Locality, detection efficiencies, and probability polytopes
aa r X i v : . [ qu a n t - ph ] A ug Locality, detection efficiencies, and probability polytopes
J. Wilms , , Y. Disser , G. Alber , I. C. Percival , Fakult¨at f¨ur Physik, Universit¨at Wien, 1090 Wien, Austria Institut f¨ur Angewandte Physik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany Department of Physics, Queen Mary, University of London, London E1 4NS, United Kingdom Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom (Dated: October 24, 2018)We present a detailed investigation of minimum detection efficiencies, below which locality can-not be violated by any quantum system of any dimension in bipartite Bell experiments. Lowerbounds on these minimum detection efficiencies are determined with the help of linear programmingtechniques. Our approach is based on the observation that any possible bipartite quantum corre-lation originating from a quantum state in an arbitrary dimensional Hilbert space is sandwichedbetween two probability polytopes, namely the local (Bell) polytope and a corresponding nonlocalno-signaling polytope. Numerical results are presented demonstrating the dependence of these lowerbounds on the numbers of inputs and outputs of the bipartite physical system.
PACS numbers: 03.65.Ud,42.50.Xa
I. INTRODUCTION
Despite numerous recent experimental tests for viola-tions of locality by quantum theory, such as the experi-ment by Weihs et al. [1], we still do not know for certainwhether or not the laws of physics are entirely local [2].This is because so far no single experiment has closedboth the locality and the detection loophole simultane-ously.In general, classical correlations between two spacelikeseparated experimenters, Alice (A) and Bob (B), obeylocality constraints, which can be expressed in terms of(generalized) Bell inequalities. According to Bell’s theo-rem [3], these inequalities can be violated if the relevantcorrelations are produced by ideal measurements of anentangled quantum system, whose quantum state is re-quired to originate in the common backward light coneof A and B. This is weak nonlocality. Strong nonlocal-ity would be the non-existent correlations produced bysignaling faster than the speed of light.In a typical two-photon Bell experiment the polariza-tion state of a pair of entangled photons is measuredindependently by A and B. Preferred states for such ex-periments are pure two-photon states of maximum en-tanglement, the so-called Bell states.One of the last remaining major problems on the wayto a loophole-free test of locality is the detection loophole,which comes from photon detection efficiencies being toosmall [4, 5]. In this context Eberhard [6] recognized thatweakly entangled pure two-photon states yield a maxi-mized tolerance to detection inefficiency. Using numeri-cal optimization he was thus able to reduce the criticaldetection efficiency in the two-photon Bell experiment to-wards a theoretical limit of 2/3 under the assumption ofidentical detection efficiencies for A and B. Later authors[4, 5] have applied his method to other experiments of theBell type and achieved theoretical values of the criticaldetection efficiency as low as 0 .
43 for the extreme asym-metric case in which only either A’s or B’s detection is perfect.In view of these results on minimum detection efficien-cies two major questions arise. Firstly, it is not clearwhether these results also apply to Bell experiments inwhich the dichotomic variables measured by A and B re-sult from quantum observables and quantum states inarbitrarily high dimensional Hilbert spaces. Secondly, itis unclear what influence symmetric and asymmetric de-tection efficiencies have on cases in which more than twophysical quantities are measured on A’s and B’s sides oron cases in which the observables have more than twopossible outcomes. It is the main purpose of this paperto address these open questions.For this purpose an efficient way is developed for de-scribing local bipartite correlations with the help of prob-ability polytopes and linear programming. It is knownthat the relevant local polytopes can be described ef-ficiently in terms of their vertexes, which can be ob-tained for any experimental setup of any number of in-puts and outputs, as described here in Sec. II. Thus, anytest of locality reduces to an inclusion test determiningwhether a given set of probabilities is located outside orinside the relevant local polytope. In addition, with thehelp of a second class of probability polytopes which de-scribe nonlocal no-signaling correlations [7] it is possibleto obtain lower bounds on minimum detection efficien-cies for bipartite Bell experiments. These latter proba-bility polytopes include all correlations of bipartite quan-tum systems of any dimension and thus yield dimension-independent lower bounds on detection efficiencies. Firstresults of such lower bounds are presented for inefficien-cies of arbitrary symmetry and for bipartite locality testswith dichotomic variables which involve random choicesof A’s and B’s observables from a set of up to four ele-ments.In addition, some new results for lower bounds onhigher numbers of outputs are presented. Finally, it isdemonstrated that the 1-norm (used here as the distance)between the point defined by the observed probabilitiesand the relevant local polytope represents a convenientway of quantifying violations of locality in the presence ofexperimental uncertainties. This distance can be deter-mined in a straightforward way by linear programming.This paper is organized as follows: Sec. II summarizesrelevant and already known results on classical correla-tions, classical transfer functions, and their relation toprobability polytopes. The local polytopes and nonlocalno-signaling polytopes are introduced. These describeclassical local correlations, and classical nonlocal corre-lations which fulfill the no-signaling condition, respec-tively. It is shown how these polytopes can be repre-sented in terms of their vertexes or equivalently in termsof inequalities for their facets.In Sec. III these two types of polytope are used to de-termine lower bounds on minimum detection efficiencieswhich still allow for a violation of locality by quantumsystems. Sec. IV finally demonstrates how the 1-normdefining the distance of a given probability distributionfrom the relevant local polytope can be determined withthe help of linear programming.
II. CLASSICAL CORRELATIONS ANDPROBABILITY POLYTOPES
In this section basic concepts involved in the descrip-tion of classical bipartite correlations are summarized.For this purpose transfer functions and probability poly-topes are introduced [8, 9, 10]. In particular, the localBell polytope L and the nonlocal no-signaling polytope P are discussed in detail. A. Transfer functions and transition probabilities
Given a classical deterministic system with discrete in-puts x and outputs a , the output is a definite function F of the input: a = F ( x ). Thereby the transfer function ofthe system, F , specifies a single transition from x to a forevery input x . If the system may be stochastic, then thebehavior of the system has to be described in terms ofthe transition probabilities P ( a | x ), which define a pointin a transition probability space whose coordinates arethese probabilities. Since, for a given input, the totalprobability of an output must be unity, the probabilitiessatisfy the normalization condition P a P ( a | x ) = 1. Fora deterministic system with transfer function F the prob-abilities are P ( a | x ) = δ ( a, F ( x )), with possible values 0or 1. In terms of these particular probabilities an arbi-trary transition probability of a stochastic system can berepresented by [9] P ( a | x ) = X F P ( F ) δ ( a, F ( x )) (1)with P ( F ) denoting the probability with which the de-terministic transfer function F governs the correlationsunder consideration. If there are N ( x ) possible values for the input x and N ( a ) possible values for the output a , there are N ( a ) N ( x ) possible transfer functions, but only N ( x ) × N ( a ) transi-tion probabilities, so usually there are many more trans-fer functions than there are transition probabilities andthe expansion in terms of transfer functions is not gener-ally unique. The sum of Eq.(1) is over all transfer func-tions, but if there are constraints on them, it can be overa subset of F .A typical bipartite Bell experiment testing locality,such as the one described by Eberhard [6], involves atwo-photon source distributing one photon to Alice (A)and the other photon to Bob (B). The subsequent ex-periment performed by A and B may be considered asan input-output system, in which A’s input x is a choiceof angle for the measurement of a photon polarizationand her output a is the result of the measurement, + or − , depending on whether the polarization is found to beparallel or perpendicular to the chosen angle. Similarlyfor B with input y and output b . In the simplest case Aand B each have a choice of two angles only, a differentpair for A and for B. So each of them has 2 inputs and 2outputs, resulting in 4 inputs and 4 outputs for the wholesystem. In generalizations of bipartite Bell experimentsthe number of inputs as well as the number of possibleoutputs of A and B may also be larger. Notice that theoutputs are classical events which result from quantummeasurements. Since the transition probabilities are allprobabilities of these classical events, the analysis of aBell experiment does not depend in any way on quan-tum theory, although the design of such an experimentclearly does.Assuming locality means that for deterministic sys-tems A’s output can only depend on her input, and thesame for B. So a transfer function F for the whole systemis made up of one transfer function for A and one for B: F = ( F A , F B ), where a = F A ( x ) and b = F B ( y ). Thisis the locality constraint on transfer functions, which ingeneral reduces their possible number significantly.Thus if the numbers of possible inputs and outputs ofA are denoted by N ( x ) and N ( a ) and of B by N ( y ) and N ( b ) respectively, the total number of local transfer func-tions is given by N ( a ) N ( x ) × N ( b ) N ( y ) and the numberof corresponding local transition probabilities is given by N ( x ) × N ( a ) × N ( y ) × N ( b ), although the latter are notindependent.So for any classical theory of a Bell experiment, thetransition probabilities must be obtainable from some lo-cal transfer function probabilities using a basic equationof the form P ( ab | xy ) = X F A ,F B P ( F A , F B ) δ ( a, F A ( x )) δ ( b, F B ( y )) . (2) facets verticesinequalities FIG. 1: The two possible representations of a polytope by itsvertexes ( V -representation) and by inequalities characterizinghalf-spaces ( H -representation). B. Probability polytopes and their representations
A convex polytope in a space of dimension D is a gen-eralization of a convex polygon in 2-space or of a convexpolyhedron in 3-space. It can be defined as all thosepoints whose coordinates are a weighted sum of the co-ordinates of its vertexes, with non-negative weights thatsum to unity. So by the fundamental equation (1), if wetreat P ( F ) as weights, the point given by the transitionprobabilities of a system with transfer functions F liesin the probability polytope whose vertexes are definedby these transfer functions. This defines the so calledvertex- or V -representation of the probability polytope.There is an alternative representation of this prob-ability polytope, the so called half-space- or H -representation, in terms of a set of inequalities, each ofwhich defines a half space. A central theorem of poly-tope theory [10] states that any polytope can alwaysbe described either by its V -representation or by its H -representation. Figure 1 illustrates this equivalence of thetwo possible representations schematically in the simplecase of a triangle.
1. Local probability (Bell) polytopes
Let us consider the special case of local classical cor-relations in more detail. These correlations are of cen-tral interest for the analysis of Bell experiments thattest locality. Provided there are N ( x ) and N ( y ) pos-sible values of inputs x and y and N ( a ) and N ( b ) possi-ble values for the outputs a and b of A and B, there are N ( x ) × N ( y ) × N ( a ) × N ( b ) possible local transition prob-abilities and N ( a ) N ( x ) × N ( b ) N ( y ) possible local transferfunctions. Each of these transfer functions defines a ver-tex of the corresponding local probability or Bell poly-tope L , thus yielding its V -representation. Any physicalsystem whose transition probabilities are located outsidethis Bell polytope is nonlocal.Whereas the V -representation of a Bell polytope canbe obtained in a straightforward way from the trans-fer functions, the determination of its corresponding H -representation is a considerably more difficult numeri-cal problem for polytopes with large numbers of vertexes [11]. The half-spaces of the H -representation of a Bellpolytope can be divided into several classes. The least in-teresting class of inequalities expresses the non-negativityconditions of all probabilities involved, i.e. P ( ab | xy ) ≥ ∀ ( abxy ) , (3)and the corresponding normalization conditions, i.e. X ab P ( ab | xy ) = 1 ∀ ( xy ) . (4)A more interesting class of constraints are the no-signaling equalities , X a P ( ab | x y ) = X a P ( ab | x y ) ∀ ( x x yb ) , X b P ( ab | xy ) = X b P ( ab | xy ) ∀ ( xy y a ) , (5)which follow because for a local system no signals maybe sent from A to B or from B to A .The third and most interesting inequalities of the H -representation are the Bell inequalities themselves. Be-cause the transformation from V - to H -representation isdifficult, they are known only in special cases, so thesearch for new families of Bell inequalities is still an ac-tive research area [11, 12, 13, 14].Our subsequent discussion will focus on the V -representation of Bell polytopes, as they can be deter-mined from the relevant transfer functions in a straight-forward way. Furthermore, by taking into accountconstraints on the transition probabilities arising fromconservation of probability and from locality, the V -representation of Bell polytopes can be obtained effi-ciently in a reduced basis. This fact was realized earlieralready by Pitovski [15, 16, 17]. Let us start from theobservation that the full space of local transition proba-bilities is N ( x ) × N ( a ) × N ( y ) × N ( b )-dimensional. Due toconservation of probability these transition probabilitiesfulfill the N ( x ) × N ( y ) relations X a,b P ( ab | xy ) = 1 . (6)Thus, for each choice of input ( x, y ) by A and B one out-put, say ( a N ( a ) , b N ( b ) ), can be eliminated by this lineardependence. Furthermore, if A’s output is equal to a N ( a ) all joint transition probabilities involving this output canbe expressed as P ( a N ( a ) b | xy ) = P B ( b | xy ) − X a = a N ( a ) P ( ab | xy ) , (7)where P B ( b | xy ) = P a P ( ab | xy ) is B’s marginal transi-tion probability. Because of the no-signaling constraints(5) this marginal transition probability cannot dependon A’s choice of input x , i.e. P B ( b | xy ) ≡ P B ( b | y ). Ananalogous argument applies to B, i.e. P ( ab N ( b ) | xy ) = P A ( a | xy ) − X b = b N ( b ) P ( ab | xy ) (8)with P A ( a | xy ) ≡ P A ( a | x ). Thus, the marginal and jointtransition probabilities which do not contain the outputs a N ( a ) or b N ( b ) are linearly independent and span the fullspace of local transition probabilities of a Bell polytope,so they form a basis of dimension D = N ( x ) × ( N ( a ) −
1) + N ( y ) × ( N ( b ) −
1) + N ( x ) × N ( y ) × ( N ( a ) − × ( N ( b ) −
1) (9)for the Bell polytope. As a result the Bell polytope isgiven by all these linearly independent marginal and jointtransition probabilities fulfilling the conditions X a = a N ( a ) P A ( a | x ) ≤ , X b = b N ( b ) P B ( b | y ) ≤ ,P ( ab | xy ) = P A ( a | x ) P B ( b | y ) (10)for a = a N ( a ) , b = b N ( b ) and all possible N ( x ) × N ( y )inputs. Furthermore, the vertexes of the Bell polytopeare given by all those points in this probability spacewhose coordinates assume the values 0 and 1 only andwhich are consistent with relations (10).
2. Nonlocal no-signaling probability polytopes
In an experiment with detection efficiency close to theideal, it would be possible to demonstrate the correlationsof weak nonlocality, but so far there has always been atleast one loophole [18]. Further, Bell’s theorem itself isincomplete, since it is based on the unproved assumptionthat such detection is possible [19, 20].Thus, it is of interest to determine minimum detectionefficiencies which still allow for a violation of locality byquantum theory provided the local measurements of Aand B are separated by a spacelike interval. For the de-termination of these minimum detection efficiencies a de-tailed knowledge of the set of correlations produced byentangled quantum systems is required. Unfortunately,a complete characterization of all possible correlations ofbipartite local measurements of quantum systems doesnot yet exist [21].However, for any given number of inputs and outputsnonlocal no-signaling polytopes P can be constructedwhich include all possible bipartite quantum correlationsof quantum systems of arbitrary dimensions. Therefore,the boundary of the region representing all possible bi-partite quantum correlations is sandwiched between theboundaries of a nonlocal no-signaling polytope P and itscorresponding Bell polytope. For any given set of inputsand outputs, this enables one to obtain lower boundson minimum detection inefficiencies which still allow theobservation of nonlocal features of quantum systems andwhich are independent of the dimension of the quantumsystem and the associated choice of quantum observablesgenerating these correlations.In the case of N ( x ) × N ( y ) inputs and N ( a ) × N ( b ) out-puts, the associated nonlocal no-signaling polytope is de-fined by all joint transition probabilities P ( ab | xy ) which are constrained only by the no-signaling conditions (5).It should be stressed that these no-signaling conditionsare weaker than locality because they do not necessarilyimply that the underlying transfer functions are local.Thus, in general the no-signaling conditions are alsocompatible with transfer functions of the form F =( F A , F B ). As with the local Bell polytopes of Sec. II B 1,the nonlocal no-signaling polytopes can be described con-veniently in the reduced basis formed by all marginaland joint transition probabilities which do not containoutputs a N ( a ) or b N ( b ) and whose dimension is given byrelation (9). In this reduced basis the no-signaling con-straints (5) are already taken into account provided themarginal and joint transition probabilities fulfill the con-sistency constraints X a = a N ( a ) P ( ab | xy ) ≤ P B ( b | y ) , X b = b N ( b ) P ( ab | xy ) ≤ P A ( a | x ) (11)for all inputs ( x, y ). These inequalities follow from Eqs.(7) and (8) and the no-signaling constraints (5). Sothe requirement of no-signaling is weaker than local-ity. Furthermore, these inequalities indicate that thenonlocal no-signaling polytopes are defined in a natu-ral way in the H -representation. Thus, for large dimen-sions of the reduced basis obtaining the corresponding V -representation from this H -representation is a difficultnumerical problem that limits the number of inputs andoutputs considerably for which this conversion can beachieved. III. DETECTION INEFFICIENCIES
Based on the previously discussed local and no-signaling polytopes L and P , in this section lower boundson minimum detection efficiencies are obtained belowwhich violations of locality cannot be observed in bipar-tite Bell experiments. The dependence of these lowerbounds on the numbers of inputs and outputs and onsymmetry is explored. It should be emphasized that fora given number of inputs and outputs these lower boundson minimum detection efficiencies apply to correlationsoriginating from arbitrary bipartite quantum states andobservables of arbitrary dimensional Hilbert spaces. Arelated problem, namely the determination of maximumpossible values of detection efficiencies which still guar-antee locality, has recently been investigated by Bigelow[22] with the help of linear programming techniques forsome special cases of correlations originating from two-and three-qubit systems. Contrary to our approach thisinvestigation does not involve the nonlocal no-signalingpolytope so that its resulting conclusions apply only tocorrelations which originate from two- and three-qubitquantum systems and from particular choices of quan-tum observables.One of the simplest ways to describe detection ineffi-ciencies of A and B is by a parameter η ∈ [0 ,
1] describ-ing the total efficiency of the detection systems involved.Thus, η is the probability that a detector fires if it ac-tually should. In practice, detection inefficiencies canhave different physical origins. They can originate froman imperfect photodetector, for example, which does notrespond to each photon hitting its detection surface. Al-ternatively, they may also arise from the fact that due tothe particular geometry of an experimental setup only afraction of photons propagating within a small solid an-gle is capable of hitting a photodetector at all. In thefollowing we assume that a combination of these effectsgives rise to the detection efficiencies η and η of A andB in a bipartite Bell experiment. Furthermore, these de-tection efficiencies are assumed to be independent of thepolarization of the photons hitting the photodetectors.For ideal detection, in a dichotomic Bell experimentA always receives a photon, so she needs only one de-tector to distinguish the polarizations. But for real de-tectors, a single polarization-sensitive detector makes nodistinction between the absence of a photon and a photonwith the wrong polarization, whereas two polarization-sensitive detectors can distinguish between these twocases. Similarly for B. Thus, in imperfect situations twodetectors give output events that are not possible withonly one, increasing the dimension of the relevant poly-topes. Indeed, we will demonstrate that the two casescan give rise to different lower bounds on detection effi-ciencies.First of all let us describe detection inefficiencies whereA’s (B’s) detector cannot distinguish between the no-detection event and the event a N ( a ) ( b N ( b ) )). Thus,the ideal joint transition probabilities, P , ( ab | xy ), arerelated to the corresponding imperfect joint transitionprobabilities, P η ,η ( ab | xy ), by P η ,η ( ab | xy ) = η η P , ( ab | xy ) ,P η ,η ( ab N ( b ) | xy ) = η P , ( ab N ( b ) | xy ) + η (1 − η ) X b = b N ( b ) P , ( ab | xy ) ,P η ,η ( a N ( a ) b | xy ) = η P , ( a N ( a ) b | xy ) + η (1 − η ) X a = a N ( a ) P , ( ab | xy ) ,P η ,η ( a N ( a ) b N ( b ) | xy ) = η η P , ( a N ( a ) b N ( b ) | xy ) + η (1 − η ) X b P , ( a N ( a ) b | xy ) + η (1 − η ) X a P , ( ab N ( b ) | xy ) +(1 − η )(1 − η ) (12)for all outputs ( a = a N ( a ) , b = b N ( b ) ) and inputs ( x, y ) ofA and B. For dichotomic Bell experiments with photonsthis describes situations in which A and B each use onepolarization-sensitive photodetector only which cannotdistinguish between a photon with the wrong polariza- tion and a no-detection event. In the reduced basis ofmarginal and joint transitions probabilities discussed inSecs. II B 1 and II B 2, in which the outputs a N ( a ) and b N ( b ) are eliminated, these relations reduce to the simpleform P Aη η ( a | x ) = η P A ( a | x ) , P Bη η ( b | y ) = η P B ( b | y ) ,P η η ( ab | xy ) = η η P , ( ab | xy ) (13)for all outputs ( a = a N ( a ) , b = b N ( b ) ) and inputs ( x, y ) ofA and B.If, in contrast, the no-detection event ∅ is treated asan additional output, the dimension of the relevant tran-sition probability polytope is increased. In this case theideal and imperfect transition probabilities P , ( ab | xy )and P η η ( ab | xy ) are related by P η ,η ( ab | xy ) = η η P , ( ab | xy ) ,P η ,η ( a ∅| xy ) = η (1 − η ) X b P , ( ab | xy ) ,P η ,η ( ∅ b | xy ) = (1 − η ) η X a P , ( ab | xy ) ,P η ,η ( ∅∅| xy ) = (1 − η )(1 − η ) (14)for all outputs ( a, b ) and inputs ( x, y ) of A and B. Fordichotomic Bell experiments with photons this describessituations in which A and B each use two photodetectorswhich are sensitive to two orthogonal polarizations. Byeliminating from Eqs. (14) all joint transition probabil-ities involving the outputs a N ( a ) or b N ( b ) with the helpof the marginal transition probabilities one obtains thecorresponding relations between the ideal and imperfecttransition probabilities of the reduced basis.For a given number of inputs and outputs, lowerbounds on detection efficiencies below which a violationof locality is no longer possible can be obtained fromEqs. (12), (13), and (14) by identifying the ideal tran-sition probabilities P , ( ab | xy ) with the possible corre-lations of the nonlocal no-signaling polytope P and bydetermining the critical detection efficiencies ( η , η ) atwhich the corresponding imperfect transition probabili-ties P η ,η ( ab | xy ) merge into the Bell polytope L .These critical detection efficiencies η and η determinelower bounds on the detection efficiencies below which aviolation of locality is no longer possible by the corre-sponding correlations produced by any quantum system.In general, it is unclear whether the lower bounds ob-tained on the basis of the no-signaling polytope P can bereached by any quantum system with appropriate choicesof the dimension of the Hilbert space and of the quantumobservables. But it is shown later that in the special caseof two inputs and two outputs of both A and B theselower bounds actually can be reached.Table I summarizes numerically-determined lowerbounds on detection efficiencies ( η and η ) of A andB which characterize the merging of the imperfect tran-sition probabilities P η ,η into the relevant local poly-tope L . If A and B randomly choose one of two pos-sible physical variables in their respective laboratories, ( A, B ) η = η η = η η = 1 η = 1add. no add. add. no add. ,
2) 0.6667 0.6667 0.5000 0.5000(3 ,
2) 0.6667 0.6667 0.5000 0.5000(3 ,
3) 0.5714 0.6000 0.3333 0.3333(4 ,
3) 0.5000 0.5714 0.2500 0.2500TABLE I: Critical detection efficiencies, η and η , of Alice(A) and Bob (B) for dichotomic bipartite symmetric ( η = η )and extreme asymmetric ( η = 1) Bell experiments with vari-ous numbers of inputs and for cases in which the no-detectionevent is treated separately (add. outcome) and in which it iscombined with the event ( a , b ) (no add. outcome). i.e. case (2,2), and if they have identical detectors, i.e. η ≡ η = η (symmetric case), the resulting critical valueof η turns out to be independent of whether or not theno-detection event is treated as an additional output.This optimal lower bound obtained on the basis of theno-signaling polytope P turns out to be identical withthe minimal detection efficiency obtained previously byEberhard [6]. Eberhard’s result demonstrated that puretwo-qubit quantum states exist which are able to violatelocality down to minimum detection efficiencies of mag-nitude η = 0 . η = 1 = η of magnitude η = 0 . , η = 0 .
667 in the symmetric case and below η = 0 .
500 in the extreme asymmetric case. This conclu-sion holds for arbitrary choices of two two-valued quan-tum observables of A and B and for arbitrary bipartitequantum states in arbitrary dimensional Hilbert spaces,which produce the statistical correlations.Table I also includes results on optimal lower boundsfor cases in which more than two physical variables areselected on Alice’s or Bob’s sides. It is apparent thatin general these lower bounds depend on whether or notthe no-detection event is treated as an additional out-put. Furthermore, the lower bounds of cases in whichthe no-detection event is treated as an additional out-put are always lower than or equal to cases in which theno-detection event is combined with an already existingoutput. However, for cases with more than two inputs ofAlice or Bob it is not known yet whether quantum sys-
FIG. 2: Lower bounds on detection efficiencies ( η , η ) forthree inputs on Alice’s and Bob’s sides with the no-signalevent treated as an extra output. tems exist which are capable of violating locality all theway down to these lower bounds. However, the numberof outputs on Alice’s and Bob’s sides, N ( a ) and N ( b ),puts a lower bound on the dimension D of the Hilbertspace of these quantum systems, i.e. D ≥ N ( a ) × N ( b ).In Figs. 2 and 3 lower bounds on minimum detectionefficiencies ( η , η ) are depicted for arbitrary cases be-tween the symmetric ( η = η ) and the extreme asym-metric (1 = η = η ) situation for the special case ofthree inputs and two outputs of both A and B. Irrespec-tive whether or not the no-signal event is treated as aseparate outcome one observes a cusp-like dependence inthese figures. This non-smooth dependence correspondsto a case in which, at a particular value of ( η , η ), avertex of the properly transformed nonlocal no-signalingpolytope (compare with Eqs.(12)) just coincides with avertex of the local (Bell) polytope L .We have also explored lower bounds on detection effi-ciencies for two inputs and three outputs on both Aliceand Bob’s sides. In the symmetrical case ( η = η ≡ η )the lower bound was given by η = 0 . η = 1) weobtained the lower bound η = 0 . P for larger numbers of inputs or outputs. Thisis due to the fact that no-signaling polytopes are definedin a natural way in the H -representation (compare withthe discussion of Sec. II B 2). Thus in order to deter-mine lower bounds on detection efficiencies one has toconvert the nonlocal no-signaling polytope from its H -representation into its V -representation, which becomesvery difficult numerically for such cases. FIG. 3: Lower bounds on detection efficiencies ( η , η ) forthree inputs on Alice’s and Bob’s sides with the no-signalevent combined with the output ( a , b ). vv FIG. 4: Schematic representations of identical violations v ofrelevant Bell inequalities. Note the different distances fromthe polytopes. IV. DISTANCE MEASURES QUANTIFYINGTHE VIOLATION OF LOCALITY
The simplest way of testing correlations for localityis to determine whether or not the relevant point X oftransition-probability space is inside the local polytope L . However, in general probabilities can be estimatedexperimentally only approximately with an uncertaintydepending on the size of the statistical sample involved.Thus, a practically useful method for determining viola-tions of locality should also quantify how far outside L the given point X is located. Therefore, it is desirableto develop methods which permit one to find the dis-tance of a given point X in transition-probability spacefrom a local polytope L , so that one can decide whetherobserved transition probabilities with their experimentaluncertainties still violate locality.A natural choice for such a distance measure is theEuclidean 2-norm distance from the nearest facet of thepolytope L which corresponds to a Bell inequality. Butalso the 1- or the ∞ -norms are possible choices. How-ever, as illustrated in Fig. 4, it definitely makes more sense to consider the “distance of X from the polytope L as a whole”. The only reasonable definition for the“distance from a polytope” is the minimum distance ofany point of L to X , which means we have to deal withan optimization problem.Let us assume L is given in D -dimensional space andthat it has r vertexes, which we denote by v i ∈ R D .Thus, any point Y ∈ L can be written as a convex com-bination of these vertexes, i.e. Y = P ri =1 w i v i withweights w i ≥ P ri =1 w i = 1. So, it is natu-ral to use the w i (or the vector w T = ( w ; w ; . . . ; w r ))as the coordinates for the optimization problem ratherthan the coordinates of Y in the actual space R D inwhich the polytope lives. This is motivated by the factthat the constraints of the polytope are given in termsof the weights w i rather than in terms of the coordinatesof the actual space. However, we still want to optimizethe distance in the actual space. In order to achieve thisfor the 1-norm, it is convenient to introduce the matrix C = ( v ; v ; . . . ; v r ) ∈ R D × r with Y = C · w . Let usalso use the abbreviations D , D and r , r for columnvectors of all zeros or ones in R D and R r , respectively.Analogously we use the notation D × D for a diagonal D × D unit matrix, and similarly r × r . The problem offinding the minimum distance between a point X ∈ R D and the local polytope L can now be formulated as thefollowing linear programming problem:Maximize − ( TD ; Tr ) · Z Subject to A · Z ≤ b with the (2 D + 3) × ( D + r ) matrix A = − D × D , C − D × D , C0 TD , − r × r TD , Tr TD , − Tr , (15)the ( D + r )-dimensional vector Z T = (¯ Z T ; w T ) and the(2 D + r +2)-dimensional vector b T = ( X T ; − X T ; 0; 1 , ; 1).As a result the 1-norm is given by ( TD ; Tr ) · Z ≡ TD · ¯ Z .A similar linear programming problem can be formu-lated in order to find the ∞ -norm. Although an analo-gous quadratic programming problem can be formulatedfor the ordinary 2-norm distance, it is worth mentioningthat the numerical solution of this quadratic problem ismuch more difficult and time-consuming than the cor-responding linear programming problem. The 2-normcan however be bounded from above and below by the 1-norm and ∞ -norm, respectively. As an example, considerFig. 5 which shows how the distance between a properlytransformed vertex of the nonlocal no-signaling polytope P (compare with Eqs. (12)) and the local polytope L varies smoothly when we vary the detection efficiency.Of course, at η = 0 . d i s t an c e f r o m l o c a l po l y t ope η ∞ -norm FIG. 5: Distance of a vertex of P from L as a function of η for two inputs and outputs of both A and B. V. SUMMARY AND CONCLUSIONS
For given numbers of inputs and outputs we have in-vestigated minimum detection efficiencies below whichlocality cannot be violated by correlations produced byany quantum system in bipartite Bell experiments. Forthis purpose lower bounds on these minimum detectionefficiencies have been obtained numerically with the helpof linear programming techniques. Our determinationof these lower bounds is based on the observation thatfor any given number of inputs and outputs any possi-ble bipartite correlation produced by a quantum systemin an arbitrary dimensional Hilbert space is sandwichedbetween the boundaries of the nonlocal no-signaling poly-tope and the Bell polytope. Thus, for imperfect detectionthe detection efficiencies at which statistical correlationsof the properly transformed nonlocal no-signaling poly-tope merge into the Bell polytope yield lower bounds onthese minimum detection efficiencies.Both the local (Bell) and the nonlocal no-signaling polytope can can be dealt with conveniently by linearprogramming. In particular, the vertex representation ofany Bell polytope can be determined in a straightforwardway. The construction of the nonlocal no-signaling poly-tope is more complicated as it is naturally defined in the H -representation.Our numerically calculated lower bounds on detectionefficiencies demonstrate that in general, with the excep-tion of two inputs and outputs of A and B, these boundsare not identical for Bell experiments with symmetricand asymmetric detection efficiencies. Furthermore, inthe case of two inputs and outputs our lower boundsagree with the minimum detection efficiencies obtainedpreviously by Eberhard [6] for two-qubit quantum corre-lations. Thus, in this case our results demonstrate thatthese minimum detection efficiencies cannot be loweredeven if one considered quantum correlations originatingfrom quantum systems of arbitrary dimensions.Our investigation constitutes a first step towards asystematic study of bipartite correlations produced byquantum systems. In general, it is still unclear to whatextent our numerically determined lower bounds can bereached by correlations of appropriately chosen quantumsystems. Further research is required to clarify this point.In addition, for numerical purposes it would be desirableto find an effective way for determining directly the V -representations of nonlocal no-signaling polytopes with-out involvement of their H -representations. This wouldalso allow the efficient treatment of cases involving manyinputs and outputs. Furthermore, our approach can alsobe adapted to the investigation of multipartite correla-tions. Acknowledgments
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