Separable quantum states do not have stronger correlations than local realism. A comment on quant-ph/0611126 of Z. Chen

0.14 MB·English

by  M. Seevinck

Separable quantum states do not have stronger correlations than local realism.
A comment on quant-ph/0611126 of Z. Chen.
∗
Michael Seevinck
Institute for History and Foundations of Science,
Utrecht University PO Box 80.000, 3508 TA Utrecht, the Netherlands
(Dated: February 1, 2008)
Chen (quant-ph/0611126) hasrecentlyclaimed “exponential violation oflocal realism bysepara-
ble states”, in the sense that multi-partite separable quantum states are supposed to give rise to
correlations and fluctuations that violate a Bell-type inequality that Chen takes to be satisfied by
localrealism. However,thiscannotbetruesinceallpredictions(includingallcorrelationsandfluc-
tuations)thatseparablequantumstatesgiverisetohavealocalrealisticdescriptionandthussatisfy
all Bell-type inequalities, and this holds for all number of parties. Since Chen claims otherwise by
presenting a new inequality, claimed to be a Bell-type one, which separable states supposedly can
violate, theremust bea flaw in theargumentation. I will expose this flaw, not merely for clarifica-
7 tion of this issue, butperhaps even more importantly since it re-teaches us an old lesson John Bell
0 taught us over 40 years ago. I will argue that this lesson provides us with a new morale especially
0 relevant to modern research in Bell-type inequalities.
2

n
a Introduction observables:
J

2 E(k ,k ,...,k )=
Chen [1] has recently claimed “exponential violation of 1 2 n

localrealismbyseparablestates”,inthesensethatmulti-
1 A (k ,λ)A (k ,λ)...A (k ,λ)p(λ)dλ. (2)
v partite separable quantum states are supposed to give Z 1 1 2 2 n n
Λ
3 rise to correlations and fluctuations that violate a Bell-
0 typeinequalitythatChentakestobesatisfiedbylocalre- Here λ is an integration variable, often denoted as the
0 alism. Theviolationisclaimedtobebyanexponentially hidden variable, the set Λ is the set of hidden variables,
1 increasing amount as the number of particles n grows andthefunctionalsAi(ki,λ)areeithertheoutcomes(+1
0
(n>2). Infact,itisclaimedtoviolatethelocalrealistic or 1) of the measurements determined by the settings
07 maximum by a factor of 2n−2. ki −and hidden variable λ in case we are dealing with a
/ deterministic theory, or they are the expectation values
h However,this cannotbe true since allpredictions(in-
(in the interval [ 1, 1]) of these outcomes in the case of
p cluding all correlations and fluctuations) that separable −
a stochastic (non-deterministic) theory, and finally p(λ)
- quantum states giverise to havea localrealisticdescrip-
t is a normalised probability distribution of the variable
n tion and thus satisfy all Bell-type inequalities, and this
λ. The Bell inequalities of Eq. (1) follow solely from the
a holds for all number of parties. Although this is known
u assumption of Eq. (2).
already, let me nevertheless show this first (for the sake
q Let us now suppose that we have an n-partite fully
of completeness of the discussion)after which I will con-
: separablequantumstate ρ,i.e., a convexsumofproduct
v tinue my comment on the work of Chen. Here, I will states: ρ= p ρ1 ρ2 ... ρn,withρi adensityoper-
Xi generalise the exposition by Z˙ukowski [2] to the n-party j j j⊗ j⊗ ⊗ j j
atorforpartPy(subsystem)i. Theindexj isasummation
case.
r or integration variable and p = 1, 0 p 1. The
a j j ≤ j ≤
Any n-party Bell-type inequality has the following correlations these states givePrise to all have the form
generic form
Aˆ (k )Aˆ (k )Aˆ (k ) :=
1 1 2 2 n n ρ
h i
Tr[ρAˆ (k ) Aˆ (k )... Aˆ (k )]= (3)
1 1 2 2 n n
c(k ,k ,...,k )E(k ,k ,...,k ) B(c), ⊗ ⊗
|k1,kX2,...,kn 1 2 n 1 2 n |≤ XpjTr1[ρ1jAˆ1(k1)]Tr2[ρ2jAˆ2(k2)]...Trn[ρnjAˆn(kn)],
(1) j
where k , k , etc. are labels that distinghuish various
1 2 with Tri[ ] the partial trace for party i (i.e., the rest of
(discrete or continuous) measurement settings that can ·
be measured on system 1, 2, etc., c(k1,k2,...,kn) are the parties is traced out), and Aˆi(ki) the operator as-
certain constantcoefficients, B(c) is the maximum value sociated with the measurement on party i with setting
obtainablebyalocalrealistictheoryfortheexpressionon ki. Eachsuchmeasurementhaspossibleoutcomes+1or
the left hand side of Eq. (1), and E(k ,k ,...,k ) is the 1, just as was the case above [3]. Since we have that
1 2 n −
‘correlation function’ for outcomes of measurement with Tri[ρiAˆi(ki)] = Aˆi(ki) ρ maxAi(ki,λ) the corre-
| | |h i | ≤
settingsk ,k ,...,k ,whichforlocalrealistictheoriesis lations of Eq. (3) can be written as in Eq. (2) and thus
1 2 n
assumedtobetheexpectationoftheproductofthelocal they can be reproduced by a local realistic theory. They
thusmustsatisfyallBell-typeinequalitiesofEq. (1),i.e.,
‘those known at present, as well as those which one day
would be derived’[2]. Note that the same holds for all
∗Electronicaddress: [email protected] fluctuations a separable state can give rise to, cf. [4].
2
Since Chen claims otherwise by presenting a new in- The local settings are anticommuting (i.e.
equality, claimed to be a Bell-type one, which separable Aˆ (k ),Aˆ (k′) = 0) since this incorporates the
states supposedly can violate, there must be a flaw in l{ocail oirthoigoina}lity of the dichotomic observables.
the argumentation. I will expose this flaw, not merely After this conversion we obtain the well-known result
for clarification of the issue, but perhaps even more im- [5] that quantum mechanics obeys the inequality
portantly since it re-teaches us an old lesson John Bell
hasmadeover40yearsago,althoughinadifferentform. ˆ 2(n−1)/2, (8)
n ρ
|hM i |≤
I willarguethis lessonprovidesuswithanew moralees-
pecially relevant to modern research in Bell inequalities. where the expectation value Xˆ := Tr[Xˆρ]. The up-
ρ
h i
perbound can be achieved by the maximally entangled
Review of Chen’s results GHZ states.
Ifwesetn=2inEq. (6)weobtaintheoriginalCHSH
Let me first present the result of Chen’s analysis. He inequality for local realism and for n = 2 in Eq. (8) we
considers the well-known Mermin-Roy-Singh-Ardehali- get the Tsirelson inequality for quantum mechanics.
Belinski˘ı-Klyshko inequality for n-parties [5]. This in- Chennow considersthe quantum mechanicaloperator
equality is characterised by a specific choise [6] of co-
ˆ := ˆ + ˆ2, (9)
efficients c(k1,k2,...,kn) in Eq. (1), where each party Vn Mn Mn
chooses between two observables (i.e., each k has two
i and shows that
possibilities) that are furthermore dichotomic ( 1 val-
±
ued). Chentakesthe inequalitytobe normalisedsothat ˆ := Tr[ρˆ ]
B(c) = 1, and chooses a specific choice of measurement hVniρ Vn
settingsk1,k2...,kn,suchthatthetwopossiblelocalset- = Tr[ρMˆn]+Tr[ρMˆ2n]
tings foreachpartyaregivenbyorthogonalvectors(i.e., = ˆ + ˆ 2+∆( ˆ ) , (10)
A (k ) A (k′)). Suppose that for this specific choice of hMniρ hMniρ Mn ρ
i i ⊥ i i
settings we define the so-called Bell polynomial [7] with∆( ˆ )= ( ˆ ˆ )2 thevarianceof ˆ in
n n n ρ ρ n
M h M −hM i i M
n(λ):= thestateρ. Forseparablequantumstatesρsep heobtains
M the bound ˆ 2n−1, whichis tightsince it canbe
c(k ,k ,...,k )A (k ,λ)A (k ,λ)...A (k ,λ). hVniρsep ≤
1 2 n 1 1 2 2 n n achieved by a separable state (see [8]).
X
k1,k2,...,kn Local realism gives Eq. (5), and Chen furthermore
(4)
claims that
It follows from A (k ,λ) = 1 for all i that for the spe-
i i
cificsettingsused−1≤Mn±(λ)≤1. Becauseoflinearity ∆(Mn)LHV := Z (Mn(λ)−hMniLHV)2p(λ)dλ
of the mean (i.e., the linear combination in Eq. (1) can
also be evaluated under the expectation value) the Bell = hM2niLHV−hMni2LHV (11)
inequality Eq. (1) (for the specific choise of coefficients 1 2 , (12)
≤ −hMniLHV
c(k ,k ,...,k )) can be reformulated as an upperbound
1 2 n
on the expectation of this Bell polynomial (λ). In- from which it follows that
n
M
deed, local realism predicts the result that
:= + 2
hVniLHV hMniLHV hMniLHV
|hMniLHV|=|Z Mn(λ)p(λ)dλ|≤1, (5) =hMniLHV+hMni2LHV+∆(Mn)LHV ≤2.
(13)
with the expectation value X := X(λ)p(λ)dλ.
h iLHV λ
This is in fact a shorthand notation foRr the Bell-type Chen thus obtains that 2 whereas
n LHV
inequality ˆ 2n−1andsincethislhaVstiboundc≤anbeachieved
hVniρsep ≤
byseparablestates,heconcludesthatforn>2theyvio-
c(k ,k ,...,k )E(k ,k ,...,k ) 1, (6)
| 1 2 n 1 2 n |≤ latethelocalrealisticinequalitybyanexponentiallylarge
X
k1,k2,...,kn factor. This conclusion contradicts the previous analysis
with the correlation E( ) given by Eq. (2). This the that the correlations in separable states can be repro-
·
Mermin-Roy-Singh-Ardehali-Belinski˘ı-Klyshko inequal- duced by local realism. So where did Chen’s analysis go
ity [5]. wrong?
The quantum mechanical counterpart of this inequal-
ity is obtainedasfollows. Choosep(λ)=δ(λ ρ) withρ Exposing the problematic relation
−
aquantummechanicalstateofnqubits,andnextsubsti-
tute the measurementfunctionals in the Bellpolynomial Let us first note that in Eq. (12) it must have been used
by the operators associatedto the measurements to give that
its quantum counterpart which is denoted by the opera-
tor Mˆn: hM2niLHV =Z (Mn(λ))2p(λ)dλ≤1, (14)
ˆ := c(k ,k ,...,k )Aˆ (k )Aˆ (k )...Aˆ (k ).
n 1 2 n 1 1 2 2 n n
M k1,kX2,...,kn whichfollowsfrom(Mn(λ))2 ≤1thatonitsturnfollows
(7) from the fact that 1 (λ) 1.
n
− ≤M ≤
3
UsingthisweseethatChen’sclaimfollowssolelyfrom the legitimate Bell-type inequality of Eq. (6) that local
the quantum mechanical inequality realism must satisfy. Indeed, all expectation values in
the Bell-type inequality Eq. (6) involve only commuting
hMˆ2niρ ≤2n−1, (15) (compatible) quantities, and no noncommuting ones. So
although (λ) cannot be thought of as an observable
n
andthefactthatEq. (15)canbesaturatedforseparable M
specifyingthesumoflocalrealisticquantitiesdetermined
states. Comparing this to Eq. (14) it would seem that
by the hidden variable λ, and as such is never measured
alreadyforn=2separablestatesgiverisetocorrelations
in an experiment, when averaged over it does allow for
that can not be reproduced by local realism.
alegitimateshorthandnotationofthe Bell-typeinequal-
However,itisofcoursenot (λ)thatismeasuredin
n ity Eq. (6). This is the reason why the hidden variable
M
any experiment, but the observables Ai(ki,λ). But this counterpart of the operator ˆ can be safely chosen to
captures only part of the problem since, as we have seen Mn
be the Bell-polynomial (λ).
MBelnl-(tλy)pecainneqnuevaelirttyh(eil.ees.sEbqe. u(6s)e)d,wtoheorbetaasi(n a l(eλg)it)i2mciatne However, I will now Margnue that from a local realistic
Mn point of view such a manouvre cannot performedfor the
not – or so I claim. What accounts for this difference?
functional ( (λ))2.
A first starting point is to note that the operator Firstly, (Mn(λ))2 is not the legitimite hidden variable
Vhˆindd=enMˆvanr+iabMlˆe2ncooufnEteqr.p(a9r)tsahsoVunld(λb)e=trManns(laλt)e+dMint2no(λit)s. caoguedntoevrperarittMgonifvethsethoepienreaqtuorali(tMyˆEn)q2..(1T4)hubsutwthheisnisavneort-
But in order for Chen’s analysis to go through, he must
a legitimite local realistic counterpart of the inequality
have translated this into (λ) = (λ)+( (λ))2.
n n n
Chenmustthus haveassumVedthatM2(λ)=(M (λ))2, Eq. (15). This can be easily seen from the following ex-
Mn Mn ample. Suppose we take n = 2 and use the observables
andsinceMˆn andMˆ2n commutethisseemsareasonable A and A′ and B and B′ for party 1 and 2 respectively.
requirement. Indeed,butonlyifthequantity (λ)can
beconsideredunproblematicallyasahiddenvMarinableob- Then if we expand (M2(λ))2 we get
servable. However, I will now argue that this is not the 1 1
( (λ))2 = (A2+A′2)(B2+B′2)+ AA′(B2 B′2)
case. 2
M 4 2 −
ofLEeqt.’s(4t)a,kaenadc—losoerneloioskreamtitnhdeeBdeolflBpoellyl’nso1m96ia6lcMritniq(uλe) +1BB′(A2 A′2), (16)
2 −
[9]onvonNeumann’s‘no-gotheorem’—itthenbecomes
apparent that the definition of (λ) uses a suspicious where the dependency ofA, A’, B, andB’on λ has been
n
additivity of incompatible obseMrvables, since it involves omitted for clarity. However, if we expand ( ˆ2)2 (and
different local setting (k =k′). Because of this the Bell using the local anticommutativity) we get M
i 6 i
polynomial of Eq. (4) can not be considered to be an
1
observable that local realism determines, and is never ( ˆ )2 = (Aˆ2+Aˆ′2) (Bˆ2+Bˆ′2) (AˆAˆ′ BˆBˆ′)
2
measured as such, since, as Bell has taught us, ”a mea- M 4 ⊗ − ⊗
1
surement of a sum of noncommuting observables cannot = (Aˆ2+Aˆ′2) (Bˆ2+Bˆ′2)+(Aˆ′′ Bˆ′′),
bemadebycombiningtriviallytheresultsofseparateob- 4 ⊗ ⊗
(17)
servations on the two terms – it requires a quite distinct
experiment. [...] Butthisexplanationofthenonadditiv-
with Aˆ′′ = [Aˆ,Aˆ′]/2i and Bˆ′′ = [Bˆ,Bˆ′]/2i some self ad-
ity of allowed values also established the nontriviality of
jointoperators(witheigenvalues 1)thatcanbethought
the additivity of expectation values. The latter is quite ±
to correspond to some well defined observables. We in-
a peculiar property of quantum mechanical states, not
deed see that Eq. (17) is structurally different from the
to be expected a priori. There is no reason to demand
localrealisticexpressionEq. (16),andtheycantherefore
it individually of the hypothetical dispersion free states
not be considered to be the counterpart of eachother.
[hidden variable states λ], whose function it is to repro-
The correct local realistic counterpart of ( ˆ )2 is ob-
duce the measurable peculiarities of quantum mechanics M2
tained by translating Eq. (17) directly into
when averaged over. ”[9]. For Bell an expression involv-
ing noncommuting spin observables such as [σ +σ ](λ)
could not be assumed to be equal to σx(λ)+σxy(λ).y M2(λ):= 41(A2+A′2)(B2+B′2)+A′′B′′, (18)
If we apply Bell’s lesson to Chen’s analysis we obtain
′′ ′′
thatmeasurementoftheBellpolynomial (λ)cannot with A and B some dichotomic 1 valued observables
n
be made by combining trivially the resuMlts of noncom- thatarethelocalrealisticcounterp±artofrespectivelyAˆ′′
muting observables. The hidden variables λ only deter- andBˆ′′. (The dependency of the righthand side quanti-
mine the outcomes A (k ,λ) of individual measurements titiesonλhasagainbeenomittedforclarity). Thefunc-
i i
(with settings k ) and not the outcomes of measurement tionalM (λ) (and not ( (λ))2) is the Bell-polynomial
i 2 2
M
of the quantity (λ) since the latter involves incom- that,whenaveragedoverandusinglinearityofthemean,
n
M
patible observables. The only function of the Bell poly- givesthe Bell-typeinequalitywhichisthecounterpartof
nomial (λ) is to allow for a shorthand notation of the quantum mechanical inequality using ( ˆ )2.
n 2
M M
the Bell-type inequalities. Indeed, when averaged over Secondly, when averagedover ( (λ))2 does not give
n
M
λ it gives the inequality Eq. (5), which by using linear- a shorthand notation for a Bell-type inequality or any
ity of the mean can be rewritten as a sum of expecta- otherconstraintwhichlocalrealismmustobey. Invoking
tion values in a legitimate localrealistic form, namely as linearity of the mean does not help here.
4
The reason for this is that we end up with the above son why the operators ˆ and ˆ2 that Chen uses can
Mn Mn
mentioned problem that Bell pointed out: we get ex- be considered to be proper quantum mechanical observ-
pectation values of the products of noncommuting (in- ables.)
compatible) observables which cannot be determined by Thus the so called Tsirelson inequality expressed as
combining measurements of the individual observables.
AˆBˆ + AˆBˆ′ + Aˆ′Bˆ Aˆ′Bˆ′ 2√2 (19)
To see this we expand n(λ))2 in ( n(λ))2 LHV, as |h iρ h iρ h iρ−h iρ|≤
M h M i
was done in Eq. (16) for n = 2. We then get terms can thus equally well be expressed in a shorthand nota-
vliokleveh[.i.n.cAomi(kpia,tλib)Alei(ekxi′p,λer)i.m.e.]nitLsHVth,awthcicohrrfeosrpoknid6=tkoi′ ilno-- tioHnoawse|vheBˆri,ρa|s≤n2o√te2d,wbyithBeBˆll=anAˆdBˆa+s cAˆitBˆed′+bAeˆf′oBˆre−inAˆ′tBhˆi′s.
cally anticommuting operators in quantum mechanics. note, this is additivity of expectation values is “a quite
Indeed, it is precisely this local noncommutativity, i.e., peculiar property of QM, not to be expected a priori”
Aˆi(ki)Aˆi(ki′) = −Aˆi(ki′)Aˆi(ki), that has no counterpart for the hidden variable states λ. Thus for Bell-type in-
for local realistic observables, which is responsible for equalitiessuchashorthandnotationinvolvestakingcare
the structural differences in Eq. (16) and Eq. (17) and of some crucial subtleties. I will now discuss three such
whichChenusestogettheexponentiallydivergingresult subtleties that should be taken into account in deriving
that h(Mn(λ))2iLHV ≤ 1 whereas hMˆ2niρ ≤ 2(n−1 (with Bell-type inequalities, and then relate them to the dis-
the latter tight for separable quantum states). But we cussion of the previous section.
have seen that for local realism these expectation values (I) Firstly, the local realistic Bell polynomials are not
ofthe products ofnoncommuting(incompatible)observ- to be regarded as observables. The danger is that be-
ables areproblematic forprecise the samereasonas why cause the operator identity in quantum mechanics ˆ =
additivity of expectation values is problematic for the AˆBˆ+AˆBˆ′+Aˆ′Bˆ Aˆ′Bˆ′ doesindeeddefineanewobsBerv-
sum of noncommuting observables: measurement of the able ˆ, one is tem−pted to formulate the hidden variable
product of noncommuting observables requires a quite countBerpart as:
distinct experiment from the experiments used to mea-
′ ′ ′ ′
(λ)=A(λ)B(λ)+A(λ)B (λ)+A(λ)B(λ) A (λ)B (λ).
sure the individual terms in the product.
B −
(20)
Thus Bell’s critique on von Neumann’s no-go theorem
However, if (λ) is not regarded as merely a shorthand
equally well applies here too: ”[...] the formal proof B
notation for the sum of the four terms in Eq. (20), but
doesnotjustifyhisinformalconclusion”[9],i.e.,although
is supposed to be the counterpart of the observable ˆ,
Chen’s proofismathematicallycorrectandassuchisin- B
Eq. (20) involves the problematic additivity of eigenval-
teresting,hisconclusionisneverthelesswantingsincethe
ues,whichcannotbe demandedoflocalrealism. Indeed,
local hidden variable theorist would not be enforced to
fourdifferentnon-compatiblesetupsareinvolvedandnot
regardtheassumptionEq. (14)asthelegitimitecounter-
just one, which the notation (λ) when considered as a
part of Eq. (15), nor to regard ( n(λ))2 as a legitimite B
M single observable could suggest.
shorthand notation for any sort of Bell-type inequality.
Thus, when deriving local hidden variable observables
Chen’s analysis thus breaks down.
that depend on the hidden variable λ, only compatible
It is thus not the strength of correlations or fluctua-
experimental setups must be considered. The difficulty
tions in separable states which ruled out local realism,
ofmeasuringincompatibleobservablesthushastobeex-
but ”[i]t was the arbitrary assumption of a particular
plicitly takeninto accountin the hidden variableexpres-
(and impossible) relation between he results of incom-
sion. This is different from quantum mechanics where
patible measurements either of which might be made on
the incompatibility structure already is captured in the
a given occasion but only one of which can in fact be
(non-)commutativitystructureoftheoperatorsthatcor-
made.”[9]
respond to the observables in question.
(II)Secondly,whenusingashorthandnotationitmust
Repercussions for modern research be possible (by for example using linearity of the mean)
on Bell-type inequalities totranslatetheshorthandnotationintoalegitimateBell-
typeinequalitywhichalocalrealistcannotbutacceptor
InmodernresearchonBell-typeinequalities(see[1,5,7, into any other legitimate local realistic constraint. As
10],howevercf. [11])oneoftenconsidersrecursivedefini- an example, note that we have seen that (λ) did al-
n
M
tionsandshorthandnotationsintermsofBellpolynomi- low for formulating a legitimite Bell inequality whereas
als [7] (e.g., see Eq. (4)) and their quantum mechanical ( (λ))2 did not.
n
M
counterparts,the socalledBelloperators. The latterare (III) Thirdly, suppose one would indeed regard the
particular linear combinations of operators that corre- functionals (λ) and (λ) to be the quantities of in-
n
M B
spondtoproductsoflocalobservables. Examplesofsuch terestandregardthemasobservables. Thefirstsubtlety
Bell-operatorsare ˆ in the multi-partite setting or for mentionedaboveshowsthatthisisunproblematiconlyif
n
example ˆ=AˆBˆ+MAˆBˆ′+Aˆ′Bˆ Aˆ′Bˆ′ for the bi-partite they are thought of as being genuine irreducible observ-
setting. IBn quantum mechanics−these operators ˆ and ables and not to be composed out of other incompatible
n
ˆ can be considered to be observables themselvMes since observables. Then to be fair to local realism from the
B
a sum of self-adjoint operators is again self-adjoint and startthe possiblevaluesofmeasurementoftheseobserv-
every self-adjoint operator is supposed to correspond to ables (λ) and (λ) in the local realist model should
n
M B
an observable. Furthermore, the additivity of operators thenbeequaltotheeigenvaluesofthequantummechan-
gives additivity of expectation values. (This is the rea- ical counterparts ˆ and ˆ. And these eigenvalues are
n
M B
5
2(n−1)/2, 2(n−1)/2, 0 and 2√2, 2√2, 0 respec- Indeed, on a different reading one can think that the re-
{ − } { − }
tively. The possible outcomes for the local realist quan- lation of Eq. (4) should be treated as defining a local
tities should equal these eigenvalues, and then no viola- realistic observable itself, since it is of this quantity that
tioncanbeseentooccur. Indeed,predictionsforasingle Chenconsidersthevarianceandtheexpectationvalueof
observation can always be mimicked by a local realistic its square. However, to treat Eq. (4) as if it was defin-
model. Furthermore, if we now go back to the problem- ing anobservable makesno difference atallfor quantum
atic operator ˆ2 that Chen considered, we see that it mechanics, but as we have seen, for local realism this
haseigenvalueMs n2(n−1), 0 ,(seefootnote[8]),whereasit makes a crucial difference. A local realist would either
{ }
was assumed that its local realistic counterpart has out- encounter the problem mentioned in the first subtlety
comes in [ 1,1], and can thus by construction, and not (andwouldthenfacetheBell-typecritique),ortheprob-
−
because of local realism, never reproduce the quantum lemmentionedinthe thirdsubtlety(andwouldthennot
outcomes. give any interesting results, i.e., local realism can then
Using these three subtleties we cannowunderstandin trivially reproduce the quantum predictions).
a different way where Chen’s analysis has gone astray:
(i) If we think of Eq. (14) as a mere shorthand notation We thus see that on both readings Chen’s analysis
of a complex summation, then the secondsubtlety is the breaks down.
majorproblem. Onthisreadingthefunctional( (λ))2
n
M
cannotbetakentogiveaconstraint(i.e.,Eq. (14))which Acknowledgements
isashorthandnotationofalegitimateBell-typeinequal-
ity. (ii)However,onecanarguethattheanalysisofChen
does not treat the Bell polynomial (λ) in Eq. (4) as IthankJosUffinkandDennisDieksforveryhelpfulcom-
n
M
merely a shorthand notation for a complex summation. ments.
[1] Z. Chen, ‘Variants of Bell inequalities’, per site’, Phys.Rev. A 64, 032112 (2002).
quant-ph/0611126 (2006). [8] Theoperator ˆ hasthespectraldecomposition ˆ =
n n
[2] Mina.lZ˙uBkeollws(k1i9,6‘4S)epIanreaqbuilaitlyitioefs’Q, uFaonutnudm. SPthaytess. v3s6.,Or5i4g1- e2i(gne−n1v)/a2lu(|eGsHarZMen+(iih)GthHeZn+va|l−ue|sGH2Z(nn−−i1h)/G2HwZin−th|),na-pnMadrttihtee
(2006). GHZ-states GHZn± =1/√2( 0±n 1n )as eigenstates
[3] Notethatinordertodistinghuishbetweenthelocalreal- and (ii) furth|ermoreithevalue|0 wii±th|degieneracy n 2.
istic observables (i.e., measurement functionals) and the Theoperator ˆ2 hasthespectraldecomposition ˆ−2 =
quantum mechanical operators, I denote the quantum 2(n−1)( 0n 0Mn n 1n 1n ),andeigenvaluesareM(i)nthe
mechanical operators that correspond to observables in value 2|(n−i1)h (d|e−ge|neraichy 2|) with separable eigenstates
the local realistic expression by a ‘hat’. This difference 0n and 1n and (ii) furthermore the value 0 with de-
will become crucial, which will become clear later on. | i | i
generacy n 2.
[4] Consider a bipartite scenario. For local realism −
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[6] See for example the paper by Chen [1] for the explicit values as was done in Eq. (1). The notation might per-
form of this inequality and the choice of coefficients haps not be so elegant, but from a conceptual point of
c(k ,k ,...,k ) heuses. However, theexplicit forms are view it is to be preferred since the subtleties and prob-
1 2 n
not relevant here. lems as discussed in this paperare explicitly avoided.
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