State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems

Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise

Multipartite entanglement detection based on generalized state-dependent entropic uncertainty relation for multiple measurements

We present the generalized state-dependent entropic uncertainty relations for multiple measurement settings, and the optimal lower bound has been obtained by considering different measurement sequences. We then apply this uncertainty relation to witness entanglement, and give the experimentally accessible lower bounds on both bipartite and tripartite entanglements. This method of detecting entanglement is applied to physical systems of two particles on a one-dimensional lattice, GHZ-Werner states and W-Werner states. It is shown that, for measurements which are not in mutually unbiased bases, this new entropic uncertainty relation is superior to the previous state-independent one in entanglement detection. The results might play important roles in detecting multipartite entanglement experimentally.Comment: 7 pages,3 figure

Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the R\'{e}nyi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis $\alpha$-entropic bounds for $\alpha\in(0;2]$. Relations in a model of detection inefficiences are obtained. In terms of R\'{e}nyi's entropies, lower bounds are given for $\alpha\in[2;\infty)$. State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered. We also obtain lower bounds in term of the so-called symmetrized entropies. The presented results for mutually unbiased bases are extensions of some bounds previously derived in the literature. We further formulate new properties of symmetric informationally complete measurements in a finite-dimensional Hilbert space. For a given state and any SIC-POVM, the index of coincidence of generated probability distribution is exactly calculated. Short notes are made on potential use of this result in entanglement detection. Further, we obtain state-dependent entropic uncertainty relations for a single SIC-POVM. Entropic bounds are derived in terms of the R\'{e}nyi $\alpha$-entropies for $\alpha\in[2;\infty)$ and the Tsallis $\alpha$-entropies for $\alpha\in(0;2]$. In the Tsallis formulation, a case of detection inefficiences is briefly mentioned. For a pair of symmetric informationally complete measurements, we also obtain an entropic bound of Maassen-Uffink type.Comment: 19 pages, no figures. The version 4 includes uncertainty relations with detection inefficiencies and brief notes on entanglement detection. New references are added. Except for the style, this version matches the journal versio