How much measurement independence is needed in order to demonstrate nonlocality?

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by  Jonathan Barrett

How much measurement independence is needed in order to demonstrate nonlocality?
Jonathan Barrett
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
Nicolas Gisin
Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
(Dated: January 28, 2011)
IfnonlocalityistobeinferredfromaviolationofBell’sinequality,animportantassumptionisthat
themeasurementsettingsarefreelychosenbytheobservers,oralternatively,thattheyarerandom
and uncorrelated with the hypothetical local variables. We study the case where this assumption
is weakened, so that measurement settings and local variables are at least partially correlated. As
1 we show, there is a connection between this type of model and models which reproduce nonlocal
1 correlations by allowing classical communication between the distant parties, and a connection
0 withmodelsthatexploitthedetectionloophole. WeshowthatevenifBob’schoicesarecompletely
2 independent,allcorrelationsobtainedfromprojectivemeasurementsonasingletcanbereproduced,
with the correlation (measured by mutual information) between Alice’s choice and local variables
n
a less than or equal to a single bit.
J
PACSnumbers:
7
2

Quantum nonlocality, whereby particles appear to in- measurement settings are not random, but are in fact
]
h fluenceoneanotherinstantaneouslyeventhoughtheyare determined by the local variables, then arbitrary corre-
p widely separated in space, is one of the most remark- lations can be reproduced [11].
- able phenomena known to modern science [1–4]. Histor-
t Herewereversetheargument. Takingforgrantedthat
n ically, this peculiar prediction of quantum theory trig-
quantum nonlocal correlations are produced, how inde-
a gered many debates and even doubts about its validity.
u pendent must the measurement settings be assumed to
Today, it is a well established experimental fact [5].
q be in order to rule out an explanation in terms of lo-
[ The profound implications of quantum nonlocality for cal variables [6]? We show that all correlations obtained

our world view remain controversial. But it is no longer from projective measurements on a singlet state can be
2
v considered as suspicious, or of marginal interest. It is reproduced with the mutual information between local
2 central to our understanding of quantum physics, and variables and the measurement setting on one side less
1 in particular it is essential for the powerful applications than or equal to one bit. This is not only of fundamen-
6 of quantum information technologies. In 1991, A. Ek- tal interest. If a cryptographic protocol is relying on the
3 ert showed how shared entanglement could enable dis- nonlocality of correlations for security, it is vital that
.
8 tantpartnerstoestablishasharedcryptographickey[8]. there is no underlying local mechanism that an eaves-
0 Ekert’s intuition is quite simple – if there are no local dropper may be exploiting. The result shows that ran-
0 variables, then no adversary could possibly hold a copy dom number generators used to determine settings must
1
of these variables – yet it came decades after the birth be assumed to have a very high degree of independence
:
v of quantum mechanics. In 2005, Barrett and co-workers from the particle source.
Xi usedthisintuitiontoshowhow,withnofurtherassump- Bell Experiments. In an experimental test of a Bell
tions about quantum theory, nonlocal correlations alone inequality, two experimenters – traditionally named Al-
r
a could ensure security of a secret key [9]. Ac´ın and co- ice and Bob – are spatially separated. They each control
workers extended this result, showing how to generate a quantum system, and the joint state of these two sys-
secretkeysusingsimplenonlocalcorrelationsthatviolate tems is an entangled state. Alice and Bob each perform
the well-known CHSH inequality [10]. Simultaneously, it a measurement on their quantum system, in such a way
was realized that Bell inequalities are the only entangle- that the measurements are spacelike separated. Alice’s
ment witnesses that can be trusted in cases where the measurementischosenfromafinitesetofpossibilities,as
dimensions of the relevant Hilbert spaces are unknown isBob’s. Callthesemeasurementsettingstheinputsand
[10]. label them x for Alice’s choice and y for Bob’s choice.
The outcome of Alice’s measurement (her output) is la-
In an experimental demonstration of quantum nonlo-
beled a and Bob’s b. By repeating this procedure many
cality, measurements are performed on separated quan-
timesandcollatingthedata,AliceandBobcanestimate
tum systems in an entangled state, and it is shown that
conditional probabilities of the form P(a,b|x,y).
the measurement outcomes are correlated in a manner
that cannot be accounted for by local variables. In order The question is then, when can the correlations pro-
to conclude that nonlocality is exhibited, it is crucial for duced in an experiment like this be explained by under-
the analysis that the choices of which measurement to lying local variables? Locality here implies that the out-
perform are freely made by the experimenters. Alterna- come of Alice’s measurement cannot be directly influ-
tively, they must be random and uncorrelated with the encedbyBob’schoiceofwhichmeasurementtoperform,
hypothetical local variables. It is well known that if the and vice versa. More precisely, let the hypothetical un-
Note the similarity with Eq. (2). The usual kind of
model, in which Alice’s and Bob’s inputs are assumed
independent of the local variables, is a special case with
? ? PCS(x,y|λ) = PCS(x,y). By Bayes’ rule, this also gives
P (λ|x,y) = P (λ), and Eq. (3) reduces to an equa-
CS CS
tion of the form of Eq. (2).
Another special case is the extreme case where the in-
x = 0 x = 1 y = 0 y= 1
puts are completely determined by λ, so that x = f(λ)
and y = g(λ) for functions f and g. Here, for all x,y,λ,
01 01 PCS(x,y|λ) = 0 or 1. Any correlations can be produced
Source of entangled by a Bell-local model of this form [11].
pairs of particles
Alice Bob In other models, x and y will be neither independent
nor determined by λ. In general there are many numeri-
cal measures of the degree to which they are correlated.
One particularly natural and simple measure is the mu-
tual information. Thus a correlated-settings model can
be characterized by the mutual information between the
FIG. 1: Schematic illustration of an experiment for testing
measurement choices x, y and λ:
quantum nonlocality.
I(x,y :λ)=H(x,y)+H(λ)−H(x,y,λ), (4)
derlying variables be denoted λ, and assume a distribu-
where H is the Shannon entropy. When x and y are
tion P(λ). Then Bell locality is the condition that
independentofλ,I(x,y :λ)=0. Ifxandyarefunctions
of λ on the other hand, then I(x,y :λ)=H(x,y). Since
P(a,b|x,y,λ)=P(a|x,λ)·P(b|y,λ). (1)
in general I(x,y : λ) ≤ H(x,y), this means that the
mutual information is maximal (with respect to a fixed
Ifthecorrelationscouldinprinciplebeexplainedasaris-
distribution over inputs P(x,y)).
ing from underlying local variables, then they can be
In the context of correlated-settings models, the ques-
written in the form
tion is no longer whether quantum correlations violate
(cid:88) a Bell inequality. The question is, how large must
P(a,b|x,y)= P(λ)·P(a|x,λ)·P(b|y,λ). (2)
I(x,y : λ) be if the correlations are to be reproduced
λ
within a Bell-local model? Alternatively, for a fixed de-
(Hereandthroughout,sumsshouldbereplacedwithinte- greeofcorrelationI(x,y :λ),doquantumcorrelationsvi-
gralswhenthevariableisnotdiscrete.) Thecorrelations olate a Bell inequality by a sufficiently large margin that
cannot be written in this form if and only if they violate the model cannot possibly be Bell-local? This depends
a Bell inequality. on the precise experiment. For example, if the input al-
Suppose that correlations are obtained which do vio- phabet consists of only two choices, as with the CHSH
late a Bell inequality. What assumptions are necessary inequality, then any correlations can be reproduced by a
to conclude that this is indicative of nonlocality? There correlated-settings model with I(x,y : λ) ≤ 1. But the
is a large literature on this topic, and a reader could do question remains open for larger input alphabets. Intu-
a lot worse than refer to Bell’s original papers [1]. But it itively one may think that for large alphabets, if Alice’s
is uncontroversial that the standard argument needs to choice is only mildly correlated with the variable λ, then
assume that the inputs x,y are freely, or at least inde- a sufficient violation of a suitable Bell inequality should
pendently and randomly chosen by Alice and Bob. still rule out locality. But we shall see that this intuition
Non-independent measurement choices. We is wrong, at least for maximally entangled qubits.
wish to analyze the case in which measurement settings A connection to communication cost. One way
can be correlated with local variables. Hence in addition tosimulatequantumcorrelations,includingnonlocalcor-
to a set of correlations P(a,b|x,y), assume a distribu- relations, is for Alice and Bob to communicate with one
tion over inputs P(x,y). A correlated-settings model is another after inputs are chosen, but before outputs are
defined by a variable λ, and an overall joint probabil- produced. Any correlations can be produced this way if
itydistributionP (a,b,x,y,λ). Thecorrelated-settings Alice and Bob share random data, and Alice simply in-
CS
model reproduces the given correlations and input dis- formsBobofherinputchoice. Hence,ifAliceischoosing
tribution if P (a,b|x,y)=P(a,b|x,y) and P (x,y)= from n possible inputs, then an obvious upper bound
CS CS A
P(x,y). The model is Bell-local if the distribution P on how many classical bits need to be communicated is
CS
also satisfies an equation of the form of Eq (1). In this givenbylogn . Theminimumnumberofbitsthatmust
A
case be communicated, on the other hand, provides a natural
way of quantifying the amount of nonlocality inherent in
(cid:88)
PCS(a,b|x,y)= PCS(λ|x,y)·PCS(a,b|x,y,λ) a given set of correlations. This interesting line of re-
λ search was started by T. Maudlin [2] and independently
= (cid:88)P (λ|x,y)·P (a|x,λ)·P (b|y,λ). (3) byTappetal. andbySteiner[12,13]. Itculminatedwith
CS CS CS
a model of Toner and Bacon, who proved that one single
λ
2
bit of communication from Alice to Bob is sufficient to = 0+H(m|µ)−H(m|µ,x,y) (6)
simulate all correlations obtained from arbitrary projec- = H(m|µ) (7)
tive (i.e. Von Neumann) measurements on two qubits in
≤ H(m). (8)
a maximally entangled state [14]. It was extended to all
two-partycorrelations,thoughignoringthemarginals,in
The first line uses the chain rule [16], the second line
[15].
followsfromthefactthatµisassumedtobeindependent
Aslightlymoreprecisedescriptionofacommunication
of the inputs x,y, and the third line holds because m is
model for simulating quantum correlations is as follows.
a function of µ and x,y. This concludes the proof.
The random data shared between Alice and Bob is a
Intuitively, λ = (µ,m) restricts Alice’s and Bob’s
variableµ,withdistributionP (µ). AssumingthatAlice
C choices to inputs (x,y) such that m = f(x,y,µ). We
communicates first, Alice sends a bit string c to Bob.
1 stress the generality of this result: the communication
Thestringc candependonµandonAlice’sinputx. It
1 model may have involved two way communication, and
may also depend on further random data held by Alice,
may have been such that the communication on a given
but this data can without loss of generality be absorbed
run is unbounded. As long as H(m) is finite, the bound
into µ. Hence assume that c = c (µ,x). Depending
1 1 is useful.
on the protocol Bob may now send a bit string c to
2 WenowapplythetheoremwithreferencetotheToner-
Alice, where again without loss of generality, we assume
Baconcommunicationmodel. Here,xandyarearbitrary
that c is a function of µ, c and y. This continues for
2 1 projective measurements. The shared random data µ
a total of k messages. The entire conversation between
takes the form of two vectors on the Bloch sphere and
AliceandBobonaparticularroundisthesequencem=
the distribution P(µ) is such that the two vectors are
c ,c ,...,c , and is a function m = f(x,y,µ) of x, y,
1 2 k independent, with each uniformly distributed. For our
and µ. Finally, Alice outputs a, where again absorbing
purposes most of the details of the model do not matter,
allrandomnessintoµ,acanbeassumedtobeafunction
andwereferthereaderto[14]foradescriptionandproof
a=g(x,µ,m) of her input, the shared random data and
that it reproduces quantum correlations for projective
the conversation. Similarly, Bob outputs b, which is a
measurements on a singlet. It suffices to know that each
function b=h(y,µ,m). The protocol then terminates.
round, the conversation m is a single bit communicated
For each pair of inputs, a communication model de-
from Alice to Bob. The bit m = f(x,µ) is a function of
fines a distribution P (a,b,m,µ|x,y), and reproduces
C Alice’s input x and µ.
correlations P(a,b|x,y) if P (a,b|x,y) = P(a,b|x,y).
C The resulting correlated-settings model also repro-
If a distribution over inputs P(x,y) is also given, the
ducesallcorrelationsfromprojectivemeasurementsona
conversation m has a well-defined distribution P(m) =
(cid:80) singlet. Ingeneral,theactualvalueofI(x,y :λ)depends
P (m|x,y)·P(x,y). Let the entropy of this distri-
x,y C onthedistributionoverinputsP(x,y). Thetheoremtells
bution be H(m).
usthatforanysuchdistribution,I(x,y :λ)≤H(m)≤1,
The connection with correlated-settings models is ex-
since m is a single bit. In particular, if Alice’s and Bob’s
pressed in the following
inputsareindependent,witheachchosenfromauniform
distributionoverallpossibledirections,itiseasytoverify
Theorem 1 Consider a fixed input distribution P(x,y),
that I(x,y :λ)≈0.85.
and a communication model with total conversation m.
Then there exists a Bell-local correlated-settings model, In a typical Bell-type experiment with a singlet, Al-
which reproduces the same correlations and P(x,y), with ice and Bob will be choosing from finite sets of measure-
I(x,y :λ)≤H(m). ments. TheToner-Baconmodelcanstillbeapplied,with
the distribution P(x,y) having support only on these fi-
Proof. Given P(x,y) and a communication model as nite sets. In this case too, I(x,y :λ)≤1. The Bell-local
above, construct a correlated-settings model as follows. correlated-settingsmodelreproducesthequantumcorre-
Define λ as the pair λ = (µ,m). In the communication lations no matter how large the input alphabets, and no
model, m is the conversation between Alice and Bob, matter what Bell inequality is being tested.
but in the correlated-settings model there is no commu- Finally, in the Toner-Bacon model, Bob’s setting is of
nication, and λ represents the underlying variable. De- course independent from the communication he receives
fine the correlated-settings model by PCS(a,b,x,y,λ) = from Alice. It follows that in the derived correlated-
P(x,y)PC(a,b,µ,m|x,y). Itiseasytoseethattheresult- settings model, Bob’s setting is independent from λ, i.e.,
ing model reproduces P(x,y) and the same correlations I(y :λ)=0.
as the communication model. Bell locality is satisfied The detection loophole. In a real Bell experiment,
since detection is inefficient. A typical analysis of the exper-
iment estimates the correlations P(a,b|x,y) from those
PCS(a,b|x,y,λ) = PC(a,b|x,y,µ,m) runs on which both Alice’s and Bob’s detectors clicked,
= P (a|x,µ,m)·P (b|y,µ,m) and simply ignores all the other runs. Such an analysis
C C
is valid, as long as it is assumed that the probability of
= P (a|x,λ)·P (b|y,λ).
CS CS
a detector clicking is independent of the hypothetical lo-
The mutual information between λ and (x,y) is cal variables. This is sometimes called the fair sampling
assumption.
I(λ:x,y) = I(µ:x,y)+I(m:x,y|µ) (5) It is possible to reproduce nonlocal correlations with
3
a model that is Bell-local, but which violates the amount of communication (as measured by the entropy
fair sampling assumption [17, 18]. Denote the event of the conversation’s distribution). With detection ef-
that Alice’s (Bob’s) detector clicks by D (D ). A ficiency models, we expect there to be a connection be-
A B
detection-efficiency model is defined by a variable λ, tweenI(x,y :λ)andthedetectionefficiencies. Exploring
with distribution P (λ), independent of x,y, and for thisconnectionisaninterestingdirectionforfuturework.
DE
each x,y a distribution P (a,b,D ,D |x,y,λ). Bell-
DE A B Conclusion. It is well known that in order to derive
localityistheconditionthatP (a,b,D ,D |x,y,λ)=
DE A B nonlocality from violation of a Bell inequality, one has
P (a,D |x,λ)·P (b,D |y,λ). Themodelreproduces
DE A DE B to assume that there is no correlation between the hypo-
correlations P(a,b|x,y) if
theticallocalvariablesλandtheexperimenters’measure-
ment choices x and y. We have investigated how much
P(a,b|x,y)=P (a,b|x,y,D ,D ). (9)
DE A B correlation could be allowed if quantum predictions are
toremainincompatiblewithlocalvariables. Surprisingly,
The efficiencies of Alice’s detectors in this model are
(cid:80) no matter how large the alphabet size of the inputs, cor-
given by P (D |x) = P (λ)P (D |x,λ), and
DE A λ DE DE A relations from projective measurements on a singlet can
similarly for Bob’s. If the detection efficiencies in a real
bereproduced,withthemutualinformationbetweenAl-
experiment are high enough, and if a Bell inequality
ice’sinputandlocalvariablesnotmorethanonebit,and
is violated by a large enough margin, then a Bell-local
Bob’s input completely independent. This deserves fur-
detection-efficiency model reproducing the correlations
therinvestigation. Futureworkshouldanalyzemoregen-
can be ruled out.
eral scenarios, involving generalized measurements, par-
GivenadistributionP(x,y),andadetection-efficiency
tialentanglement[20],higherdimensionalHilbertspaces
model reproducing correlations P(a,b|x,y), it is easy to
and more than two parties. It would also be instructive
construct a correlated-settings model which also repro-
toconsidermeasuresofcorrelationotherthanthemutual
duces P(x,y) and P(a,b|x,y). Simply define
information.
PCS(a,b,x,y,λ)=P(x,y)·PDE(a,b,λ|x,y,DA,DB). Finally, letusemphasizethechangeinparadigmsince
(10) the old EPR paper [4]. If, contrary to EPR, one accepts
It is easy to show, using Eq. (9), that the correlated- nonlocalityasafact,thennotonlycanonedeveloppow-
settings model reproduces P(x,y) and P(a,b|x,y). erful applications in quantum information science, like
The construction can be applied, for example, to the device-independentquantumkeydistribution[9,10],but
Gisin-&-Gisin detection-efficiency model [19], which re- moreover one can upper bound the lack of free choice of
producescorrelationsfromarbitraryprojectivemeasure- the players !
ments on a singlet. The efficiencies are independent of
x and y and are given by P(D ) = 1/2, P(D ) = 1.
A B
In this model λ is a vector on the Bloch sphere, and
P (λ|x,y,D ,D ) = |λ.(cid:126)x|/2π where (cid:126)x denotes the
DE A B
Bloch vector representing the measurement x. In the Acknowledgment
resulting correlated-settings model, I(x,y : λ) depends
on P(x,y). If x and y are independently and uniformly
JB is supported by an EPSRC Career Acceleration Fel-
distributed, it is easy to show that I(x,y : λ) = I(x :
lowship. NG acknowledges support from the ERC advanced
λ)≈0.28. Thisisanevenlowervaluethanwasobtained
grant QORE and the Swiss NCCR-QP. After completion of
above using the Toner-Bacon model. this work M. J. W. Hall posted a related interesting paper:
Finally, note that in the case of communication mod- arXiv:1007.5518 (PRL 105, 250404 (2010)). We thank him
els, there was a connection between I(x,y : λ) and the and C. Branciard for constructive discussions.
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4
[17] P. Pearle, Phys. Rev. D 2, 1418 (1970). munication model [14]; surprisingly, however, it is still
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[20] Fortwopartiallyentangledqubits,thereisa2bitscom-
5