Ultra-quantum coherent states in a single finite quantum system

A set of $n$ coherent states is introduced in a quantum system with $d$-dimensional Hilbert space $H(d)$. It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A $n$-tuple representation of arbitrary states in $H(d)$, analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them `ultra-quantum'. The first property is related to the Grothendieck formalism which studies the `edge' of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a `classical' quadratic form ${\mathfrak C}$ that uses complex numbers in the unit disc, and a `quantum' quadratic form ${\mathfrak Q}$ that uses vectors in the unit ball of the Hilbert space. It shows that if ${\mathfrak C}\le 1$, the corresponding ${\mathfrak Q}$ might take values greater than $1$, up to the complex Grothendieck constant $k_G$. ${\mathfrak Q}$ related to these coherent states is shown to take values in the `Grothendieck region' $(1,k_G)$, which is classically forbidden in the sense that ${\mathfrak C}$ does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.