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Coherent subspaces spanned by a finite number of coherent states areintroduced, in a quantum system with Hilbert space that has odd prime dimension$d$. The set of all coherent subspaces is partitioned into equivalence classes,with $d^2$ subspaces in each class.The corresponding coherent projectors withinan equivalence class, have the `closure under displacements property' and alsoresolve the identity. Different equivalence classes provide differentgranularisation of the Hilbert space, and they form a partial order `coarser'(and `finer'). In the case of a two-dimensional coherent subspace spanned bytwo coherent states, the corresponding projector (of rank $2$) is differentthan the sum of the two projectors to the subspaces related to each of the twocoherent states. We quantify this with `non-addditivity operators' which are ameasure of quantum interference in phase space, and also of thenon-commutativity of the projectors. Generalized $Q$ and $P$ functions ofdensity matrices, which are based on coherent projectors in a given equivalenceclass, are introduced. Analogues of the Lorenz values and the Gini index (whichare popular quantities in Mathematical Economics) are used here to quantify theinequality in the distribution of the $Q$ function of a quantum state, withinthe granular structure of the Hilbert space....

Equivalence classes of coherent projectors in a Hilbert space with prime dimension: Q functions and their Gini index