Ultra-quantum coherent states in a single finite quantum system

Abstract

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 C that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form Q that uses vectors in the unit ball of the Hilbert space. It shows that if C < 1, the corresponding Q might take values greater than 1, up to the complex Grothendieck constant kG. Q related to these coherent states is shown to take values in the ‘Grothendieck region’ (1, kG), which is classically forbidden in the sense that 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.