Measures from conical 2-designs depend only on two constants

Quantum measurements are important tools in quantum information, represented by positive, operator-valued measures. A wide class of symmetric measurements is given via generalized equiangular measurements that form conical 2-designs. We show that only two positive constants are needed to fully characterize a variety of important quantum measures constructed from such operators. Examples are given for entropic uncertainty relations, the Brukner-Zeilinger invariants, quantum coherence, quantum concurrence, and the Schmidt-number criterion for entanglement detection.

💡 Research Summary

The paper investigates a broad class of symmetric quantum measurements known as generalized equiangular measurements (GEAMs) and shows that when these measurements form a conical 2‑design, all the most relevant quantum information quantities can be expressed solely in terms of two positive constants, denoted S and C_max.

A GEAM consists of N collections (frames) of positive operators {P_{α,k}} that satisfy three conditions: (i) each frame sums to a scalar multiple of the identity, (ii) the total number of operators obeys a bound related to the Hilbert‑space dimension d, and (iii) the operators obey specific trace relations within a frame (parameter a_α, b_α, c_α) and between different frames (parameter f = 1/d). When the additional requirement that the quantity S = a_α²(b_α – c_α) is the same for every frame holds, the set becomes a conical 2‑design. In this case the second‑order tensor sum ∑{α,k} P{α,k}⊗P_{α,k} takes the simple form κ₊ I⊗I + κ₋ F, where F is the flip operator and κ₊ = μ – S/d, κ₋ = S with μ = (1/d)∑_α a_αγ_α.

A key consequence is that the index of coincidence C(ρ) = ∑{α,k} p{α,k}² (with p_{α,k}=Tr ρP_{α,k}) obeys a linear relation with the purity Tr ρ²:
C(ρ) = S