MODELO DE DETERMINANTES DE LA POBREZA PARA COLOMBIA

localhidden variables

We shall now settle the issue of the relative nega- tion¬, i.e. whether

Q11↔Q22= Q12↔Q21, or

Q11↔Q22=¬(Q12↔Q21),

for both rebits and qubits; one of these equations must be true (see theorems 5.7 and 5.9). It is

instructive to illustrate these equations by means of a question configuration analogous to a Bell scenario.

(A) Suppose Q11 ↔ Q22 = Q12 ↔ Q21 was

true. Before we determine whether it is consis- tent with our background assumptions and pos- tulates, we note that this is the case of classi- cal (or realist) logic: Ocan consistently interpret

any such configuration by means of a local ‘hid-

den variable’ model. For example, consider the case Q11 = Q22 = 1 (which is compatible with

the rules) such that Q11 ↔ Q22 = 1and, conse-

quently, also Q12 ↔ Q21 = 1. We represent the

last two equations by the following two graphs PSfrag replacements Q12 Q21 Q11 Q22 PSfrag replacements Q12 Q21 Q11 Q22

where both edges solid and of equal colour means that the corresponding questions are correlated.

O could consistently read this configuration as

follows: “Since “Q1 = Q′1” and “Q2 = Q′2”, I

would precisely have to conclude that “Q12 =

Q21”, i.e. Q12 ↔ Q21 = 1, if the individuals

Q1, Q2, Q′1, Q′2 had definite values which, how-

ever, I do not know.” For instance, Q11=Q22=

Q12↔Q21= 1would be consistent with the four

ontic states: PSfrag replacements 1 1 1 1 0 PSfrag replacements 1 1 0 0 PSfrag replacements 10 0 0 0 PSfrag replacements 1 1 0 0 .

Therefore, O could join the two graphs consis-

tently (without knowing the truth values of the individuals)

(23) i.e., draw all four edges simultaneously which cor- responds to all four edges having definite (al- beit unknown) values at the same time. Sim-

ilarly, O can interpret any other configuration

with Q11 ↔ Q22 = Q12 ↔ Q21 in terms of a

local ‘hidden variable’ model which assigns defi- nite values to the individuals and, conversely, as can be easily checked, any assignment of definite (or ontic) values to the individualsQ1, Q2, Q′1, Q′2

leads necessarily to Q11↔Q22=Q12↔Q21.

We now preclude this possibility. We note that

Q11 ↔ Q22 = Q12 ↔ Q21 can be rewritten in

terms of the individuals as

which is a classical logical identity that is al- ways true for classical Boolean logic. It states

that the brackets can be equivalently regrouped, i.e. that the order in which the XNOR ↔ is executed in the logical expression can be

switched. In classical Boolean logic, this iden- tity follows from the associativity and symme- try of the XNOR. Namely, suppose for a moment that Q1, Q2, Q′1, Q′2 had simultaneous truth val-

ues. Then Boolean logic would tell us that

(Q1 ↔Q′1)↔(Q2 ↔Q′2) = (Q1 ↔Q1′)↔Q2↔Q′2= Q1 ↔(Q2 ↔Q′1) ↔Q′₂ =Q′₂↔ Q1 ↔(Q2↔Q′1) = (Q1↔Q′2)↔(Q2 ↔Q′1). (25)

Of course, relative to O, Q1, Q2, Q′1, Q′2 do not

have simultaneous truth values. Instead, assump- tion 6 implies that, for a classical logical iden- tity to hold in O’s model, either (a) all ques-

tions in the involved logical expressions are mu- tually maximally compatible or (b) the identity follows from applying classical rules of inference to (sub-)expressions which are entirely written in terms of mutually maximally compatible ques- tions and subsequently decomposing the result logically into other questions (with possibly dis- tinct compatibility relations). However, since

Q1, Q2 and Q′1, Q′2 form maximally complemen-

tary pairs, neither is the case here. In particu- lar, the right hand side in (24) does not follow

from applying any rules of Boolean logic to the total expression Q11 ↔ Q22 or those subexpres-

sions Q1 ↔ Q′1 and Q2 ↔ Q′2 on the left hand

side which are fully composed of maximally com- patible questions.35 _{Hence, (24) violates assump-}

tion 6such that we must rule out the possibility

Q11↔Q22=Q12↔Q21.

(B) We are thus already forced to the conclu- sion that

Q11↔Q22=¬(Q12↔Q21). (26)

must hold. Indeed, its equivalent representation (Q1↔Q′1)↔(Q2 ↔Q′2)

=¬(Q1↔Q′2)↔(Q2 ↔Q′1)

does not constitute a classical logical identity

and is thus consistent with assumption 6. It 35_{The only relevant rules in this case, as in (25), follow}

from the properties of the XNOR, namely its symmetry and associativity in Boolean logic. However, it is clear that, thanks to the occurring complementarity, the argu- ments of (25) no longer apply. For instance, the first step in (25) is illegal, according to assumption 5, becauseQ2 andQ1↔Q′

1 are complementary such that they may not be directly connected.

is also consistent with assumption 5 since only maximally compatible questions are directlycon-

nected.

It is impossible forOto interpret this situation

in terms of local ‘hidden variables’ which assign

definite values to the individuals simultaneously. Namely, consider, again, the case Q11 = Q22 =

1 such that Q11 ↔ Q22 = 1 and, consequently,

now Q12↔ Q21= 0. This configuration may be

graphically represented as PSfrag replacements Q12 Q21 Q11 Q22 PSfrag replacements Q12 Q21 Q11 Q22

where one edge solid the other dashed, but same colour, means that the corresponding answers are anti-correlated. One can easily check that, in con- trast to (23), it is impossible to consistently join the two diagrams by drawing all four edges si- multaneously (which would correspond to all in- dividuals and thus all edges having definite truth values), as one would obtain a frustrated graph PSfrag replacements

<

PSfrag replacements

.(27)

O must conclude that neither the four individu-

als nor the four edges can have definite (but un- known) values all at the same time. The same verdict holds for any other configuration with

Q11↔Q22=¬(Q12↔Q21).

Since either (A) or (B) must be true by theo- rems 5.7and 5.9, and (A) violates assumption6, while (B) is consistent with both assumptions 5

and 6, we conclude that (26) is the correct rela- tion.

Note that this argument holds for both rebits and qubits and, for qubits, also for any other pairs

Qi, Qj6=iandQ′k, Q′l6=kof pairs of maximally com- plementary individuals (see theorem5.7). In fact, even more generally, the same argument can be

made for any four questions (i.e., not necessar-

ily individuals)Q, Q′, Q′′, Q′′′ which are such that Q, Q′ and Q′′, Q′′′ are maximally complementary

pairs, while Q and Q′ _{are each maximally com-} patible with both Q′′, Q′′′; also in this case one

would have to conclude that

(Q↔Q′′)↔(Q′ ↔Q′′′) =¬ (Q↔Q′′′)↔(Q′ ↔Q′′). (28)

This will become relevant later on. For now we observe that exchanging the positions of the com- plementaryQ′′, Q′′′from the left to the right hand

side introduces a negation ¬.

This relative negation ¬ in (28) precludes a classical reasoning for the distribution of truth values over O’s questions. In fact, as just seen,

it rules outlocal ‘hidden variable models’, analo-

gously to the Bell arguments. We also recall that assumption 6 was a statement about which rules of inference and logical identities are not appli-

cable to logical compositions involving mutually complementary questions. By contrast, (28) is now a first non-classical logical identity showing one possible non-classical way of reasoning in the presence of complementarity.

The correlation structure for rebits is now clear from these results; (26) implies Q33 = ¬Q˜33 for

the correlations of correlations defined in (22). This settles the fate of all possible relative nega- tions ¬ for rebits. However, these results do

not fully determine the odd and even correla-

tion structure for qubits: we still have to clarify whether there is an overall negation ¬relative to

Q33in (19,20) and more generally in theorem5.7

for other permutations of non-intersecting edges. This difference between rebits and qubits results again from the fact that Q33 is defined as the

correlation of the individuals Q3, Q′3 for qubits,

while it is a correlation of correlations for rebits. This has a remarkable consequence: rebit theory is its own ‘logical mirror image’, while qubit the- ory’s ‘logical mirror image’ is distinct from qubit theory. This topic will be deferred to section 5.4

because we firstly need to understand the ques- tion structure for the N = 3 case in order to discuss the odd and even correlation structure of two qubits further.