5.3 Three gbits
5.3.2 Independence and compatibility for three qubits
We have to delve into some technical details on the independence and compatibility structure to explain and understand monogamy and the en- tanglement structure of three qubits. These re- sults will also be used to prove the analogous re- sults by induction in section5.5.1 for N qubits.
Individual questions from qubit A are maxi-
mally compatible with the individual questions from qubits B and C, etc. But what about the
compatibility of bipartite and tripartite correla- tions? The compatibility structure of bipartite correlations of a fixed qubit pair is clear from lemma 5.5, but we have to investigate compat- ibility of bipartite correlation questions involving all three qubits.
Lemma 5.10. QiAjB and QlBkC are maximally
complementary if j 6= l. On the other hand, QiAjB and QjBkC are maximally compatible and
it holds
QiAjB ↔QjBkC =QiAkC.
The analogous statements hold for any permuta- tion of A, B, C. That is, graphically, two bipar- tite correlations involving three qubits are max- imally compatible if the corresponding edges in- tersect in a vertex and maximally complementary otherwise.
Proof. Maximal complementarity of QiAjB and
QlBkC for j 6= l is proven by noting that both
are maximally compatible with and independent of QiA, the relation QjB = QiA ↔ QiAjB and
lemma 5.1.
QiAjB andQjBkC are evidently maximally com-
patible since QiA, QjB, QkC are maximally com-
patible. Moreover, QiAjB ↔QjBkC =QiA ↔(QjB ↔QjB) | {z } =1 ↔QkC =QiAkC
thanks to the associativity of ↔.
For example, Q2A2B and Q2B2C intersect in
Q2B and are thus maximally compatible, while
Q2A2B and Q1B1C do not share a vertex and are
therefore maximally complementary: PSfrag replacements A B C Q1A Q2A Q3A Q1B Q2B Q3B Q1C Q2C Q3C Q1B1C Q1A3B Q1B3C Q2A2B Q2B2C Q111 Q333 Q322 PSfrag replacements A B C Q1A Q2A Q3A Q1B Q2B Q3B Q1C Q2C Q3C Q1B1C Q1A3B Q1B3C Q2A2B Q2B2C Q111 Q333 Q322
We continue with the tripartite questions. Lemma 5.11. Qijkis maximally compatible with QiA, QjB, QkC and maximally complementary to
QlA6=iA, QmB6=jB, QnC6=kC. That is, graphically,
Qijk is maximally compatible with an individ-
ual QiA,B,C if the corresponding vertex is one of
the vertices of the triangle representing Qijk and
maximally complementary otherwise.
Proof. Qijk is by construction maximally com-
patible with QiA, QjB, QkC. On the other hand,
complementarity ofQijk andQlA6=iA is shown by
noting that both are maximally compatible with and independent of QjBkC and lemma 5.1. One
argues analogously for the individuals of qubits
B and C.
For instance, Q111 is maximally compatible
withQ1C and maximally complementary toQ2C:
PSfrag replacements A B C Q1A Q2A Q3A Q1B Q2B Q3B Q1C Q2C Q3C Q1B1C Q1A3B Q1B3C Q2A2B Q2B2C Q111 Q333 Q322
This lemma also directly implies that any in- dividual and any tripartite correlation question are pairwise independent because (1) maximally complementary questions are in particular inde- pendent, and (2) for maximally compatible ques- tion pairs such as QiA, Qijk independence argu-
Next we consider bipartite and tripartite cor- relation questions.
Lemma 5.12. Qijk is maximally compatible
with QiAjB, QiAkC, QjBkC and, furthermore, with
QmBnC, QlAnC and QlAmB for l 6= i, m 6= j
and k 6= n. On the other hand, Qijk is max-
imally complementary to QiAmB, QiAnC, QlAjB,
QjBnC, QlAkC, QmBkC for l 6= i, m 6= j and
k 6= n. That is, graphically, Qijk is maximally
compatible with a bipartite correlation if the edge of the latter is either an edge of the triangle cor- responding to Qijk or if the edge and triangle do
not intersect. Qijk is maximally complementary
to a bipartite correlation question if the edge of the latter and the triangle corresponding to Qijk
share one common vertex.
Proof. Qijk is by construction maximally com-
patible withQiAjB, QiAkC, QjBkC. Qijk=QiA ↔
QjBkC and QmBnC are also maximally compat-
ible for j 6= m and k =6 n because QmBnC is
maximally compatible with QiA and thanks to
lemma 5.5also withQjBkC. Complementarity of
Qijk and QiAmB for j 6= m follows from noting
that both are maximally compatible with and in- dependent ofQkC and lemma5.1. The reasoning
for all other cases is analogous.
To give a graphical example,Q111is maximally
compatible with Q1B1C and Q3A3C and maxi-
mally complementary to Q1A2B:
PSfrag replacements A B C Q1A Q2A Q3A Q1B Q2B Q3B Q1C Q2C Q3C Q1B1C Q1A3B Q1A2B Q3A3C Q2B2C Q111 Q333 Q322 PSfrag replacements A B C Q1A Q2A Q3A Q1B Q2B Q3B Q1C Q2C Q3C Q1B1C Q1A3B Q1A2B Q3A3C Q2B2C Q111 Q333 Q322
We still have to check pairwise independence of the bipartite and tripartite correlation questions. Lemma 5.13. Any bipartite QmBnC, QlAnC, QlAmB and any tripartite corre-
lation question Qijk are independent from one
another.
Proof. Lemma 5.12 implies that we only have to check pairwise independence of
QmBnC, QlAnC, QlAmB from Qijk for l 6= i,
m 6= j and k 6= n because maximally comple- mentary questions are by definition independent and Qijk and QiAjB, QiAkC, QjBkC are pairwise
independent. Consider therefore Qijk and QmBnC for j 6= m and k 6= n. By lemma 5.11,
QkC is maximally compatible with Qijk and by
lemma 5.2maximally complementary toQmBnC.
This implies, using the arguments from the proof of lemma 5.3, independence of Qijk, QmBnC.
The other cases follow similarly.
Lemma 5.14. The tripartite correlation ques- tions Qijk, i, j, k = 1,2,3 are pairwise indepen-
dent.
Proof. Consider Qijk and Qlmn for i 6= l. By
lemma 5.11, QiA is maximally compatible with
Qijkand maximally complementary toQlmn. Us-
ing the analogous arguments from the proof of lemma 5.3, this implies thatQijk, Qlmn are inde-
pendent. The same reasoning holds when j6=m
and k6=n.
This has an immediate consequence:
Corollary 5.15. The individualsQiA, QjB, QkC,
the bipartite QiAjB, QiAkC, QjBkC and the tripar-
tite Qijk,i, j, k= 1,2,3 are pairwise independent
and thus, thanks to assumption 7, contained in an informationally complete set QM3.
Lastly, we consider the complementarity and compatibility structure of the tripartite correla- tions.
Lemma 5.16. Qijk and Qlmn are maximally
compatible if{i, j, k}and{l, m, n}overlap in one or three indices and maximally complementary if
{i, j, k} and {l, m, n} overlap in zero or two in- dices. That is, graphically, Qijk and Qlmn are
maximally compatible if their corresponding tri- angles intersect in one vertex (or coincide) and maximally complementary if the triangles share an edge or do not intersect.
Proof. Compatibility for an overlap in all three indices is trivial. But also Qijk = QiA ↔ QjBkC
and Qimn = QiA ↔ QmBnC are clearly maxi-
mally compatible for j 6= m and k 6= n because by lemma5.5QjBkC, QmBnC are maximally com-
patible in this case. Compatibility for the other cases of an overlap ofQijkandQlmnin one index
The proof of the complementarity of Qijk and Qlmn for i 6= l, j 6= m and k 6= n follows from
lemma5.1. One may use the fact that, by lemma
5.12, both questions are maximally compatible with and independent of QjBkC and that, by
lemma5.11,QiA6=lAis maximally complementary
to Qlmn.
Similarly, one proves complementarity of
Qijk =QiA ↔ QjBkC and Qljk = QlA ↔ QjBkC
fori6=l by using that both are maximally com- patible with and independent of QjBkC. Com-
plementarity of tripartite correlations for other overlaps in precisely two indices follows by per- mutation.
For example, Q111 and Q212 intersect in the
vertex Q1B and are thus maximally compatible.
By contrast, Q111 shares the edge Q1A1B with
Q113 and does not intersect at all withQ333 such
that Q111 is maximally complementary to both.
Q113andQ333intersect in the vertexQ3C and are
therefore maximally compatible:
PSfrag replacements Q111 Q333 Q113 Q212 PSfrag replacements Q111 Q333 Q113 Q212
5.3.3 An informationally complete set for three