Fine-grained uncertainty relation and biased non-local games in bipartite and tripartite systems

0.27 MB·English

by  Ansuman Dey

Fine-grained uncertainty relation and biased non-local games in bipartite and
tripartite systems
Ansuman Dey,1,∗ T. Pramanik,1,† and A. S. Majumdar1,‡
1S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700 098, India
The fine-grained uncertainty relation can be used to discriminate among classical, quantum and
super-quantum correlations based on their strength of nonlocality, as has been shown for bipartite
and tripartite systems with unbiased measurement settings. Here we consider the situation when
twoandthreeparties,respectively,choosesettingswithbiasforplayingcertainnon-localgames. We
showanalyticallythatwhilethefine-graineduncertaintyprincipleisstillabletodistinguishclassical,
quantumandsuper-quantumcorrelationsforbiasedsettingscorrespondingtocertainrangesofthe
biasing parameters, the above-mentioned discrimination is not manifested for all biasing.
2 PACSnumbers: 03.65.Ud,03.65.Ta
1
0
2 INTRODUCTION to various underlying theories such as classical theory,
quantum theory and no-signaling theory with maximum
t
c non-locality for bipartite systems.
O Heisenberg uncertainty relation [1] infers the restric-
tion inherently imposed by quantum mechanics that Further insight into the nature of difference between

6 we cannot simultaneously predict the measurement out- various types of correlations has been recently provided
1
comes of two non-commuting observables with certainty. by the work of Lawson et al.[10]. For a class of Bell-

] Thisuncertaintyrelationwasgeneralizedforanytwoar- CHSH [11, 12] games, they introduce the situation when
h bitraryobservablesbySchro¨dingerandRobertson[2]. In thetwopartiesdecidetochoosetheirmeasurementswith
p
quantuminformationtheory,itismoreconvenienttouse bias. It has been shown that for certain range of the bi-
-
t the uncertainty relation in terms of entropy in stead of asing parameters, quantum theory offers advantage and
n
standard deviation. A lot of effort has been devoted to- surprisingly for others, it does not provide a better re-
a
u wards improving entropic uncertainty relations [3–5], es- sult than classical mechanics. This leads towards the
q pecially in terms of their practical applicability in sev- identification of situations when quantum entanglement
[ eral information processing scenarios such as quantum isindeedessentialforimplementingaparticularinforma-

1 information locking and key generation [6–8]. Recently, tion processing task. A generalization for multipartite
v a new fine-grained form of the uncertainty relation has systems is also performed in which numerical results for
0 been proposed [9] which is able to distinguish between the upper bound of the correlation function is presented
0
uncertainties inherent in various possible measurement when all parties measure with equal bias.
3
outcomes, and is linked with the degree of nonlocality of
4 In this work we investigate the connection between
0. the underlying theory. the fine-grained uncertainty relation and non-locality in
1 Though the use of entanglement in information pro- the context of biased non-local games for first bipartite
2 cessingtasksiswidelyappreciated,quantumcorrelations and then tripartite systems. Our motivation is to utilize
1
may not be advantageous compared to classical ones in the fine-grained uncertainty relation in order to deter-
:
v alltypesofsituations. Sinceentanglementisafragilere- mine the nonlocal resources necessary for implementing
i source,thequestionastowhenpreciselyquantumnonlo- this particular information processing task of winning a
X
cality following from entanglement is necessary for prac- biased game played by two or three parties. Here we
r
a tical applications, is rather important. In this context, make use of the formalism developed by Oppenheim and
the application of the fine-grained uncertainty relation Wehner [9] for bipartite systems, and its subsequent ex-
could be particularly relevant. The fine-grained uncer- tension to the case of tripartite systems [13]. In case of
taintyrelationisabletodiscriminatebetweenthedegree bipartite systems, correlations are expressible in terms
of nonlocality in classical, quantum and super-quantum of Bell-CHSH [11, 12] form without ambiguity and can
correlationsofbipartitesystems,aswasshownbyOppen- be used efficiently for above mentioned task of discrim-
heimandWehner[9]inthecontextofaclassofnon-local ination between classical, quantum and super-quantum
retrievalgamesforwhichthereexistsonlyoneanswerfor theories. The scenario for the tripartite case is however,
any of the two parties to come up with in order to win. a bit different as there is an inherent non-uniqueness re-
Themaximumprobabilityofwiningtheretrievalgameis gardingthechoiceofcorrelationsproposedbySvetlichny
equaltotheupperboundoftheuncertaintyrelationand [14] and Mermin [15]. It has been shown [13] that the
thisquantifiesthedegreeofnonlocalityoftheunderlying Svetlichny-type correlations can discriminate among the
physicaltheory. Thisupperboundcouldthusbeusedto classical,quantumandno-signalingtheoryusingthefine-
discriminate among the degree of nonlocality pertaining grained uncertainty principle, whereas the inequality ex-
2
tracted from the Mermin-type correlation is unable to The fine-grained uncertainty relation [9] may be now
perform the same task. In our present analysis we use invoked by noting that for every setting s and the corre-
the approach proposed by Bancal et al.[16] in order to sponding outcome a of Alice one may formally denote a
calculate the upper bound of the Svetlichny function an- stringx =(x1 ,x1 ,...,xn )oflengthn=|T |deter-
s,a s,a s,a s,a
alytically in case of the biased tripartite game. We are miningthewinninganswerb=xt forBob;{s}∈S and
s,a
thusabletopresentwithoutusingnumericalmethodsour {t}∈T,S andT beingthesetofAlice’sandBob’sinput
results on the ranges of biasing parameters when quan- settings, respectively. Alice and Bob receive the binary
tum correlation are beneficial for non-local tasks. questions s,t ∈ {0,1} (i.e. representing two different
measurement settings on each side) with corresponding
probabilities (for p and q (cid:54)= 0,1 the game is non-local)
FINE-GRAINED UNCERTAINTY AND BIASED and they win if their respective outcomes a,b ∈ {0,1}
NONLOCAL GAMES
satisfy the condition (2). Before starting the game (a
biased CHSH-game), Alice and Bob communicate and
We begin with the description of the scheme of the discuss their strategy, i.e., choice of the bipartite state
gametobeplayedwithinthebipartitesystem. Thesitu- ρ they are sharing and their measurements. They are
AB
ationissuchthatthetwoparties,namely,AliceandBob not allowed to communicate once the game starts. The
share a state ρAB which is emitted and distributed by a probability of winning the game for a physical theory
source. Alice and Bob are spatially separated enough so described by bipartite state (ρ ) is given by [9],
AB
thatnosignalcantravelwhileexperimenting. Aliceper-
(cid:88) (cid:88)
forms either of her measurements A1 and A2 and Bob, Pgame(S,T,ρAB)= p(s,t) p(a,b=xts,a|s,t)ρAB
eitherofB andB atatime. Thesemeasurementshav-
1 2 s,t a
ingtheoutcomes+1and−1,canbechosenbyAliceand (3)
Bobwithoutdependingonthechoicemadebytheothar. When Pgame(S,T,ρ ) is less than 1, the outcome of
AB
The CHSH inequality [12] the game is uncertain. The value of Pgame is bound by
particular theories. For the unbiased case (i.e., p(s,t) =
1 1
E(A B )+E(A B )+E(A B )−E(A B )≤ (1) p(s).p(t) = 1.1 = 1), the upper bounds of this value in
4 1 1 1 2 2 1 2 2 2 2 2 4
classical,quantumandno-signalingtheoryare 3, 1+ √1
4 2 2 2
holdsforanylocalhiddenvariablemodelandcanbevio- and 1 respectively. The form of p(a,b = xt |s,t) in
latedwhenmeasurementsaredoneonquantumparticles terms of the measurements on the bipartites,satate ρρAB is
AB
prepared in entangled states. Here E(AiBj) is the av- given by,
erage of the product of outcomes of Alice and Bob with
(cid:88)
i,j = 1,2. The above inequality refers to the scenario p(a,b=xt |s,t) = V(a,b|s,t)(cid:104)(Aa⊗Bb)(cid:105)
s,a ρAB s t ρAB
when the two parties have no bias towards choosing a
b
particular measurement. (4)
In the following picture, describing the biased game where,Aa = I+(−1)aAs isthemeasurementoftheobserv-
s 2
[10]theintentionofAliceistochooseA1withprobability ableAscorrespondingtothesettingsgivingtheoutcome
p(0(cid:54)p(cid:54)1)andA2withprobability(1−p). Bobintends aatAlice’sside;Bb = I+(−1)bBt isameasurementofthe
to choose B and B with probabilities q(0 (cid:54) q (cid:54) 1) t 2
1 2 observable B corresponding to the setting t giving the
t
and (1−q), respectively. The measurements and their
outcome b at Bob’s side and V(a,b|s,t) filters the win-
outcomesarecodedintobinaryvariablespertainingtoan
ning combination and is given by,
input-output process. Alice and Bob have binary input
variables s and t, respectively, and output variables a V(s,b|s,t) = 1 iff a⊕b=s.t
and b, respectively. Input s takes the values 0 and 1
= 0 otherwise. (5)
when Alice measures A and A , respectively. Output a
1 2
takesthevalues0and1whenAlicegetsthemeasurement Pgame(S,T,ρ ) can now be calculated using the
AB
outcomes +1 and −1, respectively. The identifications Eqs.(3)-(5) with the given probabilities of different mea-
are similar for Bob’s variables t and b. Now, the rule surementsofAliceandBob. Forthebipartitestateρ ,
AB
of the game is that Alice and Bob’s particles win (as a the expression of Pgame is given by
team) if their inputs and outputs satisfy
1
Pgame(S,T,ρ )= [1+(cid:104)CHSH(p,q)(cid:105) ] (6)
a⊕b=s.t (2) AB 2 ρAB
where ⊕ denotes addition modulo 2. Input questions s withCHSH(p,q)=[pqA ⊗B +p(1−q)A ⊗B +(1−
1 1 1 2
and t have the probability distribution p(s,t) (for sim- p)qA ⊗B −(1−p)(1−q)A ⊗B ] being the form of
2 1 2 2
plicity we take p(s,t) = p(s)p(t) where p(s = 0) = p, CHSH-function after introducing bias.
p(s = 1) = (1−p), p(t = 0) = q and p(t = 1) = (1−q) ThemaximumprobabilityPgame ofwinningthegame
in our case). isobtainedbymaximizingthefunction(cid:104)CHSH(p,q)(cid:105)for
3
differenttheories. Suchmaximizationwasfirstperformed It has been of much interest to consider the super-
intheliteraturefortheunbiased[17]scenarioandsubse- quantum correlation or the no-signaling theory [18]. For
quently, for the biased case as well [10] which we follow no-signaling, the score of the game is
by treating it in two halves of the ranges of the parame-
(cid:88) (cid:88)
ters p and q. First, consider the case of p,q ≥ 1/2. The Pgame|no−signaling = p(s,t) p(a,b|s,t)
maximum
classical maximum is obtained using an extremal strat- s,t a,b
egy where the values of all the observables are +1 giving =pq+(1−p)q + p(1−q)+(1−p)(1−q)
the maximum value of the above CHSH-function to be
=1 (12)
1−2(1−p)(1−q). With this classical maximum, the
winning probability is given by
giving the same upper bound of the fine-grained uncer-
tainty relation in spite of the game being biased. we will
Pgame(S,T,ρ )|classical =1−(1−p)(1−q) (7)
AB maximum now proceed to case of tripartite system.
This reduces to the value 3 for the unbiased case when
4
p=q = 1.
2 A BIASED TRIPARTITE SYSTEM
For considering the quantum strategy, Lawson et al.
[10] divide the parameter space in two regions of [p,q]
Wewillnowconsiderabiasednon-localtripartitegame
space with the first region corresponding to 1 ≥ p ≥
with Alice, Bob and Charlie as players. Similar to the
(2q)−1 ≥ 1 (region-1). In this region,
2 bipartite case, Alice, Bob and Charlie have their input
binary variables (or questions) s, t and u (s,u,t ∈ 0,1)
(cid:104)CHSH(p,q)(cid:105)≤1−2(1−p)(1−q) (8)
corresponding to their respective two different measure-
ments settings, and output binary variables (or answers)
thus leading to
a, b and c (a,b,c ∈ 0,1) corresponding to their respec-
tive outcomes of measurements. Given a rule (i.e., the
Pgame(S,T,ρ )|region =1−(1−p)(1−q) (9)
AB 1 winning condition) of the game, the maximum winning
probability (having the established correspondence with
One sees that the upper bound is the same value as that
the upper bound of the fine-grained uncertainty relation
achieved by classical theory, and hence, quantum corre-
[13]) can be calculated by considering the various possi-
lation (entanglement) offers no advantage over classical
bilities of outcomes (along with the measurements) sat-
correlation in performing the specified task in this re-
isfying the rule.
gion. This result could be restated as follows. If the bias
We consider a full-correlation box (namely, Svetlichny
parametersareregulatedinthisregion,wecannotdiffer-
Box [14]) for which all one and two party correlations
entiatebetweenclassicalandquantumcorrelationsusing
vanish [19]. The game is won if the answers satisfy
the upper bound of the fine grained uncertainty relation
corresponding to the biased non-local game in context.
a⊕b⊕c=st⊕tu⊕us . (13)
Now,letusconsidertheotherregion1≥(2q)−1 >p≥
12 (region-2), which gives the bound Inthiscase,AliceintendstomeasurewithhersettingA1
√ (cid:112) (cid:112) with probability p (i.e., p(s=0)=p) and A2 with prob-
(cid:104)CHSH(p,q)(cid:105)≤ 2 q2+(1−q)2 p2+(1−p)2. ability(1−p)(i.e.,p(s=1)=(1−p)). BobmeasuresB
1
(10) andB withprobabilitiesqand(1−q)respectively(hence,
2
This value is greater than the classical bound. So, the p(t = 0) = q and p(t = 1) = (1−q)). Charlie measures
regulation of the biasing parameters in this region dis- withhisoperatorC withprobabilityrandC withprob-
1 2
criminatesamongclassicalandquantumcorrelation. The ability (1−r) (∴ p(u = 0) = r and p(u = 1) = (1−r)).
upperboundofthefine-graineduncertaintyrelation(i.e., They share the state ρ , and they can communicate
ABC
themaximumchanceofwinningthegame)isinthiscase their measurement settings before the game starts.
given by, The winning probability is quantified as,
Pgame(S,T,ρ ) | quantum Pgame(S,T,U,ρ )
AB maximum ABC
= 21[1+√2(cid:112)q2+(1−q)2 (cid:112)p2+(1−p)2] (11) = (cid:88)p(s,t,u)(cid:88)p(a,b,c=xus,t,a,b|s,t,u)ρABC(14)
s,t,u a,b
This also reduces to the unbiased value of [12 + 2√12] for wherep(s,t,u)=p(s)p(t)p(u)istheprobabilityofchoos-
p=q = 1. The quantum strategy for winning this game ing the measurement settings s by Alice, t by Bob and
2
is detailed in ref.[10]. The treatment is similar for the u by Charlie from their respective sets S, T and U.
otherregionwherep,q < 1 asthesituationissymmetric p(s,b,c|s,t,u) isthejointprobabilityofgettingout-
2 ρABC
[10]. comes, a, b and c for corresponding settings s, t and u
4
given by, that. Now, according to the form of (18) let us tem-
porarily change our point of view towards the game. It
p(a,b,c = xu |s,t,u)
s,t,a,b ρABC is the input setting of Charlie that defines which ver-
= (cid:88)V(a,b,c|s,t,u)(cid:104)Aa⊗Bb⊗Cc(cid:105) (15) sion of the CHSH-game Alice and Bob are playing. As-
s t u ρABC sume for a moment, that when Charlie’s input is C ,
c 1
Alice and Bob play the standard biased CHSH-game
where Aas, Btb and Cuc are the measurements (with the and when Charlie’s input is C2, they play CHSH(cid:48). Al-
forms given in the treatment of bipartite system) corre-
ice and Bob are together (and separated from Charlie)
sponding to the setting s and outcome a at the Alice’s
and being unaware of Charlie’s measurements, produce
side, setting t and outcome b at Bob’s side and setting u
any bipartite non-local probability distribution. Hence
and outcome c at Charlie side. V(a,b,c|s,t,u) equals 1
Alice and Bob are effectively playing the average game
onlywhencondition(13)issatisfied; otherwise, 0. Using r|CHSH(p,q)|+(1−r)|CHSH(cid:48)(p,q)|. If Alice or Bob
the condition (13) and Eq.(15), Eq.(14) simplifies to
stay separated and any of them is with Charlie to know
1 about the measurements, they will not be able to pro-
Pgame(S,T,U,ρABC,p,q,r)= 2[1+(cid:104)S(p,q,r)(cid:105)ρABC] duce results better than the local bound. For the region
(16) p,q,r ≥ 1, the classical maximum is calculated to be,
2
where S(p,q,r) is the Svetlichny function modified with
the introduction of bias, given by Smax =1−2(1−p)(1−q) (20)
S(p,q,r) giving
= pqrA ⊗B ⊗C +pq(1−r)A ⊗B ⊗C
1 1 1 1 1 2 Pgame(S,T,U,ρ )|classical =1−(1−p)(1−q) (21)
ABC maximum
+p(1−q)rA ⊗B ⊗C +(1−p)qrA ⊗B ⊗C
1 2 1 2 1 1
−p(1−q)(1−r)A ⊗B ⊗C which reduces to the value 3 for the unbiased game (r
1 2 2 4
is averaged off due to both the Bell functions possessing
−(1−p)q(1−r)A ⊗B ⊗C
2 1 2 the same classical maximum).
−(1−p)(1−q)rA ⊗B ⊗C
2 2 1 Inordertotreatthequantumoptimizationweconsider
−(1−p)(1−q)(1−r)A ⊗B ⊗C . (17) that the three parties share a three-qubit Greenberger-
2 2 2
Horne-Zeilinger (GHZ) state |ψ(cid:105) = √1 |000(cid:105) + |111(cid:105)
Tofindthemaximumprobabilityofwinning(whichisthe 2
(for the unbiased case, the maximum violation of the
upper bound of fine-grained uncertainty relation as pre-
SvetlichnyfunctionoccursfortheGHZstate[20]). Char-
sentedinEq.(14)),weneedtomaximize(cid:104)S(p,q,r)(cid:105) .
ρABC lie needs to choose two measurements in a way that he
The case when all the three parties are quantum-
preparesamaximallyentangledtwoqubitstateforAlice
correlated, has been handled only numerically in this
and Bob. One such choice is, C = σ and C = −σ
context [10]. We will however, perform this maximiza- 1 x 2 y
tion analytically using the scheme of bipartition model- which prepare the states |φ±(cid:105) = √12|00(cid:105) + |11(cid:105) and
ing [16]. This method is based on the fact that maximal |φ˜±(cid:105) = √1 |00(cid:105)+i|11(cid:105) respectively, for Alice and Bob.
2
quantum violation for the tripartite Svetlichny inequal- Note that
ity has been shown [16] even when the system does not
feature genuine tripartite nonlocality, i.e., only two of I⊗UB[|φ˜±(cid:105)(cid:104)φ˜±|]=[|φ±(cid:105)(cid:104)φ±|] (22)
the three parties are correlated in a nonlocal way. The
where U is a unitary rotation on the Bob’s qubit, given
Svetlichny function S(p,q,r) can be rearranged as B
by
S(p,q,r)
(cid:18) (cid:19)
1 0
= r[CHSH(p,q)]⊗C +(1−r)[CHSH(cid:48)(p,q)]⊗C U = (23)
1 2 B 0 −i
(18)
and consequently, the equivalence of optimizations of
where, CHSH(p,q) and CHSH(cid:48)(p,q) is realized as,
CHSH(p,q) = [pqA ⊗B +p(1−q)A ⊗B
1 1 1 2 |Tr[CHSH.|φ (cid:105)(cid:104)φ |]|
± ±
+(1−p)qA ⊗ B −(1−p)(1−q)A ⊗B ]
2 1 2 2 = |Tr[(I⊗U )CHSH(cid:48)(I⊗U†)|φ˜ (cid:105)(cid:104)φ˜ |]| (24)
CHSH(cid:48)(p,q) = [pqA ⊗B −p(1−q)A ⊗B B B ± ±
1 1 1 2
−(1−p)qA ⊗ B −(1−p)(1−q)A ⊗B ] So, instead of Alice and Bob playing with two kinds of
2 1 2 2
CHSH games, we may now think of the situation as only
(19)
the standard CHSH-game being played that is averaged
Here CHSH(p,q) is the traditional form of CHSH- over the scenarios when Bob rotates unitarily his qubit
polynomial and CHSH(cid:48)(p,q) is an equivalent form of before measurement and when he does not. The unitary
5
rotationpreservesthenonlocalpropertyofthestatecaus- todiscriminate[13]amongclassical,quantumandsuper-
ing no discrepancy. In the region p,q,r ≥ 1, the maxi- quantum correlations. But in the presence of bias, using
2
mumvalueof(cid:104)S(p,q,r)(cid:105)iscalculated(usingaprocedure a bipartition model [16] we observe here that there is
formaximizingCHSH(p,q)similartothebipartitecase) a zone specified by the biasing parameters where even
to be, the Svetlichny inequality cannot perform this discrimi-
nation. The extent of non-locality that can be captured
(cid:104)S(p,q,r)(cid:105)|GHZ(cid:105)|1max =1−2(1−p)(1−q) (25) by the fine-grained uncertainty principle thus turns out
to be regulated by the bias parameters. Our approach,
for the region 1 ≥ p ≥ (2q)−1 ≥ 12 which is the same in spite of featuring a narrower range of the biasing pa-
as the classically achieved upper bound. Here S is not a rameters providing quantum advantage, serves the pur-
functionofrbecausethenonlocalstrengthofAlice’sand pose of developing an analytical approach to explore the
Bob’ssystemsareidenticalforthetwodifferentmeasure- connection between biased nonlocal retrieval games and
ments of Charlie. For the region 1 ≥ (2q)−1 > p ≥ 12, the upper bound of fine-grained uncertainty capturing
one obtains thenonlocalstrengthsofvariouscorrelations. Analytical
√ (cid:112) (cid:112) generalizationstomultipartynonlocalgamesmayindeed
(cid:104)S(p,q,r)(cid:105) |2 = 2 q2+(1−q)2 p2+(1−p)2
|GHZ(cid:105) max be feasible using this approach.
(26)
Acknowledgements: ASM acknowledges support from
Theboundinthesecondregionisgreater,andhence,the
the DST project no. SR/S2/PU-16/2007.
quantum correlation dominates here. The expression for
maximum winning probability is in this case given by
Pgame(S,T,U,ρ )|quantum
ABC maximum
= 1[1+√2(cid:112)q2+(1−q)2(cid:112)p2+(1−p)2] (27) ∗ [email protected]
2 † [email protected]
‡ [email protected]
For every r((cid:54)= 0,1) there is the same patch in the p−q
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