Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quamtumness, Generalized Spherical 2-Design and Symmetric
  Informationally Complete POVM   C. A. Fuchs and M. Sasaki defined the quantumness of a set of quantum states
in \cite{Quantumness}, which is closely related to the fidelity loss in
transmission of the quantum states through a classical channel. In
\cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum
quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the
space. He left an open problem, asking whether fewer than $d^2$ states can
achieve this bound. Recently, in a different context, A. J. Scott introduced a
concept of generalized $t$-design in \cite{GenSphet}, which is a natural
generalization of spherical $t$-design. In this paper, we show that the lower
bound on the quantumness can be achieved if and only if the states form a
generalized 2-design. As a corollary, we show that this bound can be only
achieved if the number of states are larger or equal to $d^2$, answering the
open problem. Furthermore, we also show that the minimal set of such ensemble
is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an
equivalence relation between SIC-POVM and minimal set of ensemble achieving
minimal quantumness.

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The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys