Uncertainty equality for SU(N) observables enabling the experimentally friendly detection of k-inseparability via purity measurements
We derive an exact uncertainty relation for arbitrary quantum states of finite-dimensional Hilbert spaces. For any given $k$-partition of a $d$-dimensional multipartite system, we introduce the total uncertainty as the sum of the uncertainties associated with all possible tensor products of local $\mathrm{SU}(N)$ observables, where each observable acts on the corresponding subsystem. We show that the total uncertainty exactly equals the algebraic sum of the global state purity and the purities of all possible state reductions. For systems containing at least one single-qubit subsystem, this equality implies saturation of the Robertson-Schr\"odinger uncertainty inequality, with the missing term needed for saturation equal to the bipartite qubit-environment entanglement for a pure global state, or to the qubit two-R\'enyi entropy for a mixed global state. Leveraging on these results, we show how for any finite-dimensional multipartite system the Hilbert-Schmidt squared norm of the correlation matrix $t$ can be expressed exclusively in terms of the global and reduced state purities. We then derive a correlation matrix-based necessary condition for $k$-separability of arbitrary finite-dimensional quantum states and show, in the case of $n$ qubits, how it is related to a necessary criterion for Bell nonlocality in scenarios with two dichotomic measurements per party. For sufficiently large systems the purity-based formulation of the $k$-separability criterion always yields an exponential advantage over the direct evaluation of the $t$-matrix norm, allowing for a more efficient practical verification of multipartite entanglement and nonlocality via simple experimental schemes based on purity measurements. Our results shed some further light on the intimate and intricate relation between correlations, entropies, uncertainties, and the entanglement certification and detection problem.