Conjectured strong complementary-correlations tradeoff

Abstract

We conjecture new uncertainty relations which restrict correlations between results of measurements performed by two separated parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from MaassenUffink uncertainty relation and is much stronger than Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for validity of conjectured uncertainty relations and prove them for large classes of states and observables.

Entropic uncertainty relations and their applications

“…Following the resolved conjectures by Grudka et al (2013); Kraus (1987); and Renes and Boileau (2009), we point to a recent open conjecture by Schneeloch et al (2014). They ask if for any bipartite quantum state ρ AB ,…”

“…Notice that (231) uses the same overlap c as appearing in the Maassen-Uffink uncertainty relation (31). However, Grudka et al (2013) realized that this often leads to a fairly weak bound. They noted that the complementarity of the mutual information should depend not only on the maximum element c of overlap matrix [c xz ] (see (32) for its definition), but also on other elements of this matrix.…”

Strong majorization entropic uncertainty relations

“…when U is close to the Fourier matrix, the most accurate results are obtained by the hybrid bounds B RPZ1 and B RPZ3 . In the opposite case, when U contains some entries of modulus close to unity, the direct-sum majorization bound (26) is generically better than all other bounds. Since when c 1 is close to 1/d the bounds B RPZ1 and B RPZ3 seem to be not worse than other known bounds, and the bound (26) is always not worse than the tensor-product majorization bound established in [4,5] it is then fair to say that the collection of results derived in this work, provides the best set of bounds currently available.…”

“…Strong majorization entropic uncertainty relations, established in this work, can be used complementarily for various problems in the theory of quantum information. Specific applications include separability conditions and characterization of multipartite entanglement [24], estimation of mutual information [6] in context of the Hall's information exclusion principle [25,26], and for improved witnessing of quantum entanglement and measurements in presence of quantum memory [8].…”

Improved entropic uncertainty relations and information exclusion relations

“…We used our second result to prove a conjecture by Grudka et al [16], which strengthened Hall's information exclusion principle [12]. Hall's scenario considered the case where Y is a classical register and we want to bound the sum I(X : Y ) + I(Z : Y ).…”