Symmetric informationally complete positive-operator-valued measures: A new computer study

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In his doctoral dissertation (Ref. 31, p. 65), following the derivation of SIC-POVMs for dimensions $2,…,7$⁠, Zauner made the following observation: “Die Beispiele für $b=2,3,4,5,6,7$ legen die folgende Vermutung nahe. Für alle $b≥2$ gibt es im $([b/3]+1)$-dimensionalen Eigenraum zum Eigenwert 1 der $b×b$ Matrix $Z$ Vektoren, welche mit dem Ansatz (3.14) maximale Quantendesigns mit Grad 1 erzeugen. Für $b=3m+2$ gibt es im gleichdimensionalen Eigenraum zum Eigenwert $α$ ebensolche Vektoren.” In the present context, this quotation can be translated as follows [$Z$ is the transpose of $Z$⁠, in fact, but we will follow Appleby (Ref. 9)]: “the examples for dimension $d=2,3,4,5,6,7$ give rise to the following conjecture. For all dimensions $d≥2$⁠, the eigenspace of dimension $⌊d/3⌋+1$ with eigenvalue 1 of the $d×d$ matrix $Z$ [see Eq. (3.9)] contains fiducial vectors of a Weyl–Heisenberg covariant SIC-POVM. For $d=3m+2$⁠, the eigenspace of the same dimension with eigenvalue $e2πi/3$ contains fiducial vectors as well.”

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Solutions for fiducial vectors were found by minimizing the left hand side of Eq. (4.2), the cost function, until the bound on the right hand side was met. This was performed using the MATLAB^® Optimization Toolbox™ (Ref. 54) with the cost function and its derivatives implemented in C and linked in. The solutions were then further refined to 38 digits with the multiprecision capabilities of PARI/GP (Ref. 55). A small number of AMD Opteron™ 252 dual-processor machines were used, though for a significant amount of time.

See supplementary material at http://dx.doi.org/10.1063/1.3374022 for Appendixes A and B.

In each dimension $d$⁠, the current list of known $PEC(d)$ orbits of fiducial vectors is considered complete when, upon initializing each trial to a random vector under the Haar measure, the search consecutively encounters $30(n+1)$ solutions that generate one of the $n$ known orbits. Assuming each orbit is found with equal probability, the probability of missing an orbit is then no more than $e−30$⁠. Under this criteria, the list is complete for $d≤47$⁠. In dimensions $d=48,…,50$⁠, we have encountered enough of the known orbits to be confident that our list is complete here also, but the computations are ongoing.

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PARI/GP, version 2.3.4, The PARI Group, Bordeaux, , available online: http://pari.math.u-bordeaux.fr/.

Note that when $d$ is even, however, the subgroup of $ESL2(Z2d)$ that defines $S(ϕ)≤PEC(d)$ can have an order that is a multiple of $|S(ϕ)|$⁠. We have chosen all stabilizer orders to double when treated as subgroups of $ESL2(Z2d)$⁠, e.g., $|⟨Fz⟩|=2|⟨[Fz∣0]⟩|$⁠.

See Ref. 17 (Conjecture 4) and note that $Fz$ and $JFzJ$ belong to different conjugacy classes in $SL2(Zd¯)$ when $3∣d$⁠.

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When expressing the elements of the original number field in terms of radicals, we generally have to extend the field to one of larger degree. The information given in Table III is with respect to the original field.

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