4.3 Summary and Discussion
4.3.3 Number and locations of CPPs
Excluding the Dicke states with their continuous ring of CPPs, one observes that can-didates for maximal entanglement tend to have a large number of CPPs. The prime example is the case of five qubits, where the classical solution with only three CPPs is less entangled than the “square pyramid” state which has five CPPs. In Theorem22it was shown that 2n− 4 is an upper bound on the number of CPPs of positive symmetric n qubit states. In Table4.5this bound is compared to the number of CPPs of all candidates and solutions. It can be seen that the bound is obeyed by all states, including those that cannot be cast with positive coefficients. In many cases the number of CPPs comes close to the bound (n= 5, 8) or coincides with it (n = 4, 6, 7, 12). This raises the question whether the upper bound of 2n− 4 on the number of CPPs also holds for general sym-metric states. One indication in favour of this conjecture is given by Euler’s formula for convex polyhedra, which states that a convex polyhedron with n vertices can have at most 2n− 4 faces, with the bound being strict iff all faces are triangles. This is intriguing because our proof of Theorem22is of a very technical nature, where the number 2n− 4 arises in a seemingly arbitrary fashion. This hints at a deeper lying connection between the faces spanned by the MPs and the number of local maxima present in the spherical amplitude function g(θ, ϕ). We therefore formulate the following conjecture:
Conjecture 23. With the exception of the Dicke states, every n qubit symmetric state has
Table 4.5: The number of CPPs and polyhedral faces in the Majorana representation of the solutions or conjectured solutions are listed. The upper bound 2n− 4 must hold for the number of faces (due to Euler’s formula) and for the number of CPPs (due to Theorem22) of|Ψposn 〉. Entries are omitted where the underlying state is the same as the conjectured solution|Ψn〉 of the Majorana problem.
n CPPs|ψTón 〉 CPPs|ψThn 〉 CPPs|Ψposn 〉 CPPs|Ψn〉 faces|Ψn〉 2n− 4
4 4 4 4
5 3 3 5 5 6
6 8 8 8
7 3 10 10 10
8 2 2 10 10 12
9 3 3 10 10 14
10 2 8 3 8 16 16
11 1 2 2 11 16 18
12 15 20 20 20
at most2n− 4 CPPs.
What can we say about lower bounds on the number of CPPs? For maximally entangled symmetric n qubit states Corollary6predicts the existence of only two distinct CPPs, but our results show that in general there is a considerably larger number of CPPs, and that the CPPs tend to be well spread out over the sphere. This makes sense from the viewpoint of the necessary condition outlined in Corollary6, namely that it must be possible to write maximally entangled symmetric states as linear combinations of their CPSs.
For n= 10, 11 the numerically determined maximally entangled positive symmetric states do not exhibit a rotational symmetry. This is somewhat surprising, because Lemma 20 implies that the CPPs can then only lie on the positive half-circle of the Majorana sphere, thus likely resulting in an imbalance of the spherical amplitude function g(θ, ϕ). However, this imbalance is only very weakly pronounced for the positive solutions of n= 10, 11, with the non-global maxima of g(θ, ϕ) coming very close to the value at the CPPs. It was seen that both|Ψpos10〉 and |Ψpos11〉 are cast with only three nonvanishing basis states, and that Theorem13implies that shifting the MPs in a way that each horizontal MP ring assumes a rotational Z-axis symmetry would result in four nonvanishing basis states. It thus seems that, at least for positive symmetric states, a lower number of nonvanishing basis states is more favourable than Majorana representations with certain symmetry features.
It was found that for 3< n ≤ 12 the maximally entangled symmetric n qubit states are not Dicke states. This result can be easily extended to n> 12 by comparing the entanglement scaling of the equally balanced Dicke state (2.14) to e.g. the superpositions of Dicke states shown in Table4.1, or to the entanglement scaling of the symmetric
4.3. Summary and Discussion
states defined for all n in[90]. Since Dicke states are the only states whose Majorana representation exhibits a continuous rotational symmetry, we obtain the following result:
Corollary 24. For n> 3 the maximally entangled symmetric n qubit states with respect to the geometric measure have only a finite number of CPPs.
This finding is interesting in light of the question raised in Section 2.2, namely whether maximally entangled states of arbitrary multipartite systems have a discrete or continuous amount of CPSs (see also Tamaryan et al. [149,150]). The answer for the general (non-symmetric) case is not known, but the investigation of the symmetric nqubit case gives reason to believe that for most multipartite systems the maximally entangled states have only a finite number of distinct CPSs.
Chapter 5
Classification of Symmetric State Entanglement
In the previous chapter the entanglement of symmetric states was inves-tigated primarily from a quantitative point of view. Now the focus shifts towards the qualitative characterisation of symmetric states. The concepts of LOCC and SLOCC equivalence are adapted to the symmetric case, and the Degeneracy Configuration (DC), an entanglement classification scheme specifically for symmetric states, is reviewed. It is found that SLOCC opera-tions between symmetric states are described by the Möbius transformaopera-tions of complex analysis. This allows for an intuitive visualisation, as well as practical uses such as the determination of whether two symmetric states be-long to the same SLOCC class. The symmetric SLOCC and DC classes for up to five qubits are studied in detail, and representative states are derived for each entanglement class. Connections are made to known SLOCC invariants as well as related works, such as the Entanglement Families (EFs)[51] or alternative definitions of maximal entanglement[58,184].
5.1 Entanglement classification schemes for symmetric states
The entanglement classification schemes LOCC and SLOCC were already discussed in Section1.2.3. In particular, it was seen that SLOCC equivalence gives rise to a coarser partition than LOCC equivalence in the sense that every LOCC operation is also an SLOCC operation, but not vice versa. The concepts of LOCC and SLOCC equivalence are now applied to the subset of symmetric states, and a comparison is made to the Degeneracy Configuration (DC)[82], an entanglement classification scheme designed specifically for symmetric states.