Abstract
It has been shown in previous papers that classes of (minimal asymmetric) informationally-complete positive operator valued measures (IC-POVMs) in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states. The latter states may also be derived starting from the Poincaré upper half-plane model . To do this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates, some of the eigenstates of which are the sought fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen–Specker theorem.
Highlights
Nulle parole ne trouve une branche où se poser (No words find a branch where to land [1]).Out of nothing I have created a strange new universe wrote Janos Bolyai to his father in 1823.W
We discover minimal IC-positive operator valued measures (POVMs) and with Hermitian angles ψi |ψj i6= j ∈ A = { a1, . . . , al }, a discrete set of values of small cardinality l
One finds five distinct permutation groups of Index 6 corresponding to subgroups of Γ that lead to a six-dimensional informationally-complete positive operator valued measure (IC-POVM)
Summary
Nulle parole ne trouve une branche où se poser (No words find a branch where to land [1]). Pasch’s axiom (of plane geometry) was used by Hilbert to complete Euclid’s axioms. It is related to Pasch’s configuration of points and lines (in projective geometry). The space of rays is not a linear, but a projective space. It is known from Wigner’s theorem (1931). It is worthwhile to point out that many of the basic configurations of incidence geometry are associated to the commutation relations between Hermitian operators in the (generalized). Some projective configurations occur in the structure of informationally-complete positive operator valued measures (IC-POVMs) [10,11]. One deals with the concept of IC-POVMs, the relation to the Pauli group and the occurrence of the Kochen–Specker theorem.
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