Information Causality and Extremal Tripartite Correlations

Information Causality and Extremal Tripartite Correlations Tzyh Haur Yang,1 Daniel Cavalcanti,1 Mafalda Almeida,1 Colin Teo,1 and Valerio Scarani1, 2

arXiv:1108.2293v1 [quant-ph] 10 Aug 2011

1

Centre for Quantum Technologies, National University of Singapore, 3 Science drive 2, Singapore 117543 2 Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 We study the principle of information causality for the set of extremal correlations in the tripartite scenario. We show that all but one nonlocal extremal correlations in this scenario violate information causality. This undetected correlation is shown to satisfy any bipartite physical principle.

I.

INTRODUCTION

Much effort has been put towards the understanding of the physical origins of quantum correlations. Quantum theory was originally developed as a set of mathematical rules from which predictions on physical phenomena can be successfully obtained. Quantum theory satisfies certain physical principles, such as no-signaling [1] (i.e the impossibility of instantaneous transmission of information between two locations) and macroscopic locality [2] (i.e. the fact that a macroscopic coarse-graining of quantum correlations can be explained classically). However, these principles are not exclusive of quantum mechanics: several other theories satisfy these principles but still allow for correlations stronger than the quantum ones. Is there any physical principle that singles out quantum correlations? The principle of information causality was proposed as a solution to this question [3]. This principle limits the amount of information-gain an observer (Bob) can reach when another observer (Alice), in a different location, sends him some amount of information: if m bits are sent, then Bob cannot learn more than m bits from Alice’s system. Information causality was not proven to rule out all possible post-quantum correlations [4, 5], and it remains as an open question whether this principle is sufficient to identify bipartite quantum correlations. Information causality has already succeeded in ruling out, for instance, all correlations which are stronger than the strongest quantum correlation attained when two distant observers measure a pair of two-valued observables on their physical systems. Specifically, information causality is violated for all correlations violating the Clauser-Horn-Shimony-Holt (CHSH) Bell inequality [6] by a value superior to that allowed by quantum theory [3]. Up to now, the search for physical principles defining quantum correlations focused essentially in the bipartite case. However, recent results have shown that the structure of multipartite correlations is richer than its bipartite correspondent [7]. For instance, nontrivial nonlocal games with no quantum advantage over classical theory were found [8] and the generalization of Gleason’s theorem [9] for the multipartite case was shown to be problematic [10]. It is thus tempting to determine if information causality is able rule out post-quantum multipartite correlations. In the present paper we study information causality in the simplest scenario where multipartite correlations arise. We consider three observers (Alice, Bob and Charlie), each performing two spacelike separated measurements (x, y, z = {0, 1}), with two possible outcomes (a, b, c = {0, 1}), on their local subsystems. The obtained correlations between the parties’ outcomes, conditioned to the choice of observables, is described by the joint probability distribution P (abc|xyz). Here we assume that the no-signaling principle holds, i.e. the choice of P observable by one of the parties cannot affect the probability distribution of the remaining: P (ab|xyz) = c P (abc|xyz) = P (ab|xy), for any a, b, x, y, z and permutation of the parties. We check the principle of information causality in the presence of extremal correlations in this scenario (i.e. those correlations from which all other no-signaling correlations can be obtained by statistical mixing). As a result we conclude that information causality is able to rule out all extremal no-signaling correlations but one. Using the criterion derived in Ref. [11], and similarly to an example pointed out there, we also show that this undetected correlation cannot be ruled out by any bipartite criterion. We organize the paper as follows. We first give a brief explanation on the set of no-signaling tripartite probability distributions and on sufficient conditions for correlations to violate information-causality. Then we show that information causality is violated by all extremal probability distributions but one. Next, the features of the only undetected probability distribution are discussed. Finally we discuss the perspectives arising from the present results, specially on the necessity of a genuine multipartite physical principle.

2 II. A.

REVIEW OF TOOLS

Extremal no-signaling tripartite probability distributions

The set of no-signaling probability distributions of tripartite systems, with binary input and output for each party, defines a polytope (no-signaling polytope) in a 26-dimensional space. This polytope has been characterized [7], and its complexity is appalling, especially taking into account that this is just the simplest non-trivial tripartite scenario. Indeed, the no-signaling polytope has 53856 extremal points, belonging to 46 different classes: one class comprises the local deterministic points, while the other 45 classes contain non-local points. The corresponding local polytope has just as many facets, forming one class of trivial constraints and 45 inequivalent Bell-type inequalities [12]. Unfortunately there is no obvious correspondence between facets and extremal no-signaling points. Little is known about the boundaries of the quantum set in this scenario: in particular, no analytical bounds have been derived and the techniques used in the bipartite case [13] would probably not be very tight. However, a convergence hierarchy of semi-definite programs is known [14] and we shall use it below. Our goal here is to determine the ability of information causality (IC from now onwards) in ruling out postquantum correlations in this tripartite scenario. We focus on the extremal points of the no-signaling polytope [19]. Following the classification given in Ref. [7], class 1 corresponds to local deterministic probability distributions, which arise in classical systems and obviously satisfy IC. We are then interested in the remaining 45 nonlocal classes. We will show that 44 of them violate IC (notice that the violation of IC by boxes of class 46, and their natural multipartite generalization, has been studied in detail in Ref. [15]). Section IV is devoted to the remaining class, namely class 4, which accounts for 126 extremal points. It will be shown that the correlations of this class not only satisfy IC but any other bipartite information principle aimed at ruling out post-quantum correlations.

B.

Sufficient criteria to violate information-causality

The principle of information-causality says that Bob cannot learn more than m bits from a distant Alice, after she sends him m bits. In order to test this principle in the presence of given correlations between the parties, we must first choose some bipartite information protocol which uses the correlations as a resource. In the original IC paper, the chosen protocol starts with Alice receiving a string-bit of size n, (a1 , . . . , an ), and Bob a number between b between 1 and n. The goal is for Bob to guess the value of Alice’s bit ab . For that, Alice and Bob are allowed to share physical resources and operate locally on these, and Alice can send m classical bits to Bob. Using this scheme, it is known from Refs [3, 4] that information causality is violated whenever the parties share correlations that either: (i) achieve a larger-than-quantum violation of the CHSH inequality, i.e √ CHSH = E00 + E01 + E10 − E11 > 2 2 ; (1) where Exy = P (a = b|xy) − P (a 6= b|xy); or (ii) violate the quadratic inequality proposed by Uffink [16], i.e. E00 + E10


2

+ E01 − E11


2

> 4.

(2)

We will use these sufficient criteria for the violation of IC to derive our results.

III.

EXTREMAL TRIPARTITE CORRELATIONS VIOLATING INFORMATION CAUSALITY A.

Approach

We have seen that information-causality is a principle formulated for bipartite systems. In order to test if this principle can exclude post-quantum tripartite distributions, we will study the bipartite distributions arising from them when a pair of parties are put together (defining bipartitions A|BC, AB|C or B|AC). Moreover, in order to fit in the sufficient criteria for violation of IC mentioned earlier, the two parties that are together must receive a single bit as input and are asked to output a single bit as well. Since the tripartite box, as such, requires two bits as inputs and outputs two bits, for the group with two parties, some processing must take place: such a

3 processing is called wiring [17]. Like any other processing, a priori wiring may result in some loss of information. However, for the purpose of ruling out extremal points, it will prove powerful enough. More formally now: for each tripartite box P (a, b, c|x, y, z), a bipartition and a wiring define an effective box Peff (a′ , b′ |x′ , y ′ ), where all the inputs and outputs [20] are considered to be in {0, 1}. The goal is to find, for each P , a bipartition and a wiring such that Peff violates IC, according to the sufficient criteria in Sec. III. B.

Result and examples

The results are summarized in Table I: only the points of class 1 (local deterministic points, as expected) and of class 4 (to which we shall devote the next section) are found not to violate IC. For the others, a bipartition and a wiring can be found, such that either (1) or (2), or both, happen. We consider two distinct types of wiring, which we illustrate by examples. Without lost of generality we group Bob and Charlie. In the first type of wiring, the inputs yz of the tripartite box will only be a function of the effective input y ′ given to the block BC. Take the wiring on box 44, characterized by the relation a + b + c = xyz, as an example. This specific wiring is depicted by Fig. 1. The inputs in the tripartite box are defined by x = x′ , y = y ′ and z = 1. Therefore, the party holding both B and C uses the input y ′ only for B and uses a fixed input for C. By choosing outputs a′ = a and b′ = b + c, one realizes a′ + b′ = x′ y ′ . In other words, box 44 actually is able to realize an effective bipartite Popescu-Rohrlich box [1], known to violate IC maximally [3]. In the second type of wiring, one of the inputs z or y additionally depends on the output of the other party in the block. This necessarily imposes a time order in the use of the tripartite box: if, for example, the input z depends on the output b, Charlie can only use his part of the box after receiving the information from Bob’s outcome. A few of the classes of extremal points require this type of wiring to violate IC; its study is slightly more complicated. Take as an example the points of the class 3. The explicit form of one of its representatives can be read from Table 1 of [7]: P (a, b, c|x, y, z) =

 1 1 + (−1)a+b δx,0 + (−1)a+c δx,1 δz,0 + (−1)a+b+c δx,1 (δy,0 − δy,1 ) δz,1 . 8

(3)

The bipartition and wiring are sketched in Fig. 2. Here we group A and C. The input x′ is first used as the input for C, z = x′ , and this leads to an outcome c. The input for A is then chosen as x = z + zc, and the outcome a is used as final outcome a′ . In order to work out this example, notice that the wiring relation x = z + zc explicitly reads: if c = 0, then x = z; if c = 1, then x = 0 independently of z. So: Peff (a′ , b′ |x′ , y ′ ) = P (a, b, c = 0|x = z, y, z) + P (a, b, c = 1|x = 0, y, z)   1 1 = 1 + (−1)a+b δz,0 + (−1)a+b (δy,0 − δy,1 ) δz,1 + 1 + (−1)a+b 8 8 i i 1h 1h a+b δz,0 + (δy,0 − δy,1 ) δz,1 ≡ 1 + (−1)a+b Ex′ y′ = 1 + (−1) 4 2 2 where we recall that x′ = z and y ′ = y. From this last expression, one finds E00 = E01 = E10 = 1 and E11 = 0, whence CHSH = 3. # Wiring 1 2 3 x = z + zc 4 5 x = z + zc 6 x=1 7 y =1+x 8 z = ax 9 y =1+x 10 x=0 Continued on the

a′ b′ CHSH Quadratic b c 4 8 a b 3 5 a b 3 5 a+b c 4 8 a+b c 4 8 b c 3 5 a+b c 4 8 a+b c 4 8 next page. . .

4 TABLE I: continued. # Wiring a′ b′ CHSH Quadratic 11 z = ax a+c b 3 5 12 z = ax a+c b 3 5 13 y =1+z a b+c 40/9 14 y =1+x a+b c 10/3 52/9 15 x=0 a+b c 4 8 16 y=1 a+b c 40/9 17 x=0 a+b c 4 8 18 z=0 a b+c 3 5 19 z=1 a b+c 3 9/2 20 z=0 a b + c 16/5 26/5 21 z=1 a b+c 3 9/2 22 z = 1 + a + ax a + c b 40/9 23 y =1+x a+b c 4 8 24 x=1 a+b c 4 8 25 z=1 a b + c 10/3 52/9 26 z=0 a b+c 40/9 27 z=1 a b+c 3 5 28 x=1 a+b c 4 8 29 z=1 a b + c 10/3 52/9 30 z=0 a b + c 18/5 114/25 31 y=1 a+b c 14/5 4 32 z=0 a b + c 18/5 114/25 33 y=1 a+b c 14/5 116/25 34 z=0 a b + c 10/3 50/9 35 z=0 a b + c 10/3 50/9 36 z=1 a b + c 7/2 49/8 37 z=0 a b + c 7/2 25/4 38 z=0 a b + c 10/3 52/9 39 z=0 a b + c 10/3 52/9 40 z=0 a b+c 3 5 41 z=0 a b+c 3 5 42 z=0 a b+c 3 5 43 z=1 a b + c 26/7 340/49 44 z=1 a b+c 4 8 45 z=1 a b+c 4 8 46 z=1 a b+c 4 8 TABLE I: Violation of bipartite IC as detected by either the CHSH inequality [eq. 1] or the quadratic inequality [eq. 2], or both. The table follows the conventions of Table 2 of [7]: both the settings x, y, z and the outcomes a, b, c take the values 0 or 1. All the sums are to be taken modulo 2. The bipartitions are implied by the outputs a′ , b′ : for instance, if b′ = b + c, clearly the bipartition must be A|BC. Notice that the inequality which is violated may not necessarily be (1) or (2), but one of their equivalent forms under relabeling of the parties and/or the inputs and/or the outputs.

IV.

CLASS 4: EXTREMAL NO-SIGNALING DISTRIBUTIONS SATISFYING ANY BIPARTITE CRITERION

We turn now to a more detailed study of the class 4 of extremal point. So far, we have suggested that those extremal points do not violate bipartite IC with wirings; ultimately, we shall see that they cannot violate any form of bipartite information-theoretical principles aiming to single out quantum (or even local) correlations. Before turning to this, let us describe the points in this class and some of their properties.

5

FIG. 1: Bipartition and wiring that lead to the violation of IC by the extremal points of class 44. In this wiring, A will input x = x′ and output a′ = a. On the other partition however, the input of C will always be z = 1, while the input of B is y = y ′ ; the final output is b′ = b + c.

FIG. 2: Bipartition and wiring that lead to the violation of IC by the extremal points of class 3. In this wiring, B will input y = y ′ and output b′ = b. On the other partition however, first the input z = x′ is used for C; the corresponding output c is used to define the input x of A according to x = x′ + x′ c; the final output of this party is a′ = a.

A.

Description of the points

The points in class 4 have a very precise structure. As its representatives, we choose the one that is obtained from the version in Table 1 at [7] by changing y → 1 − y. In this formulation, the class is defined by the following deterministic correlations a0 + b 1 b 0 + c1 c0 + a1 a0 + b 0 + c0 a1 + b 1 + c1

= 0, = 0, = 0, = 0, = 1,

(4)

and random statistics for any other combination. Here, ax here refers to the output of A when its input is x and similarly for the B and C. It is clear from these equations that any cyclic permutations of (A, B, C) will have the

6 same statistics. This point is clearly non-local, because the sum of the first four correlations would imply a1 + b1 + c1 = 0, contradicted by the last one. It cannot be realized with measurement on quantum states, as it can be shown by using a modified version of the semi-definite hierarchy defined in Ref. [14]; see Appendix A for details. It has another remarkable property, namely, these correlations cannot be created by distributing arbitrary many bipartite PR-boxes between the parties [18]. Finally, what we are going to prove in the following subsection yields as a corollary that the converse also holds: this point cannot be used to create a bipartite PR-box.

B.

No violation of any bipartite criterion

We now show that extremal points in class 4 satisfy IC and any other bipartite information principle aimed at singling out quantum correlations. The main idea is to prove that the probability distributions belonging to this class are local for any bipartition, even after any local wirings. A sufficient criterion was provided in Ref. [11]: if P (a, b, c|x, y, z) belongs to the set of time-ordered bi-local (TOBL) probability distributions, then any possible bipartite distributions produced from P (a, b, c|x, y, z) are local. A probability distribution belongs to TOBL if it can be written as X P (a, b, c|x, y, z) = pλ P (a|x, λ) PB (b|y, λ) PC (c|y, z, λ), (5) λ

=

X

p′λ P ′ (a|x, λ) PB′ (b|y, z, λ) PC′ (c|z, λ),

(6)

λ

for bipartition A|BC, and analogously for AB|C and B|AC. Here pλ is the probability distribution over some variable λ. We see that the local model considers either signaling from Bob to Charlie (Eq. 5) and signaling from Charlie to Bob (Eq. 6) but never both at the same time. Also, Alice is unaware of the direction of signaling inside bipartition BC. We will now show that extremal points belonging to class 4 indeed belong to TOBL. Without loss of generality, let us partition the representative point (4) according to A|BC. Let the hidden variable λ shared by the three parties be two bits (λ0 , λ1 ) drawn with equal probabilities p(λ0 , λ1 ) = 41 . We will first construct the model A|B → C, where B signals to C. On receiving the variable λ, A will output λx depending on her input, x. B will output λ0 + λ1 if y = 0 and λ0 if y = 1. C will output λ1 if c = 0; if c = 1, he will output λ0 + λ1 if b = 0, and λ0 + λ1 + 1 if b = 1 (here is where the signalling B → C is used). In other words, we have the following instructions a = λ0 + (λ0 + λ1 )x, b = λ0 + λ1 + λ1 y, c = λ1 + (λ0 + y)z .

(7)

For the case A|B ← C, the instructions are a = λ0 + (λ0 + λ1 )x, c = λ1 + (λ0 + 1)z, b = λ0 + (λ1 + z)(1 − y) .

(8)

Both sets of instructions can be verified to reproduce (4). Since the correlation is cyclic, any bipartitions of ABC can be written correspondingly in the way specified by in Eqs (5) and (6). This shows that class 4 indeed belongs to TOBL, and thus does not violate any Bell inequality in any bipartitions even after wirings. Since extremal points in class 4 have local (classical) statistics for any bipartition we consider, we conclude they can never violate any form of bipartite information-theoretical principles aiming to single out quantum (or even local) correlations. A non-extremal point with the same property has been found in another region of the polytope [11], specifically above the “Guess-Your-Neighbor-Input” tripartite inequality defined by P (000|000) + P (110|011) + P (011|101) + P (101|110) ≤ 1 .

(9)

7 Notice that this conclusion extends at least to all points which are convex combinations of those examples and some local deterministic points: therefore, the non-quantum correlations that cannot be ruled out with bipartite criteria form sets of non-zero measure. To conclude, we show that our class of extremal points does not violate any GYNI inequality and therefore our example and the one in Ref. [11] exhibit intrinsically different kinds of nonlocality. First, we recall that the most general form of tripartite GYNI inequalities [8] is X (10) q(x1 x2 x3 )P (x2 x3 x1 |x1 x2 x3 ) ≤ max (qx1 ,x2 ,x3 + qx¯1 ,¯x2 ,¯x3 ) x1 ,x2 ,x3

x1 ,x2 ,x3 =0,1

P where x1 ,x2 ,x3 q(x1 , x2 , x3 ) = 1 and the upper bars denote negation. For the points in class 4, the probabilities of the outputs conditioned on the inputs are never larger than 1/4: P (a1 , a2 , a3 |x1 , x2 , x3 ) ≤ 41 . So, in the best case, these correlations achieve the value 1 4

X


(qx1 ,x2 ,x3 + qx¯1 ,¯x2 ,¯x3 )

(11)

x1 ⊕x2 ⊕x3 =0

at the polynomial in Eq.10. Clearly, this value can never be larger than the local bound maxx1 ,x2 ,x3 (qx1 ,x2 ,x3 + qx¯1 ,¯x2 ,¯x3 ) and nonlocal boxes in class 4 are unable to violate any GYNI inequality. V.

PERSPECTIVES AND CONCLUSIONS

We have set out to apply the principle of information causality to multipartite correlations. As a first step, we have taken the only existing form to test IC, which involves a bipartite task, and checked its violation on the extremal points of the simplest tripartite no-signalling polytope. IC detects that 44 out of 45 classes of non-local extremal points which are not quantum; the remaining class, which can also be proved not to be quantum by other means, escapes this and any other bipartite criterion. IC remains a powerful criterion to rule out no-signalling correlations which cannot be achieved with quantum physics. But it started out with a more ambitious conjecture, namely, as a physical principle which might identify the set of quantum correlations exactly. Generalizations of IC beyond the basic CHSH scenario cannot be said to have brought clarification: instead, we are discovering extremely complex structures. These are not devoid of interest and deserve further studies; certainly, for instance, it will be very instructive to find a meaningful multipartite task that detects the non-quantumness of our example and of the one reported in Ref. [11]. However, ultimately, completely different approaches will probably have to be found, in order to prove the main conjecture.

Acknowledgments

We acknowledge illuminating discussions with Antonio Ac´ın, Nicolas Brunner, Rodrigo Gallego, Miguel Navascu´es and Lars W¨ urflinger. This work was supported by the National Research Foundation and the Ministry of Education, Singapore.

Appendix A: Testing the quantumness of tripartite correlations

We want to show that probability distributions of class 4 cannot be obtained by measuring a quantum state of arbitrary dimension. The criteria described in Ref. [14] for bipartite scenarios can be extended to the multipartite scenario in a rather pedestrian way, which has no guarantee (to our knowledge) of tightness or convergence, but is sufficient for our purpose here. In fact, already the first instance of semi-definite programming leads to the conclusion. Specifically, consider the following 19 indices: • j = 1: 1;

8 • j = 2, ..., 7: a0 , a1 , b0 , b1 , c0 , c1 ; • j = 8, ..., 11: a0 b0 , a0 b1 , a1 b0 , a1 b1 ; • j = 12, ..., 15: a0 c0 , a0 c1 , a1 c0 , a1 c1 ; • j = 16, ..., 19: b0 c0 , b0 c1 , b1 c0 , b1 c1 . Then one constructs the 19 × 19 symmetric matrix M whose entries are the corresponding correlation coefficients (for convenience of notation, in this appendix we use rather a, b, c ∈ {−1, +1}). For instance, Mjj = 1, M12 = M21 = ha0 i, M78 = M87 = ha0 b0 c1 i etc. Some of these entries, those in which the same party appears twice with different inputs (for instance, M23 = M32 = ha0 a1 i) are not defined by the probability distribution P : in those places, one leaves a free variable. All in all, there are 69 free variables left in M . Now, if P can be obtained by measuring a quantum state, there always exists a set of values for the free variables such that M ≥ 0; conversely, if one finds that M < 0 for all possible values of the free variables, then definitely P is outside the quantum set. This problem is just the semi-definite problem of finding the largest λ such that M − λI ≥ 0 with I the 19 × 19 identity matrix. The defined entries for each extremal point can be directly read from Table 1 of Ref. [7]. The result of the optimization for points of class 4 is λ ≈ −0.2361. This conclusively proves that these points are not in the quantum set. Notice that we have run the same software to check all the extremal points, and we find of course that all the non-local ones are not in the quantum set, as proved already by the fact that they violate bipartite IC.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994). M. Navascu´es and H. Wunderlich, Proc. Roy. Soc. Lond. A 466, 881-890 (2009). M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, M. Zukowski, Nature 461, 1101 (2009) J. Allcock, N. Brunner, M. Pawlowski, V. Scarani, Phys. Rev. A 80, 040103 (2009). D. Cavalcanti, A. Salles, and V. Scarani, Nat. Comm. 1, 136 (2010). J.F. Clauser, M. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1969). S. Pironio, J.-D. Bancal, V. Scarani, J. Phys. A: Math. Theor. 44 (2011) 065303. M. L. Almeida et al., Phys. Rev. Lett. 104, 230404 (2010). A. Gleason, J. Math. Mech. 6, 885 (1957). A. Ac´ın et al., Phys. Rev. Lett. 104, 140404 (2010). R. Gallego, L. W¨ urflinger, A. Ac´ın, M. Navascu´es, arXiv:1107.3738 ´ C. Sliwa, Phys. Lett. A 317, 165-168 (2003). T.H. Yang, M. Navascu´es, L. Sheridan, V. Scarani, Phys. Rev. A 83, 022105 (2011) M. Navascu´es, S. Pironio, A. Ac´ın, Phys. Rev. Lett. 98, 010401 (2007); New J. Phys. 10, 073013 (2008); S. Pironio, M. Navascu´es, A. Ac´ın, SIAM J. Optim. 20, 2157 (2010) Y. Xiang, W. Ren, arXiv:1101.2971 J. Uffink, Phys. Rev. Lett. 88, 230406 (2002). The term has been introduced in: J. Allcock, N. Brunner, N. Linden, S. Popescu, P. Skrzypczyk, T. Vertesi, Phys. Rev. A 80, 062107 (2009); the notion is a rather natural one and had been used previously in a non-systematic way. The proof (S. Pironio, private communication) is similar to the one used for a five-partite distribution in: J. Barrett, S. Pironio, Phys. Rev. Lett. 95, 140401 (2005) It is enough to discuss the violation of IC for one representative of each class, since all correlations in the same class are equivalent under relabeling of parties, inputs or outputs. Notice that our outputs a, b, c were written a ˆ, ˆb, cˆ in Ref. [7], where the letters a, b, c were used to denote outcomes belonging to {−1, +1}.