ELFT HÍRADÓ

[Janos Asboth <asboth.janos AT ttk.bme.hu>]

Meghívó

BME Elméleti Fizika Szeminárium,

ápr. 11 péntek 10h15,
1111 Bp., Budafoki út 8., BME FIII. magasföldszint 1, szemináriumi szoba

Solomon Brhanu Samuel (BME Elm. Fiz. tanszék):
Classification of Numerical SIC-POVMs in Dimensions n<8

The Symmetric Informationally Complete Positive Operator-Valued Measures
(SIC-POVMs) are known to exist in all dimensions <= 151 and many larger
dimensions as high as 39604. All known solutions with the exception of the
Hoggar solutions are covariant with respect to the Weyl-Heisenberg group
and in the case of dimension 3, it has been proven that all SIC-POVMs are
Weyl-Heisenberg group covariant. In this work[1], we explore this by
SIC-POVMs in dimensions 4-7 without the assumption of group covariance. We
introduce two functions with which SIC-POVM Gram matrices can be generated
without the group covariance constraint and analytically show that the
SIC-POVM Gram matrices exist on critical points of the surfaces defined by
the two functions on a subspace of Hermitian matrices. In dimensions 4 to
7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of
solution. Thus, we define an equivalent class that relates Gram matrices
based on the trivial symmetries of the two functions. In dimensions 4 to 7,
we generated $\{1.7\times10^6,1.1\times10^5,169,50\}$ Gram matrices,
respectively. For each of the Gram matrices, we generate the symmetry group
of their respective disjoint sets.  In all cases, the symmetry group was
isomorphic to a subgroup of the Clifford group containing the
Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions
4-6, All constructed solutions belong to a single equivalence class of Gram
matrices whereas in dimension 7, we find two distinct families of
SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM
strongly suggests that the functions don't have a non-trivial symmetry.
Furthermore, in all the dimensions tested, the only equivalence classes
found correspond to the distinct stabilizers of fiducial vectors of each
dimension.

[1]: S B Samuel and Z Gedik 2024 J. Phys. A: Math. Theor. 57 295304
Minden érdeklődőt szeretettel várunk.
Asbóth János,
 szemináriumi koordinátor