
Itisworthnotingthattherearemanynaturalschemes can write
(cid:0) (cid:1)
that achieve the optimum:
n
• Ifp(i)=δi,j forsomeparticularj ∈{1,...,n},and p(m|Px)=a0(m)+ [ai,0(m)δxi,0+ai,1(m)δxi,1]
p (m)p (m)=0, then Alice has simply encoded i=1
j,0 j,1 X
(11)
the value of x in her message.
j
6
where we have defined nonnegative coefficients
parametric quarter wave plates
downconversion
a (m)=2pˆ(m), a (m)=0 if sgn(pˆ(m))≥0, source
i,0 i i,1 i detector
a (m)=0, a (m)=−2pˆ(m) if sgn(pˆ(m))<0;
i,0 i,1 i i
beam beam
and we have implicitly defined a constant a0(m), which fiber displacer displacer
we presently show is also nonnegative. To do this, we polarization filter
controller
define an n-bit string z(m) that encodes the signs of the
Alice Bob
Fourier coefficients. Specifically, z(m) is defined by
FIG. 2: Experimental set-up. The parametric downconver-
1 if sgn(pˆ(m))≥0,
i
z (m)≡ sionsourceprovidessinglephotonstotheexperiment. Detec-
i
(0 if sgn(pˆi(m))<0. tioneventsintheexperimentarecountedincoincidencewith
thedownconversion trigger photon (not shown).
It follows from this definition that
a (m)δ +a (m)δ =0
i,0 zi(m),0 i,1 zi(m),1
(BiBO) to generate pairs of 820 nm, horizontally polar-
for all i, and consequently that
ized single photons from a 410 nm, 60 mW continuous-
p(m|P )=a (m), wave diode laser. A 10 nm FWHM interference filter
z(m) 0
is used to reject background light. In the experiment,
which establishes that a (m)≥0. we obtained coincidence rates (2.5 ns window) of ap-
0
Finally,weshowthatEq.(11)canbeputintotheform proximately23100pairs/sinthe 2-bitscheme and15200
of Eq. (10). By the normalization of the distribution pairs/s in the 3-bit scheme.
p(m|Px), we have Although we chose to implement the experiment with
a heralded mode of a downconversionsource, similar re-
n
sults could also have been obtained with weak coherent
1= p(m|P )= a (m)+ a (m),
x 0 i,xi states(usingthesamemeasurements)[15]. Inbothcases,
m m i=1 m
X X XX onemustpostselectonnotfindingthevacuum(implying,
for all x. Defining A0 = ma0(m) and Ai,xi = incidentally,thatthedetectorloopholeisnotsealed[14]),
mai,xi(m), we have P and in both cases there is a small amplitude for more
than one photon and hence a small amount of leaked
P n
parity information. Our choice was motivated by differ-
A + A =1,
0 i,xi ences in ideal performance – it is only for the downcon-
i=1
X versionschemethattheleakageofparityinformationcan
forallx,whichimpliesthat n A isindependentof be eliminated in principle (through the use of true sin-
i=1 i,xi
x and in particular of x . We deduce that glephotonsheraldedbyefficientnumber-resolvingdetec-
i
P tors). Nonetheless, this ideal has not yet been realized.
A =A
i,0 i,1
foralli. Eq.(10)nowfollowsfromEq.(11)byidentifying
p(0)=A [1] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966); S. Kochen
0
and E. P. Specker,J. Math. Mech. 17, 59 (1967).
p(i)=A =A
i,0 i,1 [2] R. W. Spekkens,Phys. Rev.A 71, 052108 (2005).
[3] J. S.Bell, Physics 1, 195 (1964).
and
[4] A. Cabello and G. Garcia-Alcaine, Phys. Rev. Lett. 80,
1797 (1998); C. Simon, C. Brukner and A. Zeilinger,
p (m)=a (m)/p(0)
0 0 Phys. Rev. Lett. 86, 4427 (2001); J.-A. Larsson, Euro-
p (m)=a (m)/p(i). phys.Lett. 58, 799 (2002).
i,b i,b
[5] E. F. Galv˜ao, Ph.D. thesis, arXiv:quant-ph/0212124v1.
if p(0),p(i)6=0. [6] S. Wiesner, Sigact News 15, 78 (1983).
[7] A.Ambainis,A.Nayak,A.Ta-Shma,andU.Vazirani,in
Proceedingsofthe31stAnnualACMSymposiumonthe
Theory of Computing (ACM Press, NewYork,1999).
APPENDIX B: EXPERIMENTAL DETAILS
[8] M.Hayashiet al.,NewJournalofPhysics8,129(2006).
[9] J. F. Clauser, M.A. Horne, A. Shimony and R. A. Holt,
A schematic of the experimental set-up is provided in Phys. Rev.Lett. 23, 880 (1969).
Fig.2. Weusedtype-Idownconversioninbismuthborate [10] M. A.Rowe et al., Nature409, 791 (2001).
7
[11] Y.Hasegawa et al.,Nature425, 45 (2003). [15] TheWignerrepresentation would notprovideaclassical
[12] D.A. Meyer, Phys.Rev. Lett. 83 3751 (1999). statisticalmodeloftheexperimentbecausetherepresen-
[13] N.D. Mermin, arXiv:quant-ph/9912081. tation of themeasurements would be nonpositive.
[14] P.Pearle, Phys.Rev.D, 2, 1418 (1970).