Independence and compatibility for three rebits

3 qubit case. We shall thus be briefer here. The reader exclusively interested in qubits may jump to section 5.4.

According to definition 3.5, O’s questions to

the three rebit system must contain the 6 indi- viduals QiA, QjB, QkC, 12 bipartite correlations

QiAjB, QiAkC, QjBkC,i, j, k= 1,2, and3bipartite

correlations of correlations Q3A3B, Q3A3C, Q3B3C.

To render this set informationally complete, we have to top it up with tripartite questions. There are now 8 tripartite correlations Qijk, i, j, k =

1,2, of the kind (30, 31) and, moreover, 6 tri- partite correlations of a rebit individual question with the question Q33 asking for the correlation

of bipartite correlations of the other rebit pair

Qi33:=QiA ↔Q3B3C,

Q3j3 :=QjB ↔Q3A3C,

Q33k :=Q3A3B ↔QkC, i, j, k = 1,2.

Given that the individuals Q3A, Q3B, Q3C do not

exist for rebits there is now also no tripartite cor- relation Q333.

5.3.7 Independence and compatibility for three rebits

The independence, compatibility and comple- mentarity structure for the questions not involv-

ing an index i, j, k = 3 directly follows from the qubit discussion as lemmas5.10–5.16also hold in the present case for i, j, k = 1,2. But we now have to clarify the question structure once two indices are equal to 3 (an odd number of indices

cannot be 3). The status of any such purely bi- partite relations was clarified in section 5.2such that here we have to consider the case that all three rebits are involved.

Lemma 5.18. Q3B3C is maximally complemen-

tary toQiAjB, i, j= 1,2, and maximally compat-

ible with Q3A3B. Furthermore,

Q3A3B ↔Q3B3C =Q3A3C. (36)

The same holds for any permutations of A, B, C.

Proof. One proves complementarity ofQiAjB and

Q3B3C = Q1B2C ↔ Q2B1C by employing that

both are maximally compatible with and inde- pendent ofQiA and lemma5.1. This also requires

invoking thatQjB andQ3B3C are maximally com-

plementary by lemma5.8.

Compatibility of Q3A3B and Q3B3C fol-

lows indirectly. Namely, Q1A2B, Q2B1C and

Q2A1B, Q1B2C are two maximally compatible

pairs by lemma 5.10. We can then apply the reasoning around (28) to find

Q3A3B ↔Q3B3C = (Q1A2B ↔Q2A1B)↔(Q1B2C ↔Q2B1C)

=¬((Q1A2B ↔Q2B1C) | {z } =Q1A1C ↔(Q1B2C ↔Q2A1B) | {z } =Q2A2C ) =¬(Q1A1C ↔Q2A2C) =Q3A3C.

The last equality holds thanks to theorem 5.9. The same reasoning applies to any permuta- tion ofA, B, C.

Next, we discuss the tripartite correlations of individuals with the questionsQ33asking for the

correlation of bipartite correlations of rebit pairs. Lemma 5.19. Qi33is maximally compatible with

QiA and maximally complementary toQjB, QkC.

The same holds for any permutation of A, B, C. Proof. Compatibility of Qi33 with QiA is true

by construction. Complementarity of QjB and

Qi33 = QiA ↔ (Q1B2C ↔ Q2B1C) follows from

the observation thatQi33andQjB are both max-

imally compatible with and independent of QiA

and a similar reasoning to the previous proof. Lemma 5.20. Qi33is maximally compatible with

Q3B3C, QjBkC, j, k = 1,2, and QlAkC, QlAjB for

i6=l. On the other hand,Qi33is maximally com-

plementary to QiAjB, QiAkC and Q3A3B, Q3A3C.

Furthermore, Qijk is maximally compatible with Q3A3B. The same holds for any permutation of

A, B, C.

Proof. Compatibility of Qi33 with QjBkC and

Q3B3C, as well as compatibility of Qijk with Q3A3B is obvious. Compatibility of Qi33 with

QlAkC for i6=l follows indirectly by noting that

QiA, QlA and Q3B3C, QkC are two pairs of max-

imally complementary questions, but that the

questions in one pair are maximally compatible with both questions of the other. In this case the reasoning of (28) applies and entails the cor- relation of QiA with Q3B3C must be maximally

compatible with the correlation ofQlA withQkC.

Compatibility of Qi33 with QlAjB follows simi-

larly.

Complementarity of Qi33 and QiAjB follows

from the fact that both are maximally compati- ble with and independent ofQiA and using argu-

ments as in previous proofs. Likewise, Qi33 and

Q3A3B follows similarly by noting that both are

maximally compatible with Q3B3C.

Lemma 5.21. Any of Qi33, Q3j3, Q33k is

independent from any bipartite correla- tion question QiAjB, QiAkC, QjBkC and

any bipartite correlation of correlations questionQ3A3B, Q3A3C, Q3B3C. Qijk is also

pairwise independent from the latter. Further- more, the Qijk and Qi33, Q3j3, Q33k are pairwise

independent.

Proof. The proof is completely analogous to the proofs of lemmas 5.3,5.13 and 5.14.

As before this has an important consequence. Corollary 5.22. The individualsQiA, QjB, QkC,

the bipartite correlations QiAjB, QiAkC, QjBkC,

the bipartite correlations of correlations Q3A3B, Q3A3C, Q3B3C, the tripartite correla-

i, j, k = 1,2, are pairwise independent and thus, thanks to assumption 7, part of an informationally complete set QM3.

The compatibility and complementarity struc- ture of questions involving the ‘correlation of cor- relations’ Q33is analogous to lemma 5.16.

Lemma 5.23. Qijkis maximally compatible with Qi33, Q3j3, Q33k and maximally complementary

to Ql33, Q3m3, Q33n for i6= l, j 6=m and k 6=n.

Furthermore, Qi33 is maximally compatible with

Q3j3, Q33k, but Q133 and Q233 are maximally

complementary. The analogous result holds for all permutations of A, B, C.

Proof. Compatibility of Qijk with

Qi33, Q3j3, Q33k follows from the fact that

the constituents of the latter QiA, Q3B3C, . . .

are maximally compatible with Qijk. Comple-

mentarity of, e.g., Qijk and Ql33 for i 6= l can

be shown by noting that both are maximally compatible with and independent of Q3B3C and

lemma 5.1. Compatibility of, say, Qi33 and

Q3j3 can be demonstrated by using (28) and

noting that QiA, Q3A3C and QjB, Q3B3C are two

pairs of maximally complementary questions which are such that each question in one pair is maximally compatible with both questions of the other pair. Finally, Q133 and Q233 are

maximally complementary because both are maximally compatible withQ3B3C andQ1A, Q2A

are maximally complementary.

This finishes our considerations of the inde-

pendence and complementarity structure of three